N=3的自旋链上二比特量子信息传输的调控

N =3的自旋链上二比特量子信息传输的调控 Control of Two-Qubit Quantum Information Transfer in N =3 Spin Chain first-author 王清亮，男，副教授，博士，现从事量子信息传输方面的研究。E-mail：wangql19790511@163.com 王清亮，男，副教授，博士，现从事量子信息传输方面的研究。E-mail：wangql19790511@163.com wangql19790511@163.com corresp E-mail： shamemei2@163.com E-mail： shamemei2@163.com shamemei2@163.com 忻州师范学院 物理系，山西 忻州 034000 忻州师范学院 物理系，山西 忻州 034000 Department of Physics， Xinzhou Teachers University， Xinzhou 034000， Shanxi， China Department of Physics， Xinzhou Teachers University， Xinzhou 034000， Shanxi， China 利用对角化哈密顿矩阵的方法，研究了中心对称、两端反向的弱磁场对 N =3的自旋链上二比特量子信息完美传输的影响。此时，实现二比特量子信息完美传输的时间条件不仅仅取决于体系自身，还取决于该外加磁场。借助该磁场的影响，提出了对 N =3的自旋链上二比特量子信息完美传输进行量子调控的理论方案：缩短信息完美传输的时间可通过增大近邻格点磁场的差值来实现，反之，延长信息完美传输的时间则可通过减小该差值实现。 The influence of a particular weak magnetic field for the perfect transfer of two-qubit quantum information in the spin chain ( N =3) was studied by the method of the diagonalized Hamiltonian matrix, where the magnetic field is symmetric on the center of the spin chain and its direction is opposite on each side. The time condition for perfect transfer of two-qubit quantum information is not only determined by the system itself, but also by the nature of the external magnetic field in this case. Based on the influence of the magnetic field, a theoretical scheme was proposed to control the perfect transfer of the two-qubit quantum information in the spin chain ( N =3). In order to shorten the perfect transfer time, it can be realized by increasing the difference value of the magnetic field between the nearest lattice points; conversely, extending the perfect transfer time of information can be achieved by reducing the difference value. 量子信息传输 量子比特 自旋链 量子调控 quantum information transfer quantum bit spin chain quantum control 0　 引 言 1 量子信息理论是量子计算机理论的重要分支，在量子信息处理中，有关量子信息在自旋链信道上传输的研究已经取得了相当大的进展 ［ 1 1 ~ 4 4 1~4 ］ 。在诸多可以传输量子信息的物理信道中，自旋链信道被视为最具优势的信息传输信道，近年来在该体系中进行量子信息的传输已成为量子信息研究领域的热点问题之一 ［ 4 4 ~ 8 8 4~8 ］ 。两端开放的一维自旋链网络可视作相互作用量子比特的组合，在仅考虑了自旋链各格点间近邻相互作用的经过修正的Heisenberg- XX 模型上，任意多态量子信息的完美传输都可以仅由系统的动力学演化来实现。理论研究表明，无论链的长度是多少，其完美传输得以实现的时间条件均为固定取值，即一旦该条件得以满足，无论链有多长，总可以实现信息的完美传输 ［ 6 6 ~ 12 12 6~12 ］ 。但其不足之处也同样明显，在量子计算机中真正进行量子信息传输时，往往要求能够人为控制信息传输的时间，随心所欲地进行量子信息的传输，而这正是自旋链上信息的完美传输所无法做到的 ［ 9 9 ~ 17 17 9~17 ］ 。以此为背景，本文引入一种特定的弱磁场，研究这种弱磁场对 N =3的自旋链上二比特量子信息传输的影响，并进一步借助该影响研究如何实现这种信息传输的量子调控，从而人为控制信息的完美传输。 1　 N =3的自旋链上二比特量子信息的完美传输 1 在讨论磁场对 N =3的自旋链上二比特量子信息传输的影响之前，首先引入未加磁场时实现二比特量子信息完美传输的条件 ［ 1 1 ~ 5 5 1~5 ］ 。考虑经修正的Heisenberg- XX 模型自旋链，其哈密顿量 ［ 1 1 ， 2 2 ］ 为 <math display="block" id="M1"><mi>H</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>J</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mfenced separators="|"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msubsup><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>-</mo></mrow></msubsup><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268124&type= 55.54133606 8.46666718 其中， <math id="M2"><msubsup><mrow><mi>J</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mo>=</mo><mroot><mrow><mi>n</mi><mo stretchy="false">(</mo><mi>N</mi><mo>-</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268135&type= 32.93533325 4.82599974 是两个（第 n 个和第 n +1个）近邻格点间的耦合强度；泡利算符角标 n 和 n +1意味着分别仅对第 n 和第 n +1个格点处的自旋态作用。本文主要讨论格点数为3的自旋链，因而这里 N =3，用格点数 N 表示自旋链的长度。 在进行二比特量子信息传输时，可在自旋链的一端制备二比特量子信息 <math id="M3"><mi>α</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">00</mn></mrow></mfenced><mo>+</mo><mi>β</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">11</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268143&type= 21.75933456 4.23333359 ，这里 <math id="M4"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>α</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>β</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268151&type= 23.53733444 5.84200001 ，即初始信息已归一化，此时整个自旋链体系的状态为 <math display="block" id="M5"><mfenced open="|" close=">" separators="|"><mrow><mi>Ψ</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>α</mi><mfenced open="|" close=">" separators="|"><mrow><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">00</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced><mo>+</mo><mi>β</mi><mfenced open="|" close=">" separators="|"><mrow><msub><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268158&type= 48.59866714 4.82599974 其中，A和B分别表示自旋链的两个端点，A表示信息输入端，B表示信息传输的终端。 