General approach to quantum mechanics as a statistical theory

Abstract

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 and Phys. Rev. A 96, 022117] 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. Firstly we provide a brief modernized discussion of the general framework of phasespace 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 Schrödinger'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.

Visualization of correlations in hybrid discrete—continuous variable quantum systems

“…where P Ŵ ( )is the displaced parity operator for some parameterization of phase space Ω. The displaced parity operator is defined through displacing a generalized parity operator [24], and for the CV Wigner function is [57] is the standard CV displacement operator written using the annihilation and creation operators, â and â † , respectively. Note that we have introduced the subscript f, for 'field', to indicate CV systems.…”

“…as the displacement of the vacuum state, ñ 0 f | , generating a new coherent state bñ f | . As shown in [23,24], a similar approach to(2) can be used to generate Wigner functions for arbitrary quantum systems. For two-level DV systems, for example,…”

“…where the generalized parity, P â , for a single, two-level, system is s P = + 3 2 a z  (ˆˆ) [23,24,28], for a full derivation of the kernel see [58]. Note that the subscript a here indicates that this is a state for the 'atom', or DV system.…”

“…Understanding the nature of the correlations between systems is central to harnessing quantum effects for information processing; the methods presented here reveal the nature of these correlations, allowing a clear visualization of the quantum information present in these hybrid discrete-continuous variable quantum systems. The methods presented here could be viewed as a form of quantum state spectroscopy.Bloch sphere [20][21][22][23][24][25][26]. For example, there have been various proposals put forward that use a continuous Wigner function to reveal correlations between DV systems [26][27][28].…”

“…Bloch sphere [20][21][22][23][24][25][26]. For example, there have been various proposals put forward that use a continuous Wigner function to reveal correlations between DV systems [26][27][28].…”