The Global Indicator of Classicality of an Arbitrary N -Level Quantum System
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THE GLOBAL INDICATOR OF CLASSICALITY OF AN ARBITRARY N -LEVEL QUANTUM SYSTEM V. Abgaryan,∗ A. Khvedelidze,† and A. Torosyan‡
UDC 512.816.2, 530.145
It is commonly accepted that the deviation of the Wigner quasiprobability distribution of a quantum state from a proper statistical distribution signifies its nonclassicality. Following this ideology, we introduce a global indicator QN for quantifying the “classicality-quantumness” correspondence in the form of a functional on the orbit space O[PN ] of the adjoint action of the group SU(N ) on the state space PN of an N -dimensional quantum system. The indicator QN is defined as the (+) relative volume of the subspace O[PN ] ⊂ O[PN ] where the Wigner quasiprobability distribution is (+) positive. The algebraic structure of O[PN ] is revealed and exemplified by the case of a single qubit (N = 2) and a single qutrit (N = 3). For the Hilbert–Schmidt ensemble of qutrits, the dependence of the global indicator on the moduli parameter of the Wigner quasiprobability distribution is found. Bibliography: 18 titles.
1. Introduction Over the past decades, a number of witnesses and measures of the nonclassicality of quantum systems have been formulated (see, e.g., [1–3]). Most of them are based on the primary impossibility of a classical statistical description of quantum systems. In particular, the nonexistence of positive definite probability distributions serves as a certain indication of the nonclassicality of a physical system.1 In the present note, we will focus on the problem of quantifying the nonclassicality of quantum systems associated with a finite-dimensional Hilbert space by studying the nonpositivity of the Wigner quasiprobability distributions (the Wigner function, or, in short, WF) [6–9]. Our treatment is based on the recent publications [10, 11], where the Wigner quasiprobability (ν) distribution W (ΩN ) of an N -level quantum system is constructed via the dual pairing, W(ν) (ΩN ) = tr [ Δ(ΩN | ν)] ,
(1)
of a density matrix , which is an element of the quantum state space PN = {X ∈ MN (C) | X = X † ,
X ≥ 0,
tr (X) = 1},
(2)
and an element of the dual space Δ(ΩN | ν) ∈ P∗N , the so-called Stratonovich–Weyl (SW) kernel. The dual space P∗N is defined as2 P∗N = {X ∈ MN (C) | X = X † , tr (X) = 1, tr X 2 = N }, (3) ∗ Yerevan Physics Institute, Yerevan, Armenia; Czech Technical University, Prague, Czech Republic; Joint Institute for Nuclear Research, Dubna, Russia, e-mail: [email protected]. †
Razmadze Mathematical Institute, Iv. Javakhishvili Tbilisi State University and Georgian Technical University, Tbilisi, Georgia; Joint Institute for Nuclear Research, Dubna, Russia, and People’s Friendship University of Russia (RUDN University), Moscow, Russia, e-mail: [email protected]. ‡
Joint Institute for Nuclear Research, Dubna, Russia, e-mail: [email protected].
1Furthermore, the negativity of quasiprobability distributions was shown to be a resource for quantum
computation [4, 5]. 2The algebraic equations in (3) define a family of s-parameter SW kernels.
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