二比特量子信息的完美传输即为在链自身动力学演化的作用下，将初态信息 <math id="M6"><mi>α</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">00</mn></mrow></mfenced><mo>+</mo><mi>β</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">11</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268165&type= 21.92866707 4.23333359 经过一定的时间传输至链的另一端，在传输的过程中信息的保真度取最大值1，从而使得体系状态变为 <math display="block" id="M7"><mfenced open="|" close=">" separators="|"><mrow><mi>Ψ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><mi>α</mi><mfenced open="|" close=">" separators="|"><mrow><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">00</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced><mo>+</mo><mi>β</mi><mfenced open="|" close=">" separators="|"><mrow><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">11</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268172&type= 48.17533112 4.82599974 由于 <math id="M8"><mfenced open="|" close=">" separators="|"><mrow><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">00</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268179&type= 11.09133339 4.82599974 是自旋链体系哈密顿量算符零本征值的本征态，在演化算符的作用下其叠加系数 <math id="M9"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268189&type= 1.94733334 2.79399991 不会改变，因此在信息传输的过程中，确定保真度的演化概率幅仅仅取决于 <math id="M10"><mfenced open="|" close=">" separators="|"><mrow><msub><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268201&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268197&type= 11.00666714 4.82599974 的演化。文献［ 5 5 ］给出了该概率幅的表达式 <math display="block" id="M11"><mi>F</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mfenced open="<" close="|" separators="|"><mrow><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">11</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">{</mo><mo>-</mo><mi mathvariant="normal">i</mi><mi>λ</mi><mi>H</mi><mi>t</mi><mo stretchy="false">}</mo><mfenced open="|" close=">" separators="|"><mrow><msub><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>λ</mi><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268210&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268205&type= 80.94133759 9.56733322 其中， t （s）表示实现信息完美传输的时间； <math id="M12"><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo>-</mo><mi mathvariant="normal">i</mi><mi>λ</mi><mi>H</mi><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268214&type= 19.64266586 3.72533321 为态的演化算符； <math id="M13"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268222&type= 1.77800000 2.79399991 是为平衡量纲而引入的常量，可令 <math id="M14"><mi>λ</mi><mo>=</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mi mathvariant="normal">ℏ</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268229&type= 12.78466702 3.30199981 ， <math id="M15"><mi mathvariant="normal">ℏ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268238&type= 1.94733334 2.79399991 为普朗克常量， <math id="M16"><mi mathvariant="normal">ℏ</mi><mo>=</mo><mn mathvariant="normal">1.054</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">34</mn></mrow></msup><mi mathvariant="normal">J</mi><mo>⋅</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268247&type= 33.27399826 3.04799986 。显然，要想实现二比特量子信息的完美传输，需满足传输的保真度为1，即 <math display="block" id="M17"><mfenced open="|" close="|" separators="|"><mrow><mi>F</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268257&type= 16.42533302 4.74133348 从而可得出实现信息完美传输所满足的时间条件为 <math display="block" id="M18"><mi>t</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268265&type= 9.31333351 7.53533316 由（6）式可知，实现信息完美传输的时间条件仅仅由常量 <math id="M19"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268273&type= 1.77800000 2.79399991 决定，与自旋链的长度和格点数目无任何关系，表明：在未加磁场时，只需满足上述时间条件即可实现二比特量子信息在自旋链上的完美传输，而该条件仅由体系自身的动力学演化属性决定 ［ 1 1 ~ 5 5 1~5 ］ ，因此任何人都无法根据需要调整信息完美传输的条件，从而对其进行人为地调控。 