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

Russell P Rundle,Mark J Everitt

doi:10.1002/qute.202100016

Abstract

AbstractThe 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.

Highlights

The phase-space formulation of quantum mechanics has recently seen taking a quantum wavefunction and representing it in phase space—the Wigner funcincreased use in testing quantum technologies, including methods of tion
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 Subsequently, in the 1940s, both Gronewold[3] and Moyal[4] developed Wigner’s phase-space function to fit around the earlier language Weyl used to represent quantum mechanics in phase space
The phase-space formulation is just one of many approaches introduced the representation of discrete quantum systems to consider quantum mechanics, where the well-known into phase-space methods.[5]

Summary

Introduction

The phase-space formulation of quantum mechanics has recently seen taking a quantum wavefunction and representing it in phase space—the Wigner funcincreased use in testing quantum technologies, including methods of tion. 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 Subsequently, in the 1940s, both Gronewold[3] and Moyal[4] developed Wigner’s phase-space function to fit around the earlier language Weyl used to represent quantum mechanics in phase space. They did this by developing tools to quantum technologies are provided, with a focus on discrete quantum transform any arbitrary operator into the systems, qubits in particular. The phase-space formulation is just one of many approaches introduced the representation of discrete quantum systems to consider quantum mechanics, where the well-known into phase-space methods.[5]

Objectives

Methods

Conclusion