Theory of Quantum Phase Space: Foundations and Applications

• URL: http://arxiv.org/abs/2606.02675v1
• Date: Mon, 01 Jun 2026 12:43:13 GMT
• Title: Theory of Quantum Phase Space: Foundations and Applications
• Abstract summary: This article provides a concise review of quantum phase space theory.<n>It begins with its foundational principles and the properties of standard quantum quasi-probability distributions.<n>We discuss the intrinsic limitations of these distributions, such as the appearance of negative values and phase-space blurring.
• Score: 2.5213900963983096
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This article provides a concise review of quantum phase space theory, beginning with its foundational principles and the properties of standard quantum quasi-probability distributions, specifically the Wigner, Husimi Q, and Glauber--Sudarshan P functions. We discuss the intrinsic limitations of these distributions, such as the appearance of negative values and phase-space blurring. A significant portion of this review highlights recent theoretical developments, particularly the quantum Wannier basis. This approach establishes a unitary mapping between the Hilbert space and a discretized phase space, yielding a genuine probability distribution in phase space and thereby providing a basis-dependent entropy for pure quantum states. Furthermore, we examine Bourgain's nonperiodic basis as a theoretical framework to circumvent the constraints imposed by the Balian--Low theorem. These developments provide practical tools for numerical studies based on the quantum Wannier basis, as well as conceptual benchmarks for understanding the localization limits of orthonormal phase-space representations.

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