基于量子电路的门限量子秘密共享方案

基于量子电路的门限量子秘密共享方案 Threshold Secret Sharing Scheme Based on Quantum Circuit 1 first-author 李志慧，女，教授，博士生导师，主要研究方向为有限域和密码学。 E⁃mail:lizhihui@snnu.edu.cn 李志慧，女，教授，博士生导师，主要研究方向为有限域和密码学。 E⁃mail:lizhihui@snnu.edu.cn lizhihui@snnu.edu.cn 1 1 陕西师范大学 数学与信息科学学院，陕西 西安710119 陕西师范大学 数学与信息科学学院，陕西 西安710119 College of Mathematics and Information Science, Shaanxi Normal University,Xi’an 710119，Shaanxi, China College of Mathematics and Information Science, Shaanxi Normal University,Xi’an 710119，Shaanxi, China 对 <math id="M1"><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118000&type= 7.78933334 3.04799986 门限量子秘密共享方案参与者人数 <math id="M2"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118008&type= 1.60866666 2.45533323 为任意的情形(即不局限于 <math id="M3"><mi>n</mi><mo>=</mo><mn mathvariant="normal">4</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo> </mo><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118015&type= 15.07066631 2.53999996 ， <math id="M4"><mi>k</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118022&type= 9.31333351 2.87866688 )进行了研究，根据量子电路作用在量子态上的性质构造了一个 <math id="M5"><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118030&type= 7.78933334 3.04799986 门限量子秘密共享方案，进而由Clifford群门和 <math id="M6"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118036&type= 2.28600001 2.45533323 门组成的可证明安全的量子电路进行安全性评估。 最后，对不诚实的参与者攻击进行了安全性分析。 We study a <math id="M7"><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118045&type= 7.78933334 3.04799986 threshold quantum secret sharing scheme, where the number of participants is arbitrary (the number of participants is not limited to <math id="M8"><mi>n</mi><mo>=</mo><mn mathvariant="normal">4</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>k</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118052&type= 25.73866653 3.04799986 ). And a threshold quantum secret sharing scheme is constructed according to the properties of quantum circuits acting on quantum states，in which the quantum circuit is a provably secure quantum circuit composed of Clifford group gates and <math id="M9"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118061&type= 2.28600001 2.45533323 gates. Finally，we use the property of the quantum circuit to analyze the security against dishonest participant attacks. 秘密共享方案 量子电路 量子计算 secret sharing scheme quantum circuit quantum computation 0　 引 言 1 随着网络信息的快速发展，保密的重要性越来越受到人们的关注。量子计算是通过调控量子信息且遵循量子力学规律的一种新型计算模式，它使得量子比特依次通过不同的量子逻辑门来达到编程的目的，通过组合不同的量子逻辑门可以实现各种量子算法。从计算效率的角度来说由于存在量子力学叠加性，已有某些量子算法在解决问题时比传统的通用计算机速度要快。 在量子密码学领域，安全的量子算法已经开始得到人们的关注，如安全多方计算、盲计算和可验证委托计算 [ 1 1 , 2 2 , 3 3 , 4 4 , 5 5 , 6 6 , 7 7 , 8 8 , 9 9 , 10 10 ] 。最早的量子算法是Deutsch和Jozsa [ 11 11 ] 提出，该量子算法表现出经典计算机不具备的计算能力，但却不具有实用性。 直到1994年，Shor 提出了带有革命性的算法 [ 12 12 ] ，该算法的发现激发了人们对于量子计算机研究的热情，同时也推动了量子密码、量子通信的发展。 1999年，Hillery等人 [ 13 13 ] 首次提出了一个量子秘密共享方案，该概念的出现为量子态秘密的安全提供了有效的途径。之后，基于量子特点的量子秘密共享方案得到长久的关注和发展 [ 14 14 , 15 15 , 16 16 , 17 17 , 18 18 , 19 19 , 20 20 , 21 21 ] 。所谓 <math id="M10"><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118068&type= 8.97466660 3.55599999 门限量子秘密共享方案中，一个 <math id="M11"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118079&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118076&type= 1.26999998 2.79399991 位比特的量子态秘密分发给 <math id="M12"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118082&type= 1.86266661 2.79399991 个参与者，满足任何少于 <math id="M13"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118090&type= 1.69333339 2.79399991 个参与者密谋合作都不能重建秘密量子态，而任何 <math id="M14"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118098&type= 1.69333339 2.79399991 个参与者均可重构秘密量子态 [ 22 22 , 23 23 , 24 24 , 25 25 , 26 26 ] 。 在量子电路中，对量子态执行酉变换，其作用相当于在量子态上执行逻辑门后所起的作用。在任意维的Hilbert空间中，如果对应量子态的酉变换都可以通过一个逻辑门组中的元素组合来实现，称这样的逻辑门组为通用量子逻辑门组(universal gate)，下文简称通用门。已证明，Toffoli门可实现任意维Hilber空间所有的酉变换 [ 27 27 ] ，但它是三位量子门，从物理实现的角度来说比较困难。 然而，不管怎样，像经典计算通用逻辑门组一样，量子计算的通用门有其不同的表现形式，正如文献[ 28 28 ]指出: <math id="M15"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118106&type= 2.62466669 2.79399991 门，受控相位门以及Toffoli门中的任意一个与Clifford群门都可以形成一组通用门。 而本文考虑的是由Clifford群门和 <math id="M16"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118114&type= 2.62466669 2.79399991 门组成的通用门。 Ouyang等人 [ 29 29 ] 提出了一个 <math id="M17"><mfenced separators="|"><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118120&type= 8.63599968 3.97933340 门限量子秘密共享方案，其中 <math id="M18"><mi>n</mi><mo>=</mo><mn mathvariant="normal">4</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>k</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118128&type= 29.80266571 3.47133350 。本文考虑更一般的情形即对任意的正整数 <math id="M19"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118140&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118135&type= 1.86266661 2.79399991 ，提出了 <math id="M20"><mfenced separators="|"><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118142&type= 8.63599968 3.97933340 门限量子秘密共享方案，进而利用由离散的Clifford群门和 <math id="M21"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118150&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118147&type= 2.62466669 2.79399991 门组成的量子电路对不诚实参与者攻击进行安全性评估。 1　 文献[ 29 29 ]方案回顾 1 1) 准备阶段 2 在 图1 图1 中，第一列前 <math id="M22"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118155&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118152&type= 1.26999998 2.79399991 位初始化为量子秘密 <math id="M23"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118157&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118156&type= 1.69333339 2.79399991 ，用红色表示；后 <math id="M24"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118160&type= 1.10066664 2.79399991 位初始化为魔幻态 <math id="M25"><mi>τ</mi><mo> </mo><mo>=</mo><mfrac><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="bold-italic">X</mi><mo>+</mo><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mn mathvariant="normal">2</mn><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118165&type= 29.54866600 8.88999939 令 <math id="M26"><mi>N</mi><mo> </mo><mo>=</mo><mo> </mo><mi>s</mi><mo> </mo><mo>+</mo><mo> </mo><mi>t</mi><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118172&type= 17.18733406 2.96333337 其余列初始化为最大混合态 <math id="M27"><mfrac><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118180&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118176&type= 4.48733330 7.53533316 其中 <math id="M28"><mi mathvariant="bold-italic">I</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118182&type= 16.17133331 3.21733332 都是常见的Pauli阵。 红色表示初始化为量子秘密；绿色表示初始化为魔幻态 <math id="M49"><mi>n</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118195&type= 7.87400007 2.53999996 participants 2) 加密阶段 2 当 <math id="M29"><mi>x</mi><mo> </mo><mo>∈</mo><mo> </mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>N</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118210&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118208&type= 27.17800140 3.97933340 时， <math id="M30"><mi mathvariant="bold-italic">U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118215&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118213&type= 2.96333337 2.79399991 (见 图2 图2 )作用于第 <math id="M31"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118217&type= 2.03200006 2.79399991 行量子比特，因此 <math id="M32"><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>⊗</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118222&type= 7.02733326 3.04799986 作用于 <math id="M33"><mi>N</mi><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118226&type= 4.74133348 2.87866688 位量子态，将量子秘密编码成高度混合态。 