Abstract Measures of delocalization in phase space are analyzed using Rényi entropies, especially two of which play an important role in characterizing extension and shape of distributions: the linear entropy related to the participation number and the Shannon-entropy. The difference of these two, termed as structural entropy, has been successfully applied in a large variety of physical situations and for various mathematical problems. A very similar quantity has coincidentally been used as a measure of complexity by some other authors. Hereby we show that various semiclassical phase space representations of quantum states can be well described by the structural entropy providing a transparent picture in relation to the thermodynamic description. Thermodynamic and quantum fluctuations are analytically treated for the special case of harmonic oscillators invoking the Einstein model of heat capacity. It is demonstrated that the thermal uncertainty relations are linked to the delocalization over the phase space. For respective limits of zero temperature implying quantum behavior or infinite temperature implying classical behavior we also show which quantities remain useful. As a byproduct the thermal extension of the phase space distribution can be calculated that is directly related to a decoherence parameter introduced by Zurek in a different context.
The Wehrl entropy or the "classical" entropy associated with a quantum system is the entropy of the probability distribution in phase space, corresponding to the Husimi Q-function in terms of coherent states.In the present paper, we shall focus our attention on the examination of the Wehrl entropy for both the pure and the mixed (thermal) states of the pseudoharmonic oscillator (PHO).The choice of the PHO is interesting because this oscillator is an intermediate between the ideal one-dimensional harmonic oscillator (HO-1D) and the more practical anharmonicone.
The quantum-mechanical joint (quasi-) density in phase space (the Wigner distribution) for thermodynamic equilibrium is studied in the semiclassical limit h(cross) to 0. The approximation is of an asymptotic (WKB) type and nonperturbative in character. The equilibrium phase-space distribution can be expressed in terms of classical paths satisfying certain well defined conditions. These paths are complex-valued, i.e. classically forbidden. The resulting semiclassical expression for the equilibrium Wigner phase-space density agrees with the exact quantum result for the analytical solvable case of harmonic potentials for all temperatures.
For a symmetric
N-quDit system described by a density matrix
ρ, we construct a one-parameter
s family
Fρ(s) of quasi-probability distributions through generalized Fano multipole operators and Stratonovich-Weyl kernels. The corresponding phase space is the complex projective
CPD−1=U(D)/U(D−1)×U(1), related to fully symmetric irreducible representations of the unitary group
U(D). For the particular cases
D=2 (qubits) and
D=3 (qutrits), we analyze the phase-space structure of Schrödinger
U(D)-spin cat (parity adapted coherent) states and we provide plots of the corresponding Wigner
Fρ(0) function. We examine the connection between non-classical behavior and the negativity of the Wigner function. We also compute the generalized heat kernel relating two quasi-probability distributions
Fρ(s) and
Fρ(s′), with
t=(s′−s)/2 playing the role of ``time'', together with their twisted Moyal product in terms of a trikernel. In the thermodynamic limit
N→∞, we recover the usual Gaussian smoothing for
s′>s. A diagramatic interpretation of the phase-space construction in terms of Young tableaux is also provided.
Wigner representation In classical optics (Born and Wolf, 1999), the state of the electromagnetic oscillator is perfectly described by the statistics of the classical amplitude α . The amplitude may be completely fixed (then the field is coherent), or α may fluctuate (then the field is partially coherent or incoherent). In classical optics as well as in classical mechanics, we can characterize the statistics of the complex amplitude?or, equivalently, the statistics of the components position q and momentum p introducing a phase space distribution W ( q , p ). (As explained in Section 3.1, the real and the imaginary part of the complex amplitude α can be regarded as the position and the momentum of the electromagnetic oscillator.) The distribution W ( q , p ) quantifies the probability of finding a particular pair of q and p values in their simultaneous measurement. Knowing the phase space probability distribution, all statistical quantities of the electromagnetic oscillator can be predicted by calculation. In this sense the phase space distribution describes the state in classical physics. All this is much more subtle in quantum mechanics. First of all, Heisenberg's uncertainty principle prevents us from observing position momentum simultaneously and precisely. So it seems there is no point in thinking about quantum phase space. But wait! In quantum mechanics we cannot directly observe quantum states either. Nevertheless, we are legitimately entitled to use the concept of states as if they were existing entities (whatever they are). We use their properties to predict the statistics of observations.
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.