2　 弱磁场对 N =3自旋链上二比特量子信息完美传输的影响 1 这里，将自旋链置于一个空间分布随格点逐差排列且不随时间变化的弱磁场中，设磁场方向沿着自旋 z 方向，则（1）式所示的体系哈密顿量增加了一项磁场作用项，变为 <math display="block" id="M20"><mi>H</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>J</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mfenced separators="|"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msubsup><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>-</mo></mrow></msubsup><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></mrow></mfenced></mrow><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></munderover></mstyle><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>z</mi></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268281&type= 72.30532837 8.38199997 其中， <math id="M21"><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268292&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268288&type= 3.64066648 3.72533321 （T）（ n =1，2，3）表示第 n 个格点处磁感应强度的数值。 具体而言，设定弱磁场方向沿 z 轴的同时，其大小沿格点逐差变化值为 <math id="M22"><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268300&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268297&type= 6.43466663 3.30199981 ，其中， B （T）表示近邻格点磁场的差值，且各格点磁场取值的大小关于链中心对称，即自旋链各格点处的磁场值分别为 <math display="block" id="M23"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268305&type= 42.41799927 7.53533316 相应地，链的哈密顿量为 <math display="block" id="M24"><mtable columnalign="left"><mtr><mtd><mi>H</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>J</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow/></msubsup><mfenced separators="|"><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msubsup><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>-</mo></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>-</mo></mrow></msubsup><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></mrow></mfenced></mrow><mo>+</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mo>-</mo><mfrac><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>z</mi></mrow></msubsup><mo>+</mo><mn mathvariant="normal">0</mn><msubsup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>z</mi></mrow></msubsup><mo>+</mo><mfrac><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mi>z</mi></mrow></msubsup></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268312&type= 59.52066422 19.13466644 对于二比特量子信息传输而言，自旋链体系的哈密顿量被限制在其第三子空间中，该空间是由自旋链所有第二激发态作为基矢而张成的。第二激发态指的是自旋链上两个格点的自旋处于向上而其他格点的自旋处于向下的状态，显然对于 N =3的自旋链而言，其第二激发态共有 <math id="M25"><msubsup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268321&type= 11.17599964 3.97933340 个，分别是 <math id="M26"><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">110</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268332&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268328&type= 7.53533316 3.97933340 、 <math id="M27"><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">101</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268336&type= 7.53533316 3.97933340 和 <math id="M28"><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">011</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268343&type= 7.53533316 3.97933340 ，可以张成三维的第三子空间，该空间中，基矢分别可选为 <math display="block" id="M29"><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">110</mn></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><mo> </mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">101</mn></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><mo> </mo><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">011</mn></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268351&type= 58.84333038 13.46199894 从而可得（9）式所示哈密顿量的矩阵形式为 <math display="block" id="M30"><mi mathvariant="bold-italic">H</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mi>B</mi></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mi>B</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268360&type= 38.35400009 28.78666687 根据第一节的讨论，形如 <math id="M31"><mi>α</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">00</mn></mrow></mfenced><mo>+</mo><mi>β</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">11</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268368&type= 21.75933456 4.23333359 的量子信息，其传输的概率幅可由下式决定 <math display="block" id="M32"><mtable columnalign="left"><mtr><mtd><mi>F</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mfenced open="<" close="|" separators="|"><mrow><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">11</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">{</mo><mo>-</mo><mi mathvariant="normal">i</mi><mi>λ</mi><mi mathvariant="bold-italic">H</mi><mi>t</mi><mo