3) 共享阶段 2 <math id="M34"><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>⊗</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118246&type= 7.02733326 3.04799986 加密后的 <math id="M35"><mi>n</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118250&type= 2.20133328 2.79399991 列混合量子态，第 <math id="M36"><mi>y</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118255&type= 2.20133328 3.72533321 个参与者持第 <math id="M37"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118259&type= 1.86266661 3.72533321 列混合量子态，其中 <math id="M38"><mi>y</mi><mo> </mo><mo>∈</mo><mo> </mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118263&type= 25.99266624 4.06400013 。 4) 解密阶段 2 (a) 收集 <math id="M39"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118268&type= 1.86266661 2.79399991 个参与者持有的量子态； (b) 用酉阵 <math id="M40"><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>⊗</mo><mi>N</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>†</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118273&type= 11.68400002 3.81000018 作用于 <math id="M41"><mi>n</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118278&type= 2.20133328 2.79399991 列共享的混合量子态，丢弃其余列，剩下的第一列含有量子态秘密。 令 <math id="M42"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118286&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118284&type= 26.50066566 3.97933340 ，当 <math id="M43"><mi>n</mi><mo>=</mo><mn mathvariant="normal">4</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>k</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118292&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118290&type= 29.80266571 3.47133350 时，有结论 [ 13 13 ] : <math display="block" id="M44"><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>,</mo><mi>σ</mi><mo>∈</mo><mi mathvariant="bold-italic">P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118296&type= 48.68333054 4.06400013 本文将考虑参与者人数 <math id="M45"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118302&type= 1.86266661 2.79399991 为任意情形(即不局限于 <math id="M46"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">4</mn><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118307&type= 18.20333290 2.96333337 ， <math id="M47"><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118315&type= 10.92199993 3.38666677 )， 并给出新的结论。 2　 改进后的量子秘密共享方案 1 2.1　量子电路及其相关结果 2 本节在文献[ 29 29 ]的基础上进行推广和改进，利用由Clifford群门和 <math id="M52"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118326&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118321&type= 2.62466669 2.79399991 门组成的可证明安全的量子电路图进行分析，给出电路图相应的结论。 在 图1 图1 中，第一列量子比特的前 <math id="M53"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118329&type= 1.26999998 2.79399991 位是将要共享的量子态秘密，后 <math id="M54"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118340&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118337&type= 1.10066664 2.79399991 位是辅助量子态, y 表示加密后第 y 列量子态比特，令 <math id="M55"><mi>N</mi><mo> </mo><mo>=</mo><mo> </mo><mi>s</mi><mo> </mo><mo>+</mo><mo> </mo><mi>t</mi><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118344&type= 17.18733406 2.96333337 用 <math id="M56"><msub><mrow><mi>R</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118352&type= 3.89466667 3.72533321 表示第 <math id="M57"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118358&type= 2.03200006 2.79399991 行量子比特，其中: <math id="M58"><mi>x</mi><mo>∈</mo><mo> </mo><mfenced open="{" close="" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>,</mo><mfenced open="" close="}" separators="|"><mrow><mo> </mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>N</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118364&type= 27.68600082 3.97933340 ， <math id="M59"><mi>y</mi><mo> </mo><mo>∈</mo><mo> </mo><mfenced open="{" close="" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>,</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118371&type= 22.35199928 4.06400013 <math id="M60"><mfenced open="" close="}" separators="|"><mrow><mo> </mo><mo> </mo><mi>n</mi><mo> </mo><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118375&type= 10.83733368 4.06400013 。 在 图2 图2 中，当 <math id="M61"><mi>i</mi><mo> </mo><mo> </mo><mo>∈</mo><mo> </mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo> </mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118386&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118382&type= 25.90799904 3.97933340 时， <math id="M62"><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118389&type= 3.72533321 3.81000018 表示第 <math id="M63"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118396&type= 1.10066664 2.79399991 列对应酉阵，令 <math id="M64"><mi mathvariant="bold-italic">A</mi><mo> </mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi mathvariant="bold-italic">V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>;</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118406&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118402&type= 29.54866600 4.14866638 <math id="M65"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118412&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118409&type= 4.48733330 3.81000018 表示第 <math id="M66"><mi>n</mi><mo> </mo><mo>+</mo><mo> </mo><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118416&type= 8.46666718 2.96333337 列对应酉阵。 令 <math id="M67"><mi mathvariant="bold-italic">B</mi><mo> </mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118423&type= 31.41133308 4.14866638 则 <math id="M68"><mi mathvariant="bold-italic">U</mi><mo> </mo><mo>=</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118429&type= 13.46199894 2.79399991 。 根据 图1 图1 和 图2 图2 给出的量子电路，我们给出如下几个性质。 性质1 当 <math id="M69"><mi>σ</mi><mo>=</mo><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118436&type= 8.04333305 2.79399991 时，有 <math display="block" id="M70"><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>+</mo></mrow></msup><mo> </mo><mo>=</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>,</mo><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118444&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118441&type= 51.39266586 4.06400013 性质2 当 <math id="M71"><mi>σ</mi><mo>=</mo><mi mathvariant="bold-italic">X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118447&type= 9.22866726 2.79399991 时，有 <math display="block" id="M72"><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>+</mo></mrow></msup><mo> </mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>,</mo><mo> </mo><mi>n</mi><mo>=</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>,</mo><mi>n</mi><mo> </mo><mo>=</mo><mn mathvariant="normal">2</mn><mi>k</mi></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><mi>k</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118457&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118454&type= 76.36933136 9.82133293 证 首先证明下式成立 <math display="block" id="M73"><mi mathvariant="bold-italic">A</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>,</mo><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118462&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118460&type= 52.23933029 4.06400013 已知 <math display="block" id="M74"><mi mathvariant="bold-italic">A</mi><mo> </mo><mo>=</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo> </mo><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo> </mo><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118469&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118466&type= 62.39933395 4.06400013 , <math id="M75"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">A</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo> </mo><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><mo>·</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>⊗</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo></mtd></mtr><mtr><mtd><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo> </mo><mo>⊗</mo><msup><mrow><mi>X</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><mo> </mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo> </mo><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118476&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118472&type= 73.74466705 27.60133362 当 <math id="M76"><mi>n</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118478&type= 2.20133328 2.79399991 是奇数时，易知 <math id="M77"><mi>n</mi><mo>=</mo><mn mathvariant="normal">3,5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118485&type= 12.69999981 3.21733332 时,(3)式成立，故归纳假设 <math id="M78"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118491&type= 18.20333290 2.96333337 时,(3)式成立。