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Journal of Physics A Mathematical and General

Quantum correlations and the extended phase space

Chee Ching Chong, Apostol Vourdas

Published Nov 13, 2001• 6 citations
Publisher

Abstract

An extended phase space x – p – X – P is introduced for one-mode systems, where x – p is the position–momentum plane and X – P is the quantum correlations plane. The Wigner functions W ( x , p ) and the Weyl functions ( X , P ) for arbitrary operators are studied in this space. Creation and annihilation operators for Wigner and Weyl functions are introduced and are used to perform displacements and squeezing in the x – p – X – P extended phase space. The formalism can be used for the construction of quantum states with desirable correlation properties.

DOI
10.1088/0305-4470/34/46/308

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An extended phase space x – p – X – P is introduced for one-mode systems, where x – p is the position–momentum plane and X – P is the quantum correlations plane. The Wigner functions W ( x , p ) and the Weyl functions ( X , P ) for arbitrary operators are studied in this space. Creation and annihilation operators for Wigner and Weyl functions are introduced and are used to perform displacements and squeezing in the x – p – X – P extended phase space. The formalism can be used for the construction of quantum states with desirable correlation properties.
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Finite quantum systems where the position xxx and the momentum ppp take values in Z(d)\mathbb{Z}(d)Z(d) are considered. An extended phase space $x\ensuremath{-}p\ensuremath{-}X\ensuremath{-}P$, where XXX is position increment and PPP is momentum increment, is introduced. The extended Wigner function W(x,p,X,P)\mathcal{W}(x,p,X,P)W(x,p,X,P) and the extended Weyl function $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{W}}({x}^{\ensuremath{'}},{p}^{\ensuremath{'}},{X}^{\ensuremath{'}},{P}^{\ensuremath{'}})$ are defined and their properties are studied. They quantify fourth-order interference. Several examples are discussed.
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Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functions of phase space variables, leading to expressions for quantum correlation functions in terms of phase space integrals in which phase space variables for each mode—c-numbers for bosons, Grassmann numbers for fermions—replace mode annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functions, with distinct phase space variables for annihilation and creation operators; however, symmetrically ordered (Wigner type) distribution functions are also considered, along with unnormalised B distribution functions, which lead to phase space integrals for Fock state populations and coherences. Characteristic functions are first defined and then shown to be related to distribution functions via phase space integrals. The existence and symmetry properties of distribution functions—which are non-unique and non-analytic for bosons—is demonstrated using Bargmann coherent-state projectors.
Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 (2016) and Phys. Rev. A 96, 022117 (2017)] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function---the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics---extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. First we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space---putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of ``parity'' and ``displacement'' in a natural broadening of previously developed phase-space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.