stretchy="false">}</mo><mfenced open="|" close=">" separators="|"><mrow><msub><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfenced open="<" close="|" separators="|"><mrow><msub><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">11</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced><mi mathvariant="bold-italic">U</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">{</mo><mo>-</mo><mi mathvariant="normal">i</mi><mi>λ</mi><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi></mrow></msup><mi>t</mi><mo stretchy="false">}</mo><msup><mrow><mi mathvariant="bold-italic">U</mi></mrow><mrow><mo>†</mo></mrow></msup><mfenced open="|" close=">" separators="|"><mrow><msub><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268376&type= 70.52733612 11.51466751 其中， U 为幺正变换矩阵 ， <math id="M33"><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=6268384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268381&type= 5.41866684 3.04799986 为其厄米共轭矩阵， <math id="M34"><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268387&type= 6.77333355 3.04799986 是 H 在其自身表象下的对角矩阵，对角元是其本征值，满足 <math display="block" id="M35"><mi mathvariant="bold-italic">H</mi><mo>=</mo><mi mathvariant="bold-italic">U</mi><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi></mrow></msup><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=6268396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268392&type= 23.28333473 3.04799986 从而得出信息传输的概率幅可以表示为 <math display="block" id="M36"><mi>F</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mi>λ</mi><msup><mrow><mi>E</mi></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mi>t</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268401&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268400&type= 44.70399857 8.38199997 其中， <math id="M37"><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo> </mo><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268406&type= 9.14400005 4.48733330 是幺正变换矩阵 <math id="M38"><mi mathvariant="bold-italic">U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268411&type= 2.96333337 2.79399991 的第 i <math id="M39"><mo> </mo><mo> </mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268418&type= 19.04999924 3.55599999 行第 k 列的矩阵元， <math id="M40"><msubsup><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268423&type= 7.95866680 4.65666676 是矩阵元 <math id="M41"><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268432&type= 7.87400007 4.48733330 的复共轭； <math id="M42"><msup><mrow><mi>E</mi></mrow><mrow><mfenced separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268438&type= 5.07999992 3.21733332 为哈密顿量 H 在第二激发态为基矢所张成的第三子空间中的第 k 个能级，亦即三维对角矩阵 <math id="M43"><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268450&type= 6.77333355 3.04799986 的第 k 列的非零矩阵元。（11）式所示哈密顿量算符 <math id="M44"><mi mathvariant="bold-italic">H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268456&type= 3.04799986 2.79399991 的本征值为 <math display="block" id="M45"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mo>-</mo><mroot><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mo>,</mo><mo> </mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mo> </mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mroot><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mtext>  </mtext></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268467&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268462&type= 72.05133057 4.82599974 可以得到 <math display="block" id="M46"><msup><mrow><mi mathvariant="bold-italic">H</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi></mrow></msup><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mo>-</mo><mroot><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mroot><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268469&type= 55.79533386 16.67933464 这里需要注意的是，所加的磁场为弱磁场，表现为（8）式中各个格点处磁场数值极小，即 <math id="M47"><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268475&type= 6.43466663 3.30199981 或 B 为极小值，因而在进行进一步计算时，可近似采用无磁场时体系哈密顿量对角化的幺正变换矩阵： <math