下证 <math id="M79"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118497&type= 18.20333290 2.96333337 时,(3)式也成立。由归纳假设和(4)式知： <math display="block" id="M80"><mi mathvariant="bold-italic">B</mi><mo> </mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118506&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118503&type= 32.85066605 4.14866638 <math display="block" id="M81"><mi mathvariant="bold-italic">U</mi><mo> </mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118513&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118511&type= 43.34933472 4.06400013 <math display="block" id="M82"><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mo> </mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118518&type= 50.46133423 4.14866638 当 <math id="M83"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118531&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118526&type= 18.20333290 2.96333337 时, <math display="block" id="M84"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo> </mo><mo>=</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>⋯</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo> </mo><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo>⋯</mo><mo> </mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mo> </mo><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118534&type= 67.90267181 27.09333420 其中: <math display="block" id="M85"><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msup><mo>⊗</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>+</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msup><mo>⊗</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>⊗</mo><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118541&type= 55.11800385 10.07533360 , <math display="block" id="M86"><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msup><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>+</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msup><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118550&type= 54.86400223 10.07533360 则有 <math display="block" id="M87"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">3</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118556&type= 79.16333008 22.18266678 <math display="block" id="M88"><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⊗</mo><mo stretchy="false">(</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo> </mo><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>⊗</mo><mi mathvariant="bold-italic">X</mi><mo>+</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>⊗</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118564&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118561&type= 75.86133575 4.06400013 当 <math id="M89"><mi>n</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118570&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118567&type= 2.20133328 2.79399991 是偶数时， <math id="M90"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2,4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118572&type= 13.37733269 3.21733332 时，(3)式成立，归纳假设 <math id="M91"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118580&type= 11.17599964 2.79399991 时，(3)式成立。下证 <math id="M92"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118593&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118589&type= 18.20333290 2.96333337 时,(3)式也成立。由归纳假设和(4)式知： <math display="block" id="M93"><mi mathvariant="bold-italic">B</mi><mo> </mo><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118602&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118596&type= 36.40666580 4.14866638 <math display="block" id="M94"><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118605&type= 54.18666840 4.06400013 <math display="block" id="M95"><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo> </mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118612&type= 52.06999969 4.14866638 当 <math id="M96"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118622&type= 18.20333290 2.96333337 时，有 <math display="block" id="M97"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo> </mo><mo>=</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msub><mo>⋯</mo><mi mathvariant="bold-italic">W</mi><msubsup><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">I</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mo> </mo><mo>⋯</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">I</mi></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118629&type= 65.61666107 29.29466629 其中： <math display="block" id="M98"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msup><mo> </mo><mo>⊗</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>+</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⊗</mo></mtd></mtr><mtr><mtd><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>⊗</mo><mi mathvariant="bold-italic">I</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118636&type= 67.90267181 9.99066734 <math display="block" id="M99"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msup><mo> </mo><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>+</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo> </mo><mo>⊗</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118646&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118642&type= 71.28932953 10.07533360 则有 <math display="block" id="M100"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msup><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118650&type= 78.40132904 26.66999817 由数学归纳法可知(3)式成立。 性质3 当 <math id="M101"><mi>σ</mi><mo>=</mo><mi mathvariant="bold-italic">Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118661&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118657&type= 9.48266697 2.79399991 时,有 <math display="block" id="M102"><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>,</mo><mi>n</mi><mo>=</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>,</mo><mi>n</mi><mo>=</mo><mn mathvariant="normal">2</mn><mi>k</mi></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><mi>k</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118665&type= 76.36933136 9.82133293 证 令 <math id="M103"><mi mathvariant="bold-italic">Y</mi><mo> </mo><mo>=</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo> </mo><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>-</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118670&type= 27.85533333 3.97933340 ,而由 <math id="M104"><mi mathvariant="bold-italic">Y</mi><mo> </mo><mo>=</mo><mi>i</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118681&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118678&type= 12.36133385 3.97933340 <math id="M105"><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>-</mo><mi>i</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118684&type= 16.76399994 3.97933340 知（5）式显然也成立。