Discrete and Continuous Models and Applied Computational Science

2658-46702658-7149

Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)

29429

10.22363/2658-4670-2021-29-4-361-386

Articles

Статьи

Research Article

Parameterizing qudit states

Параметризация состояний кудита

https://orcid.org/0000-0002-5953-0140

Khvedelidze

Arsen

Хведелидзе

А.

PhD in physics and mathematics, Head of Group of Algebraic and Quantum Computations of Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear Research; Director of Institute of Quantum Physics and Engineering Technologies, Georgian Technical University; Researcher in A. Razmadze Mathematical Institute, Iv. Javakhishvili Tbilisi State University

akhved@jinr.ru

https://orcid.org/0000-0003-3817-5976

Mladenov

Dimitar

Младенов

Д.

PhD in Physics and Mathematics, Associate professor of department of Theoretical Physics of Faculty of Physics

mladim2002@gmail.com

https://orcid.org/0000-0002-4514-2884

Torosyan

Astghik

Торосян

А.

Junior Researcher in Meshcheryakov Laboratory of Information Technologies

astghik@jinr.ru

A. Razmadze Mathematical Institute Iv. Javakhishvili Tbilisi State UniversityМатематический институт им. А. Размадзе Тбилисский государственный университет им. И. Джавахишвили

Institute of Quantum Physics and Engineering Technologies Georgian Technical UniversityИнститут квантовой физики и инженерных технологий Грузинский технический университет

Meshcheryakov Laboratory of Information Technologies Joint Institute for Nuclear ResearchОбъединённый институт ядерных исследований

Sofia University “St. Kliment Ohridski”Софийский университет им. св. Климента Охридского

12112021

294

VOL 29, NO4 (2021)

ТОМ 29, №4 (2021)

36138612112021

2021

Khvedelidze A., Mladenov D., Torosyan A.

Хведелидзе А., Младенов Д., Торосян А.

http://creativecommons.org/licenses/by/4.0

https://journals.rudn.ru/miph/article/view/29429

Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in the area of developing quantum technologies, a whole set of novel tasks for improving our understanding of the structure of finite-dimensional quantum systems has appeared. In the present article we will concentrate on one aspect of such studies related to the problem of explicit parameterization of state space of an $N$ -level quantum system. More precisely, we will discuss the problem of a practical description of the unitary ${SU(N)}$ -invariant counterpart of the $N$ -level state space ${\mathcal{B}_N}$ , i.e., the unitary orbit space ${B_N/SU(N)}$ . It will be demonstrated that the combination of well-known methods of the polynomial invariant theory and convex geometry provides useful parameterization for the elements of ${B_N/SU(N)}$ . To illustrate the general situation, a detailed description of ${B_N/SU(N)}$ for low-level systems: qubit ( ${N= 2}$ ), qutrit ( ${N=3}$ ), quatrit ( ${N= 4}$ ) - will be given.

Квантовые системы с конечным числом состояний всегда были основным элементом многих физических моделей в ядерной физике, физике элементарных частиц, а также в физике конденсированного состояния. Однако сегодня, в связи с практической потребностью в области развития квантовых технологий, возник целый ряд новых задач, решение которых будет способствовать улучшению нашего понимания структуры конечномерных квантовых систем. В статье мы сфокусируемся на одном из аспектов исследований, связанных с проблемой явной параметризации пространства состояний $N$ -уровневой квантовой системы. Говоря точнее, мы обсудим вопрос практического описания унитарного пространства орбит - ${SU(N)}$ -инвариантного аналога $N$ -уровневого пространства состояний ${B_N}$ . В работе будет показано, что сочетание хорошо известных методов теории полиномиальных инвариантов и выпуклой геометрии позволяет получить удобную параметризацию для элементов ${B_N/SU(N)}$ . Общая схема параметризации ${B_N/SU(N)}$ будет детально проиллюстрирована на примере низкоуровневых систем: кубита ( ${N= 2}$ ), кутрита ( ${N= 3}$ ), куатрита ( ${N= 4}$ ).

density matrix parameterizationquantum systemqubitqutritquatritquditpolynomial invariant theoryconvex geometry

параметризация матрицы плотностиквантовая системакубиткутриткуатриткудиттеория полиномиальных инвариантоввыпуклая геометрия

The work is supported in part by the Bulgaria-JINR Program of Collaboration. One of the authors (AK) acknowledges the financial support of the Shota Rustaveli National Science Foundation of Georgia, Grant FR-19-034. DM has been supported in part by the Bulgarian National Science Fund research grant DN 18/3.

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