{"title":"解读双量子比特相空间中的交映线性变换","authors":"William K. Wootters","doi":"10.1142/s0219749924400148","DOIUrl":null,"url":null,"abstract":"<p>For the continuous Wigner function and for certain discrete Wigner functions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. That is, performing this unitary transformation is simply a matter of moving Wigner-function values around in phase space. This result holds in particular for the simplest discrete Wigner function defined on a <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mi>d</mi><mo stretchy=\"false\">×</mo><mi>d</mi></math></span><span></span> phase space when the Hilbert-space dimension <span><math altimg=\"eq-00002.gif\" display=\"inline\"><mi>d</mi></math></span><span></span> is odd. It does not hold for a <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi>d</mi><mo stretchy=\"false\">×</mo><mi>d</mi></math></span><span></span> phase space if the dimension is even. Here we show, though, that a generalized version of this correspondence does apply in the case of a two-qubit phase space. In this case, a symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.</p>","PeriodicalId":51058,"journal":{"name":"International Journal of Quantum Information","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Interpreting symplectic linear transformations in a two-qubit phase space\",\"authors\":\"William K. Wootters\",\"doi\":\"10.1142/s0219749924400148\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For the continuous Wigner function and for certain discrete Wigner functions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. That is, performing this unitary transformation is simply a matter of moving Wigner-function values around in phase space. This result holds in particular for the simplest discrete Wigner function defined on a <span><math altimg=\\\"eq-00001.gif\\\" display=\\\"inline\\\"><mi>d</mi><mo stretchy=\\\"false\\\">×</mo><mi>d</mi></math></span><span></span> phase space when the Hilbert-space dimension <span><math altimg=\\\"eq-00002.gif\\\" display=\\\"inline\\\"><mi>d</mi></math></span><span></span> is odd. It does not hold for a <span><math altimg=\\\"eq-00003.gif\\\" display=\\\"inline\\\"><mi>d</mi><mo stretchy=\\\"false\\\">×</mo><mi>d</mi></math></span><span></span> phase space if the dimension is even. Here we show, though, that a generalized version of this correspondence does apply in the case of a two-qubit phase space. In this case, a symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.</p>\",\"PeriodicalId\":51058,\"journal\":{\"name\":\"International Journal of Quantum Information\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-05-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Journal of Quantum Information\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1142/s0219749924400148\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Quantum Information","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1142/s0219749924400148","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
摘要
对于连续维格纳函数和某些离散维格纳函数来说,按照交映线性变换对维格纳函数值进行排列,就等同于对状态进行某种单元变换。也就是说,进行这种单位变换只是在相空间中移动维格纳函数值。当希尔伯特空间维数 d 为奇数时,这一结果尤其适用于定义在 d×d 相空间上的最简单离散维格纳函数。如果维数为偶数,则 d×d 相空间不成立。不过,我们在这里证明,这种对应关系的广义版本确实适用于双量子比特相空间。在这种情况下,相空间各点的交折线性排列,再加上对维格纳函数的某种重新解释,就等同于单位变换。
Interpreting symplectic linear transformations in a two-qubit phase space
For the continuous Wigner function and for certain discrete Wigner functions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. That is, performing this unitary transformation is simply a matter of moving Wigner-function values around in phase space. This result holds in particular for the simplest discrete Wigner function defined on a phase space when the Hilbert-space dimension is odd. It does not hold for a phase space if the dimension is even. Here we show, though, that a generalized version of this correspondence does apply in the case of a two-qubit phase space. In this case, a symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.
期刊介绍:
The International Journal of Quantum Information (IJQI) provides a forum for the interdisciplinary field of Quantum Information Science. In particular, we welcome contributions in these areas of experimental and theoretical research:
Quantum Cryptography
Quantum Computation
Quantum Communication
Fundamentals of Quantum Mechanics
Authors are welcome to submit quality research and review papers as well as short correspondences in both theoretical and experimental areas. Submitted articles will be refereed prior to acceptance for publication in the Journal.