根据计算易知下式成立： <math display="block" id="M106"><mi mathvariant="bold-italic">A</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>+</mo></mrow></msup><mo> </mo><mo>=</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>,</mo><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118690&type= 64.34666443 4.06400013 当 <math id="M107"><mi>n</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118695&type= 2.20133328 2.79399991 是奇数时， <math id="M108"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">3,5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118701&type= 13.37733269 3.21733332 时，(5)式成立，归纳假设 <math id="M109"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118710&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118707&type= 17.86466599 2.96333337 时，(5)式成立。下证 <math id="M110"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118717&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118714&type= 18.20333290 2.96333337 时，(5)式也成立。由归纳假设和(6)式知： <math display="block" id="M111"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⊗</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo><mo>·</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118722&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118720&type= 55.03333664 22.18266678 <math display="block" id="M112"><mtable columnalign="left"><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⊗</mo><mo stretchy="false">(</mo><mo>-</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo> </mo><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>⊗</mo><mi mathvariant="bold-italic">Y</mi><mo>+</mo></mtd></mtr><mtr><mtd><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>⊗</mo><mi mathvariant="bold-italic">Y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">3</mn></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118726&type= 44.53466797 9.99066734 当 <math id="M113"><mi>n</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118732&type= 2.20133328 2.79399991 是偶数时， <math id="M114"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2,4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118738&type= 13.37733269 3.21733332 时，(5)式成立，归纳假设 <math id="M115"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118746&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118743&type= 11.17599964 2.79399991 时，(5)式成立。下证 <math id="M116"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118749&type= 18.20333290 2.96333337 时，(5)式也成立。由归纳假设和(6)式知： <math display="block" id="M117"><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118753&type= 38.43866730 4.14866638 <math display="block" id="M118"><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118758&type= 53.17066574 4.14866638 <math display="block" id="M119"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Z</mi><mo> </mo><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo> </mo><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>·</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118764&type= 67.05599976 16.51000023 由数学归纳法可知(5)式成立。 性质4 当 <math id="M120"><mi>σ</mi><mo>=</mo><mi mathvariant="bold-italic">Z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118768&type= 8.88999939 2.79399991 时，有 <math display="block" id="M121"><mi mathvariant="bold-italic">U</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>,</mo><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118775&type= 50.88466644 4.06400013 证 根据计算易知 <math display="block" id="M122"><mi mathvariant="bold-italic">A</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>,</mo><mi>n</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118781&type= 62.06066895 4.06400013 当 <math id="M123"><mi>n</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118788&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118785&type= 2.20133328 2.79399991 是奇数时， <math id="M124"><mi>n</mi><mo>=</mo><mn mathvariant="normal">3,5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118791&type= 12.69999981 3.21733332 时，(7)式成立，假设 <math id="M125"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118797&type= 17.86466599 2.96333337 时，(7)式成立，下证 <math id="M126"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118803&type= 18.20333290 2.96333337 时，(7)式也成立。由假设和(8)式知 <math display="block" id="M127"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">I</mi><mo stretchy="false">)</mo><mo>·</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⊗</mo><mo stretchy="false">(</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo> </mo><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi><mo>-</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">3</mn></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118809&type= 71.03533173 31.83466721 当 <math id="M128"><mi>n</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118816&type= 2.20133328 2.79399991 是偶数时， <math id="M129"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2,4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118825&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118823&type= 13.37733269 3.21733332 时，(7)式成立，假设 <math id="M130"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118832&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118829&type= 11.17599964 2.79399991 时，(7)式成立，下证 <math id="M131"><mi>n</mi><mo> </mo><mo>=</mo><mo> </mo><mn mathvariant="normal">2</mn><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118836&type= 18.20333290 2.96333337 时，(7)式也成立。由归纳假设和(8)式知 <math display="block" id="M132"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">U</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">'</mi><mo stretchy="false">)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msup><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>·</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">W</mi><msub><mrow><mi mathvariant="bold-italic">'</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mo> </mo></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118841&type= 60.28266525 27.94000053 由数学归纳法可知(7)式成立。 2.2　量子秘密共享方案 2 1) 准备阶段 3 第一列前 <math id="M133"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118851&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118849&type= 1.26999998 2.79399991 位初始化为量子秘密，后 <math id="M134"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118858&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118854&type= 1.10066664 2.79399991 位初始化为魔幻态 <math id="M135"><mi>τ</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118861&type= 3.38666677 3.21733332 令 <math id="M136"><mi>N</mi><mo> </mo><mo>=</mo><mo> </mo><mi>s</mi><mo> </mo><mo>+</mo><mo> </mo><mi>t</mi><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118868&type= 17.18733406 2.96333337 其余列初始化为最大混合态 <math id="M137"><mfrac><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118876&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118873&type= 2.62466669 7.53533316 。 2) 加密阶段 3 当 <math id="M138"><mi>x</mi><mo> </mo><mo>∈</mo><mo> </mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>N</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118879&type= 27.17800140 3.97933340 时， <math id="M139"><mi mathvariant="bold-italic">U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118886&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118883&type= 2.96333337 2.79399991 作用于第 <math id="M140"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118889&type= 2.03200006 2.79399991 行量子比特，因此 <math id="M141"><mi mathvariant="bold-italic">U</mi><msup><mrow><mo> </mo></mrow><mrow><mo>⊗</mo><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118897&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118894&type= 6.68866634 3.04799986 作用于 <math id="M142"><mi>N</mi><mo stretchy="false">(</mo><mi>n</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118903&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118899&type= 15.83266640 3.55599999 位量子态，将量子秘密编码成高度混合态。 