{"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}
引用次数: 0
Abstract
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.