Parameterizing qudit states

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Abstract

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 NN-level quantum system. More precisely, we will discuss the problem of a practical description of the unitary SU(N){SU(N)}-invariant counterpart of the NN-level state space BN{\mathcal{B}_N}, i.e., the unitary orbit space BN/SU(N){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 BN/SU(N){B_N/SU(N)}. To illustrate the general situation, a detailed description of BN/SU(N){B_N/SU(N)} for low-level systems: qubit (N=2{N= 2}), qutrit (N=3{N=3}), quatrit (N=4{N= 4}) - will be given.

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1. Introduction Quantum mechanics is a unitary invariant probabilistic theory of finitedimensional systems. Both basic features, the invariance and the randomness, strongly impose on the mathematical structure associated with the state © KhvedelidzeA., MladenovD., TorosyanA., 2021 This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/ space

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About the authors

Arsen Khvedelidze

A. Razmadze Mathematical Institute Iv. Javakhishvili Tbilisi State University; Institute of Quantum Physics and Engineering Technologies Georgian Technical University; Meshcheryakov Laboratory of Information Technologies Joint Institute for Nuclear Research

Author for correspondence.
Email: akhved@jinr.ru
ORCID iD: 0000-0002-5953-0140

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

1, Ilia Chavchavadze Avenue, Tbilisi, 0179, Georgia; 77, Kostava St., Tbilisi, 0175, Georgia; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation

Dimitar Mladenov

Sofia University “St. Kliment Ohridski”

Email: mladim2002@gmail.com
ORCID iD: 0000-0003-3817-5976

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

15, Tsar Osvoboditel Boulevard, Sofia, 1164, Bulgaria

Astghik Torosyan

Meshcheryakov Laboratory of Information Technologies Joint Institute for Nuclear Research

Email: astghik@jinr.ru
ORCID iD: 0000-0002-4514-2884

Junior Researcher in Meshcheryakov Laboratory of Information Technologies

6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation

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Copyright (c) 2021 Khvedelidze A., Mladenov D., Torosyan A.

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