Overview of the Phase Space Formulation of Quantum Mechanics with Application to Quantum Technologies

Abstract

The phase‐space formulation of quantum mechanics has recently seen increased use in testing quantum technologies, including methods of tomography for state verification and device validation. Here, an overview of quantum mechanics in phase space is presented. The formulation to generate a generalized phase‐space function for any arbitrary quantum system is shown, such as the Wigner and Weyl functions along with the associated Q and P functions. Examples of how these different formulations are used in quantum technologies are provided, with a focus on discrete quantum systems, qubits in particular. Also provided are some results that, to the authors' knowledge, have not been published elsewhere. These results provide insight into the relation between different representations of phase space and how the phase‐space representation is a powerful tool in understanding quantum information and quantum technologies.

Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility

“…Let {� i ⟩} be a basis of the Hilbert space of the quantum system that consists of eigenstates of the pointer variable. The state of the system at time t can then be written as ( 47) �Ψ⟩(0) = �𝜓⟩ ⊗ �E⟩ 5 In Gibbsian statistical mechanics, it is well known that equilibrium systems coupled to a heat bath minimize a free energy functional. For the Boltzmannian case, this deserves some comment: As shown by Dean [115], a system of Langevin equations for the position of classical particles can be exactly rewritten as a Langevin equation for the microscopic one-body density.…”

“…Wigner functions provide a description of quantum systems in phase space. Since their initial development [1], they have found a significant number of applications [2][3][4][5] in fields such as atomic physics [6], quantum optics [7][8][9], visualization of quantum effects [10,11], computational electronics [12][13][14], and solid-state theory [15][16][17][18][19]. Besides these practical aspects, they are also of interest for fundamental questions in quantum mechanics such as the theory of quantum chaos [20].…”

An informationally complete Wigner function for the Tavis–Cummings model

“…We can calculate the Wigner function [13][14][15] of a system of arbitrary components by taking the expectation value of a suitable displaced parity operator over all its possible configurations-the phase space [16][17][18][19]. The total displaced parity operator for the composite system simply comprises the tensor product of the displaced parity operators for each element of the system.…”

An Informationally Complete Wigner Function For The Tavis-Cummings Model: Visualization of Cat Swapping and Quantum Correlations in Field-Many Atom Systems

“…We can calculate the Wigner function [7,8] of a system of arbitrary components by taking the expectation value of a suitable displaced parity operator over all its possible configurations -the phase space [9][10][11][12]. The total displaced parity operator for the composite system simply comprises the tensor product of the displaced parity operators for each element of the system.…”