On Families of Wigner Functions for <i>N</i>-Level Quantum Systems
A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary <i>N</i>-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl corresponden...
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MDPI AG
2021-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/6/1013 |
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| author | Vahagn Abgaryan Arsen Khvedelidze |
| author_facet | Vahagn Abgaryan Arsen Khvedelidze |
| author_sort | Vahagn Abgaryan |
| collection | DOAJ |
| description | A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary <i>N</i>-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl correspondence in the form of algebraic “master equations” for the spectrum of the Stratonovich–Weyl kernel. The later implements a map between the operators in the Hilbert space and the functions in the phase space identified by the complex flag manifold. The non-uniqueness of the solutions to the master equations leads to diversity among the Wigner quasiprobability distributions. It is shown that among all possible Stratonovich–Weyl kernels for a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mo>(</mo><mn>2</mn><mi>j</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-level system, one can always identify the representative that realizes the so-called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>-symmetric spin-<i>j</i> symbol correspondence. The method is exemplified by considering the Wigner functions of a single qubit and a single qutrit. |
| first_indexed | 2024-03-10T10:41:41Z |
| format | Article |
| id | doaj.art-1e1ec7b9997e482e91bb30eb26cf67f3 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T10:41:41Z |
| publishDate | 2021-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-1e1ec7b9997e482e91bb30eb26cf67f32023-11-21T22:53:39ZengMDPI AGSymmetry2073-89942021-06-01136101310.3390/sym13061013On Families of Wigner Functions for <i>N</i>-Level Quantum SystemsVahagn Abgaryan0Arsen Khvedelidze1Laboratory of Information Technologies, Joint Institute for Nuclear Research, 141980 Dubna, RussiaLaboratory of Information Technologies, Joint Institute for Nuclear Research, 141980 Dubna, RussiaA method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary <i>N</i>-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl correspondence in the form of algebraic “master equations” for the spectrum of the Stratonovich–Weyl kernel. The later implements a map between the operators in the Hilbert space and the functions in the phase space identified by the complex flag manifold. The non-uniqueness of the solutions to the master equations leads to diversity among the Wigner quasiprobability distributions. It is shown that among all possible Stratonovich–Weyl kernels for a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mo>(</mo><mn>2</mn><mi>j</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-level system, one can always identify the representative that realizes the so-called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>-symmetric spin-<i>j</i> symbol correspondence. The method is exemplified by considering the Wigner functions of a single qubit and a single qutrit.https://www.mdpi.com/2073-8994/13/6/1013quantum mechanics on phase spacefinite-level quantum systems<i>SU</i>(2) spin-<i>j</i> symbol correspondence |
| spellingShingle | Vahagn Abgaryan Arsen Khvedelidze On Families of Wigner Functions for <i>N</i>-Level Quantum Systems Symmetry quantum mechanics on phase space finite-level quantum systems <i>SU</i>(2) spin-<i>j</i> symbol correspondence |
| title | On Families of Wigner Functions for <i>N</i>-Level Quantum Systems |
| title_full | On Families of Wigner Functions for <i>N</i>-Level Quantum Systems |
| title_fullStr | On Families of Wigner Functions for <i>N</i>-Level Quantum Systems |
| title_full_unstemmed | On Families of Wigner Functions for <i>N</i>-Level Quantum Systems |
| title_short | On Families of Wigner Functions for <i>N</i>-Level Quantum Systems |
| title_sort | on families of wigner functions for i n i level quantum systems |
| topic | quantum mechanics on phase space finite-level quantum systems <i>SU</i>(2) spin-<i>j</i> symbol correspondence |
| url | https://www.mdpi.com/2073-8994/13/6/1013 |
| work_keys_str_mv | AT vahagnabgaryan onfamiliesofwignerfunctionsforinilevelquantumsystems AT arsenkhvedelidze onfamiliesofwignerfunctionsforinilevelquantumsystems |