3) 共享阶段 3 <math id="M143"><mi mathvariant="bold-italic">U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118906&type= 2.96333337 2.79399991 加密后的 <math id="M144"><mi>n</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118917&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118914&type= 9.22866726 2.96333337 列混合量子态，Alice持第 <math id="M145"><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118921&type= 1.86266661 2.79399991 列，第 y 列量子态分发给第 <math id="M146"><mi>y</mi><mo> </mo><mo>-</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118927&type= 9.14400005 3.72533321 个参与者，其中 <math id="M147"><mi>y</mi><mo> </mo><mo>∈</mo><mo> </mo><mfenced open="{" close="}" separators="|"><mrow><mn mathvariant="normal">2,3</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></mrow></mfenced><mi mathvariant="normal">。</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118932&type= 37.25333405 4.14866638 4) 解密阶段 3 (a) 收集 <math id="M148"><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118940&type= 8.63599968 2.96333337 个参与者的共享态； (b) 用酉阵 <math id="M149"><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>†</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118945&type= 5.41866684 3.04799986 作用于 <math id="M150"><mi>n</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118954&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118950&type= 9.22866726 2.96333337 列共享的混合量子态，丢弃其余列，剩下的第一列含有量子态秘密。 3　 共享秘密上的评估 1 为了评估作用在共享秘密上的量子电路，每个参与者只在自己持有的份额上执行量子计算，其计算是在共享和解密阶段之间进行。本文考虑由离散的Clifford门和 <math id="M151"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118957&type= 2.62466669 2.79399991 门构成的一组通用门集合(当然还有其他组合)，其中 <math id="M152"><mi mathvariant="bold-italic">T</mi><mo> </mo><mo>=</mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">4</mn></mrow></msup><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mfenced open="<" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118963&type= 36.06800079 3.97933340 。现在我们考虑作用在 <math id="M153"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118970&type= 1.26999998 2.79399991 位量子态秘密上的一序列门 <math id="M154"><mi mathvariant="bold-italic">U</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>l</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118980&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118977&type= 26.58533287 4.91066647 ， <math id="M155"><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118984&type= 15.66333389 4.14866638 都是酉阵，且对于参与者是已知的。 当 <math id="M156"><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118990&type= 3.72533321 3.72533321 是Clifford门时，每个参与者在自己持有的一列量子比特子集上执行 <math id="M157"><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118998&type= 3.72533321 3.72533321 ，因此 <math id="M158"><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119007&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119005&type= 8.63599968 2.96333337 方参与者共同实现 <math id="M159"><msubsup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi mathvariant="normal"></mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119011&type= 11.85333347 3.97933340 。 <math id="M160"><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119020&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119018&type= 3.72533321 3.72533321 是单量子比特Clifford门时如 图3 图3 所示(用 G 来表示Clifford门)， <math id="M161"><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119024&type= 3.72533321 3.72533321 是双量子比特CNOT门时如 图4 图4 所示。 当 <math id="M162"><msub><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119050&type= 3.72533321 3.72533321 是 <math id="M163"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119057&type= 2.62466669 2.79399991 门时，每个参与者在持有的量子秘密上可以执行 <math id="M164"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119064&type= 1.10066664 2.79399991 个 <math id="M165"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119070&type= 2.62466669 2.79399991 门，为了实现作用在第 <math id="M166"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119077&type= 1.10066664 3.64066648 位量子比特上的第 <math id="M167"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119083&type= 1.69333339 2.79399991 个 <math id="M168"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119091&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119089&type= 2.62466669 2.79399991 门， <math id="M169"><mi>k</mi><mo> </mo><mo>∈</mo><mo> </mo><mfenced open="{" close="" separators="|"><mrow><mn mathvariant="normal">1,2</mn><mo>,</mo></mrow></mfenced><mo>⋯</mo><mo>,</mo><mi>t</mi><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119093&type= 26.07733345 3.97933340 ，操作如 图5 图5 。 其中 <math id="M172"><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119117&type= 3.38666677 2.87866688 框表示测量装置， <math id="M173"><mi mathvariant="bold-italic">S</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119122&type= 11.26066685 3.04799986 。首先计算经过两个受控非门后的量子态：已知第 <math id="M174"><mi>N</mi><mo> </mo><mo>-</mo><mo> </mo><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119130&type= 17.27199936 2.96333337 个量子比特为魔幻态 <math id="M175"><mi mathvariant="bold-italic">T</mi><mi mathvariant="bold-italic">H</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119134&type= 12.27666664 3.97933340 假设第 <math id="M176"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119140&type= 1.10066664 3.64066648 个量子比特为 <math id="M177"><mfenced open="|" close=">" separators="|"><mrow><mi>ψ</mi></mrow></mfenced><mo>=</mo><mi>α</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>+</mo><mi>β</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119145&type= 29.46399879 4.74133348 则经过两个受控非门后的量子态为 <math display="block" id="M178"><mtable columnalign="left"><mtr><mtd><mfenced open="|" close=">" separators="|"><mrow><mi>ψ</mi></mrow></mfenced><mo>⊗</mo><mi mathvariant="bold-italic">T</mi><mi mathvariant="bold-italic">H</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mi>α</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>+</mo><mi>β</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></mfenced><mo>⊗</mo><mfrac><mrow><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">4</mn></mrow></msup><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac></mtd></mtr><mtr><mtd><mover accent="false"><mo>→</mo><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mover><mi>α</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>⊗</mo><mfrac><mrow><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">4</mn></mrow></msup><mo> </mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac><mo>+</mo><mi>β</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>⊗</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">4</mn></mrow></msup><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></mrow><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac></mtd></mtr><mtr><mtd><mover accent="false"><mo>→</mo><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mover><mfenced separators="|"><mrow><mi>α</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">4</mn></mrow></msup><mi>β</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></mfenced><mo>⊗</mo><mfrac><mrow><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></mrow><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mi>α</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">4</mn></mrow></msup><mi>β</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></mrow></mfenced><mo>⊗</mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">4</mn></mrow></msup><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119151&type= 73.74466705 53.00133133 其中， <math id="M179"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119157&type= 3.72533321 3.72533321 表示第 <math id="M180"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119162&type= 1.10066664 3.64066648 位量子比特为控制比特，第 <math id="M181"><mi>N</mi><mo> </mo><mo>-</mo><mo> </mo><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119167&type= 17.27199936 2.96333337 位量子比特为目标比特； <math id="M182"><msub><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119173&type= 3.72533321 3.72533321 表示第 <math id="M183"><mi>N</mi><mo> </mo><mo>-</mo><mo> </mo><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119179&type= 17.27199936 2.96333337 位量子比特为控制比特，第 <math id="M184"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119186&type= 1.10066664 3.64066648 位量子比特为目标比特。 