Theory of Quantum Phase Space: Foundations and Applications

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.