display="block" id="M48"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">U</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">3</mn></mrow></mfenced></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">3</mn></mrow></mfenced></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">3</mn></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mfrac><mrow><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268482&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268480&type= 48.85266495 47.83666611 联立（14）、（16）和（17）式可以得到，二比特量子信息传输的概率幅为 <math display="block" id="M49"><mi>F</mi><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mo>-</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>λ</mi><mi>t</mi><mroot><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268486&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268484&type= 45.21199799 11.93799973 要实现此演化过程中二比特量子信息的完美传输，即使信息传输的保真度保持为1，需要使态演化的概率为1，即 <math id="M50"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>N</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268491&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268488&type= 18.11866570 4.91066647 ，利用（18）式可以得出，在 N =3的自旋链上实现二比特量子信息完美传输的时间条件为 <math display="block" id="M51"><mi>t</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mi>λ</mi><mroot><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268497&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268494&type= 23.02933502 9.14400005 （19）式表明，磁场中自旋链上二比特量子信息完美传输的时间条件虽然仍与链的长度无关，但却并非仅仅取决于体系自身的物理属性，而是由系统和所设定的磁场共同决定。 3　 弱磁场对 N =3自旋链上二比特量子信息传输调控的讨论 1 上面的研究表明，将长度为 N= 3的自旋链置于方向沿 z 轴（关于自旋链中心对称）、大小沿格点逐差变化为恒定值 <math id="M52"><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268503&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268500&type= 6.43466663 3.30199981 的弱磁场中，即各格点磁场数值如（8）式所示。若在其一端制备形如 <math id="M53"><mi>α</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">00</mn></mrow></mfenced><mo>+</mo><mi>β</mi><mfenced open="|" close=">" separators="|"><mrow><mn mathvariant="normal">11</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268506&type= 21.75933456 4.23333359 的二比特量子信息，将其传输至链的另一端时，体系状态演化的概率幅如（18）式所示，满足保真度为1的完美传输的时间条件即（19）式。 结合（18）和（19）式，并与无磁场时的传输概率幅（（5）式）和传输时间条件（（6）式）及其结论相比较可知，对于长度为 N =3的自旋链而言，如果给其加上一个关于中心对称、大小沿格点逐差变化为恒定值 <math id="M54"><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268515&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268512&type= 6.43466663 3.30199981 的弱磁场，磁场对二比特量子信息的传输概率起着决定性的作用，实现完美传输的时间将不再仅仅取决于体系自身的物理属性，还由外磁场的性质所决定，该传输概率将随近邻格点磁场间隔 <math id="M55"><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268521&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268518&type= 6.43466663 3.30199981 的增大而减小，随近邻格点磁场差值 <math id="M56"><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268524&type= 6.43466663 3.30199981 的减小而增大。 利用这种特定弱磁场的影响，显然可以将传输二比特量子信息的自旋链置于这种磁场中，以（18）式和（19）式的理论计算结果为依据，通过改变近邻格点磁场差值 <math id="M57"><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268536&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268534&type= 6.43466663 3.30199981 的大小来进行信息完美传输的量子调控，根据需要来调整实现量子信息完美传输的时间：1） 为使二比特量子信息完美传输的时间缩短，可使 B 的数值增大，即增加近邻格点之间磁场差值 <math id="M58"><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268540&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268538&type= 6.43466663 3.30199981 的大小；2） 为使二比特量子信息完美传输的时间延长，可使 B 的数值减小，即降低近邻格点之间磁场差值 <math id="M59"><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268542&type= 6.43466663 3.30199981 的大小；3） 增大或降低磁场差值 <math id="M60"><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268547&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268545&type= 6.43466663 3.30199981 大小的量化数值，可由（19）式决定。 4　 结 语 1 实现量子信息完美传输的调控是量子信息理论需要解决的关键问题之一。本文首先引入了仅存在近邻格点相互作用的经过修正的Heisenberg- XX 自旋链模型，回顾了在该模型上实现二比特量子信息完美传输的时间条件，该时间条件表明完美传输仅由体系自身所决定。将传输二比特量子信息的自旋链置于关于链中心对称、大小沿格点逐差变化为恒定值 <math id="M61"><mrow><mrow><mi>B</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=6268550&type= 6.43466663 3.30199981 的弱磁场中，讨论这种磁场对长度为 N =3自旋链上二比特量子信息完美传输的影响；并以此为基础，研究得出可以借助近邻格点间隔恒定且关于中心对称的弱磁场，实现 N =3的自旋链上二比特量子信息完美传输的量子调控。 参考文献 BOSE S . 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DOI:10.14188/j.1671-8836.2020.0125 王清亮,任恒峰.N=3的自旋链上二比特量子信息传输的调控[J].武汉大学学报（理学版）,2020,66(6):561-564. DOI:10.14188/j.1671-8836.2020.0125 山西省高等学校教学改革创新项目（J2019169）；山西省教育科学“十三五”规划课题（GH-17062）；忻州师范学院院级教改课题（JGZD201602） 2020-05-19 2020-12-24 2021-01-26 武汉大学学报（理学版） 6 66