对第 <math id="M185"><mi>N</mi><mo> </mo><mo>-</mo><mo> </mo><mi>k</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119191&type= 17.27199936 2.96333337 位量子比特进行测量，若结果态为 <math id="M186"><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119196&type= 3.81000018 3.97933340 ，则第 <math id="M187"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119202&type= 1.10066664 3.64066648 位量子比特作用如下 <math display="block" id="M188"><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup><mfenced separators="|"><mrow><mi>α</mi><mo> </mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo> </mo><mo>+</mo><mo> </mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">4</mn></mrow></msup><mi>β</mi><mo> </mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mi mathvariant="bold-italic">T</mi><mfenced open="|" close=">" separators="|"><mrow><mi>ψ</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119207&type= 50.96933365 5.16466665 若结果态为 <math id="M189"><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119212&type= 3.81000018 3.97933340 ，则第 <math id="M190"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119217&type= 1.10066664 3.64066648 位量子比特作用如下 <math display="block" id="M191"><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">S</mi><mi mathvariant="bold-italic">X</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mfenced separators="|"><mrow><mi>α</mi><mo> </mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced><mo> </mo><mo>+</mo><mo> </mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">4</mn></mrow></msup><mi>β</mi><mo> </mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mi mathvariant="bold-italic">T</mi><mfenced open="|" close=">" separators="|"><mrow><mi>ψ</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119223&type= 53.00133133 5.16466665 因此，最终实现 <math id="M192"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119228&type= 2.62466669 2.79399991 门作用于第 <math id="M193"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119237&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119234&type= 1.10066664 3.64066648 位量子比特。每评估一个 <math id="M194"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119244&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119240&type= 2.62466669 2.79399991 门就会消耗一个魔幻态 [ 28 28 ] 。 实例1 <math id="M195"><mi>n</mi><mo>=</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119250&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119247&type= 9.39799976 2.79399991 时， 假设共享的量子态秘密是 <math id="M196"><mi>σ</mi><mo> </mo><mo>=</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi><mo>,</mo><mi>t</mi><mo>=</mo><mn mathvariant="normal">2</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119254&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119251&type= 37.84600067 3.55599999 则经过本文方案中酉阵 <math id="M197"><mi mathvariant="bold-italic">U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119257&type= 2.96333337 2.79399991 作用后终态为 <math display="block" id="M198"><msub><mrow><mover><mrow><mi>ρ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow></msub><mo> </mo><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">5</mn></mrow></msup><mo>⊗</mo><msup><mrow><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119264&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119262&type= 46.73600006 5.67266655 其中， <math id="M199"><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo>=</mo><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">5</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">5</mn></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">5</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119266&type= 36.32199860 9.22866726 。假设前3个参与者(不包括Alice)不诚实，第4个参与者是诚实的，则对第4个参与者持有的量子态求迹，即求包括Alice在内的4维子系统对应的约化密度矩阵： <math display="block" id="M200"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mover><mrow><mi>ρ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow></msub></mrow></mfenced><mo>=</mo><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">5</mn></mrow></msup><mo>⊗</mo><msup><mrow><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">4</mn></mrow></msup><mo>⊗</mo><msup><mrow><msup><mrow><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup><mo>·</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo> </mo><mfenced separators="|"><mrow><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><mi>τ</mi><mo>⊗</mo><mi>τ</mi></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">4</mn></mrow></msup><mo>⊗</mo><msup><mrow><msup><mrow><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup><mo>·</mo><mn mathvariant="normal">0</mn><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119274&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119271&type= 64.51599884 28.70199966 其中， <math id="M201"><msup><mrow><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup><mo>=</mo><mo> </mo><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">4</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac><mo> </mo><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">4</mn></mrow></msup><mo> </mo><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">4</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119277&type= 38.77733231 9.22866726 ，则其余3个参与者和Alice合作得不到关于量子秘密的任何信息。 实例2 <math id="M202"><mi>n</mi><mo>=</mo><mn mathvariant="normal">5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119281&type= 8.63599968 2.79399991 时，假设共享的量子态秘密是 <math id="M203"><mi>σ</mi><mo>=</mo><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119290&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119287&type= 27.43200111 3.55599999 <math id="M204"><mi>t</mi><mo>=</mo><mn mathvariant="normal">2</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119294&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119291&type= 9.65199947 3.21733332 则经过本文方案中酉阵 <math id="M205"><mi mathvariant="bold-italic">U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119297&type= 2.96333337 2.79399991 作用后终态为 <math display="block" id="M206"><msub><mrow><mover><mrow><mi>ρ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow></msub><mo> </mo><mo>=</mo><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced><mo>⊗</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">5</mn></mrow></msup><mo>⊗</mo><msup><mrow><mover accent="true"><mrow><mi>τ</mi></mrow><mo>̂</mo></mover></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119300&type= 52.57799911 4.74133348 其中， <math id="M207"><mover accent="true"><mrow><mi>τ</mi></mrow><mo>̂</mo></mover><mo>=</mo><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">6</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msup></mrow></mfrac><mo> </mo><mo>+</mo><mfrac><mrow><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">5</mn></mrow></msup><mo> </mo><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">6</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msup><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119306&type= 44.11133194 9.22866726 。