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

• Published in: Advanced quantum technologies (Online) Vol. 4; no. 6
• Main Authors: Rundle, Russell P., Everitt, Mark J.
• Format: Journal Article
• Language: English
• Published: 01.06.2021
• Subjects: phase‐space methods; quantum information; verification; Wigner function
• ISSN: 2511-9044; 2511-9044
• DOI: 10.1002/qute.202100016

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. As the development of quantum technologies advances, one is able to create ever‐increasing numbers of qubits. With this increase methods of state verification and certification become more difficult. Here, phase‐space methods are reviewed as a way to gain information from these quantum systems as well as translating already well‐known figures of merit into the phase‐space formulation.