假设前4个参与者(不包括Alice)试图获得秘密信息，则对第5个参与者持有的量子态求迹，即求包括Alice在内的5维子系统对应的约化密度矩阵 <math display="block" id="M208"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mfenced separators="|"><mrow><msub><mrow><mover><mrow><mi>ρ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow></msub></mrow></mfenced><mo>=</mo><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mfenced separators="|"><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced><mo>⊗</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">5</mn></mrow></msup><mo>⊗</mo><msup><mrow><mover accent="true"><mrow><mi>τ</mi></mrow><mo>̂</mo></mover></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced><mo>⊗</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">4</mn></mrow></msup><mo>⊗</mo><msup><mrow><msup><mrow><mover accent="true"><mrow><mi>τ</mi></mrow><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup><mo>·</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi><mi mathvariant="normal">r</mi><mo> </mo><mfenced separators="|"><mrow><mi mathvariant="bold-italic">X</mi><mo>⊗</mo><mi mathvariant="bold-italic">Y</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><mi>τ</mi><mo>⊗</mo><mi>τ</mi></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi><mo>⊗</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced><mo>⊗</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">4</mn></mrow></msup><mo>⊗</mo><msup><mrow><msup><mrow><mover accent="true"><mrow><mi>τ</mi></mrow><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup><mo>·</mo><mn mathvariant="normal">0</mn><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119313&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119310&type= 70.61199951 25.14599991 其中， <math id="M209"><msup><mrow><mover accent="true"><mrow><mi>τ</mi></mrow><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup><mo>=</mo><mo> </mo><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">5</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup></mrow></mfrac><mo> </mo><mo>+</mo><mfrac><mrow><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">4</mn></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">5</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119318&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119315&type= 46.99000168 9.22866726 则其余4个参与者和Alice合作得不到关于量子态秘密的任何信息。 4　 安全性讨论 1 一个方案如果对于内部参与者(比外部参与者拥有更多信息)都是安全的，则这个方案对于其他外部攻击者也是安全的。因此，本文针对不诚实参与者攻击进行分析。 不诚实的参与者攻击：一个 <math id="M210"><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119322&type= 8.97466660 3.55599999 门限量子秘密共享方案满足两个性质：1) 任意 <math id="M211"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119326&type= 1.69333339 2.79399991 个或者多于 <math id="M212"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119334&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119331&type= 1.69333339 2.79399991 个参与者合作可以完全恢复秘密量子态；2) 任意 <math id="M213"><mi>k</mi><mo> </mo><mo>-</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119337&type= 8.97466660 2.79399991 个或者少于 <math id="M214"><mi>k</mi><mo> </mo><mo>-</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119341&type= 8.97466660 2.79399991 个参与者合作都不能获得关于秘密量子态的任何信息。当 <math id="M215"><mi>k</mi><mo> </mo><mo>=</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119348&type= 8.72066689 2.79399991 时，第一个性质显然成立，在加密阶段可以看出编码程序是完全可逆的。 接下来讨论第二个性质。 加密前量子态秘密为 <math display="block" id="M216"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>-</mo><mi>s</mi></mrow></msup><mo> </mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119356&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119353&type= 28.70199966 7.45066643 其中 <math id="M217"><mi>σ</mi><mo>=</mo><mo> </mo><msub><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo> </mo><mo>⊗</mo><mo> </mo><msub><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo> </mo><mo>⊗</mo><mo> </mo><mo> </mo><mo>⋯</mo><mo>⊗</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119358&type= 37.76133347 4.48733330 ， <math id="M218"><mi mathvariant="bold-italic">P</mi><mo> </mo><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mi mathvariant="bold-italic">I</mi><mo>,</mo><mi mathvariant="bold-italic">X</mi><mo>,</mo><mi mathvariant="bold-italic">Y</mi><mo>,</mo><mi mathvariant="bold-italic">Z</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119369&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119365&type= 26.83933258 3.97933340 ， <math id="M219"><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119372&type= 3.55599999 3.72533321 是 <math id="M220"><mo> </mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119380&type= 5.75733376 3.04799986 中非平凡算子 <math id="M221"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119388&type= 1.69333339 2.79399991 对应的系数，当 <math id="M222"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119394&type= 1.69333339 2.79399991 是平凡Pauli算子时， <math id="M223"><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119403&type= 10.49866676 3.72533321 。 当 <math id="M224"><mi>n</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119411&type= 9.22866726 2.96333337 是奇数时，由上述性质可知，加密后终态为 <math display="block" id="M225"><msub><mrow><mover><mrow><mi>ρ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow></msub><mo> </mo><mo>=</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>-</mo><mi>s</mi></mrow></msup><mfenced separators="|"><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfenced><mo>⊗</mo><msup><mrow><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>⊗</mo><mi>t</mi></mrow></msup><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119416&type= 52.74732971 11.51466751 <math display="block" id="M226"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>-</mo><mi>s</mi></mrow></msup><mfenced separators="|"><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><mo>\</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfenced><mo>⊗</mo><msup><mrow><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>⊗</mo><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119420&type= 61.72200012 11.51466751 其中： <math display="block" id="M227"><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo>=</mo><mo> </mo><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">Y</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119425&type= 47.24399948 9.31333351 <math display="block" id="M228"><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mi>y</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><mo>\</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfenced><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><mo>\</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mi>y</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119430&type= 81.11066437 11.17599964 则 <math display="block" id="M229"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mi>y</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced open="{" close="}" separators="|"><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>-</mo><mi>s</mi></mrow></msup><mfenced separators="|"><mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><mo>\</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfenced></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>-</mo><mi>s</mi></mrow></msup><mfenced open="{" close="}" separators="|"><mrow><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mi>y</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">0</mn></mrow></mfenced><mo>=</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119434&type= 69.00333405 19.04999924 假设第 <math id="M230"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119442&type= 1.86266661 3.72533321 个参与者是诚实的，则其余 <math id="M231"><mi>n</mi><mo> </mo><mo>-</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119446&type= 9.22866726 2.79399991 个参与者合作得不到关于量子秘密的任何信息。 当 <math id="M232"><mi>n</mi><mo> </mo><mo>+</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119453&type= 9.22866726 2.96333337 是偶数时，加密后终态为 <math display="block" id="M233"><mtable columnalign="left"><mtr><mtd><msub><mrow><mover><mrow><mi>ρ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi></mrow></msub><mo> </mo><mo>=</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>-</mo><mi>s</mi></mrow></msup><mfenced separators="|"><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><mi>θ</mi><mo>⊗</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mrow></mfenced><mo>⊗</mo><msup><mrow><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>⊗</mo><mi>t</mi></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>-</mo><mi>s</mi></mrow></msup><mfenced separators="|"><mrow><mi>θ</mi><mo>⊗</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>+</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><mo>\</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><mi>θ</mi><mo>⊗</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mrow></mfenced><mo> </mo><mo>⊗</mo><msup><mrow><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>⊗</mo><mi>t</mi></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119458&type= 69.51133728 24.89200020 其中： <math display="block" id="M234"><mover><mrow><mi>τ</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo>=</mo><mo> </mo><mfrac><mrow><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="bold-italic">I</mi><mo>⊗</mo><msup><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo> </mo><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119465&type= 50.79999924 9.31333351 <math display="block" id="M235"><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mi>y</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo> </mo><mfenced separators="|"><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><mo>\</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><mi>θ</mi><mo>⊗</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mrow></mfenced><mo> </mo><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><mo>\</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><mi>θ</mi><mo> </mo><mo>⊗</mo><mo> </mo><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mi>y</mi></mrow></msub><mo> </mo><mfenced separators="|"><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mrow></mfenced><mo> </mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119469&type= 50.79999924 20.57399940 则 <math id="M236"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mi>y</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced open="{" close="}" separators="|"><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>-</mo><mi>s</mi></mrow></msup><mfenced separators="|"><mrow><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><mi>θ</mi><mo>⊗</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mi>y</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mfenced open="{" close="}" separators="|"><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>-</mo><mi>s</mi></mrow></msup><mfenced separators="|"><mrow><mi>θ</mi><mo>⊗</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>+</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><mo>\</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><mi>θ</mi><mo>⊗</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo>-</mo><mi>s</mi></mrow></msup><mfenced separators="|"><mrow><mi>θ</mi><mo>⊗</mo><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>+</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><mo>\</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup></mrow></munder><msub><mrow><mi>ω</mi></mrow><mrow><mi>σ</mi></mrow></msub><mi>θ</mi><mo>⊗</mo><mi mathvariant="normal">t</mi><msub><mrow><mi mathvariant="normal">r</mi></mrow><mrow><mi>y</mi></mrow></msub><mfenced separators="|"><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mrow></mfenced><mo> </mo></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mi>θ</mi><mo>⊗</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>⊗</mo><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119478&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119476&type= 75.52266693 45.29666519 <math id="M237"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119481&type= 1.77800000 2.79399991 为Alice持有的第一列量子比特的乘积态。同理假设第 <math id="M238"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119485&type= 1.86266661 3.72533321 个参与者是诚实的，则其余 <math id="M239"><mi>n</mi><mo> </mo><mo>-</mo><mo> </mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119489&type= 9.22866726 2.79399991 个参与者合作得不到关于量子秘密的任何信息。 不论 <math id="M240"><mi>n</mi><mo> </mo><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119494&type= 8.88999939 2.96333337 为偶数还是奇数时，不诚实参与者只能得到高度混合态，得不到关于量子秘密的任何信息。 在共享秘密上门评估也满足第二个性质。假设 <math id="M241"><mi>n</mi><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119500&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119499&type= 2.20133328 2.79399991 个参与者中只有一个是诚实方，而其余 <math id="M242"><mi>n</mi><mo> </mo><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119503&type= 8.88999939 2.79399991 个可以执行任何操作，可证明诚实方公布的结果比特是一致随机的且独立于其他参与者的行为 [ 29 29 ] 。根据门评估作用效果，得到评估第 <math id="M243"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119510&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119507&type= 1.10066664 2.79399991 个门之后系统状态形式为： χ <math id="M244"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow/></munder><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">e</mi></mrow></msub><mo>⊗</mo><mfenced separators="|"><mrow><mfrac><mrow><mi>σ</mi><mo>⊗</mo><mi>θ</mi></mrow><mrow><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>N</mi></mrow></msup></mrow></mfrac></mrow></mfenced><mo>⊗</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119511&type= 49.86866760 9.65199947 (10) 其中 <math id="M245"><mi>σ</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold-italic">P</mi></mrow><mrow><mo>⊗</mo><mi>s</mi></mrow></msup><mo>,</mo><mi>θ</mi><mo>∈</mo><msup><mrow><mfenced open="{" close="}" separators="|"><mrow><mi>I</mi><mo>,</mo><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></mfenced></mrow><mrow><mo>⊗</mo><mi>t</mi><mo>-</mo><mi>i</mi></mrow></msup><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119519&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119516&type= 42.24866486 4.91066647 <math id="M246"><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119520&type= 2.62466669 3.72533321 是一组标量， <math id="M247"><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119526&type= 6.51933336 4.57200003 是Alice持有的一列量子态， χ i 是不诚实方系统的集合。诚实方与其他系统以乘积态的形式存在，因此其余的 <math id="M248"><mi>n</mi><mo> </mo><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119532&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119530&type= 8.88999939 2.79399991 个不诚实的参与者不能恢复秘密，因为诚实方的测量结果没有传递任何有用的信息。 5　 结 论 1 本文对参与者人数 <math id="M249"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119537&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119534&type= 1.86266661 2.79399991 为任意情形(即不局限于 <math id="M250"><mi>n</mi><mo>=</mo><mn mathvariant="normal">4</mn><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119541&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119539&type= 16.93333435 2.96333337 )构造了基于量子电路的一个 <math id="M251"><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119544&type= 9.14400005 3.55599999 门限量子秘密共享方案，且进一步由Clifford群门和 <math id="M252"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119548&type= 2.62466669 2.79399991 门组成的通用门进行安全性评估。由于通用量子逻辑门有不同的表现形式，因此寻找新型的通用量子逻辑门集合以及考虑其他几类通用门构造的量子电路，并建立基于这些量子电路上的更加安全的量子秘密共享方案，是今后有待研究的问题。 参考文献 CRÉPEAU C , GOTTESMAN D ， SMITH A . 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DOI:10.14188/j.1671⁃8836.2019.02.009(Ch). LI Zhihui,BAI Haiyan,BAI Chenming.Threshold Secret Sharing Scheme Based on Quantum Circuit [J].J Wuhan Univ（Nat Sci Ed）,2019,65(2):200⁃206. DOI:10.14188/j.1671⁃8836.2019.02.009(Ch). 国家自然科学基金 (61373150，61602291)；中央高校基本科研业务费专项资金 (GK201603087) 2018-08-17 2019-04-24 2020-11-25 武汉大学学报（理学版） 2 65