Abstract The Husimi function (Q-function) of a quantum state is the distribution function of the density operator in the coherent state representation. It is widely used in theoretical research, such as in quantum optics. The Wehrl entropy is the Shannon entropy of the Husimi function, and is non-zero even for pure states. This entropy has been extensively studied in mathematical physics. Recent research also suggests a significant connection between the Wehrl entropy and many-body quantum entanglement in spin systems. We investigate the statistical interpretation of the Husimi function and the Wehrl entropy, taking the system of N spin-1/2 particles as an example. Due to the completeness of coherent states, the Husimi function and Wehrl entropy can be explained via the positive operator-valued measurement (POVM) theory, although the coherent states are not a set of orthonormal basis. Here, with the help of the Bayes’ theorem, we provide an alternative probabilistic interpretation for the Husimi function and the Wehrl entropy. This interpretation is based on direct measurements of the system, and thus does not require the introduction of an ancillary system as in the POVM theory. Moreover, under this interpretation the classical correspondences of the Husimi function and the Wehrl entropy are just phase-space probability distribution function of N classical tops, and its associated entropy, respectively. Therefore, this explanation contributes to a better understanding of the relationship between the Husimi function, Wehrl entropy, and classical-quantum correspondence. The generalization of this statistical interpretation to continuous-variable systems is also discussed.
A density of the information-theoretic Wehrl entropy in terms of the Hu- simi
Q -function is defined.The entropic density is then applied to describe the quantum properties of some optical fields including Fock states, cohe- rent and squeezed states, superposition of chaotic and coherent states, and Schr\"odinger cat and cat-like states.The entropic density is compared with both the conventional Wehrl entropy and the Husimi phase distribution.It is shown that the entropic density is a good measure of the phase-space uncertainty (noise), and the phase locking or phase bifurcation effects.The advantages of the entropic description of the superposition principle are presented.It is also demonstrated that the entropic density properly de- scribes phase randomization processes, thus can be used as a measure of decoherence.I. INTRODUCTION Quantum entropy is one of the most fundamental concepts of quantum physics [1], parti- cularly useful in quantum information [2].Quantum entropy has been applied, e.g., as a measure of quantum entanglement, qualltum decoherence, photocount statistics, quantum 数理解析研究所講究録 1142 巻 2000 年 1-22 1 optical correlations, purity of states, quantum noise or accessible information in quantum measurement (capacity of quantum channel).Quantum entropy, as a natural generalization of classical Boltzmann entropy, was propo- sed by von Neumann [3].A classical-like entropy associated with quantum fields was intro- duced by Wehrl [4] in terms of the Glauber coherent states and the Husimi
Q -function [5].A rigorous proof that the von Neumann entropy tends to the Wehrl entropy in the clas- sical limit $\hslasharrow 0$ was given by Beretta [6].The quantum von Neumann entropy can be expanded in power series of classical entropies.As was shown explicitly by Pe\v{r}inov\'a et al. [7], the first term of this expansion is the Wehrl entropy.Bu\v{z}ek et al. [8] related the von Neumann entropy with the sampling entropies based on operational approach to a phase-space measurement.They also identified the Wehrl entropy as a particular example of the sampling entropy, when the quantum ruler is represented by coherent states.Other aspects of the relation between classical and quantum-mechanical entropies have also been extensively studied (see, e.g., excellent surveys by Wehrl [9], and Ohya and Petz [1] with references included therein).The Wehrl entropy has been successfully applied in a description of different properties of quantum optical fields.In particular, it has been shown explicitly that the Wehrl entropy is a useful measure of phase-space uncertainty (quantum noise, phase-space localization, wave-packet spreading) [8,10-14], quantum interference [10], decoherence [11,15,16], ioni- zation [14], squeezing [16-18], Schr\"odinger cat formation
[13,19] or, in general, splitting of
Q -function [15].The von Neumann entropy becomes zero for all pure states, thus cannot be used in discriminating them.Paradoxically, the classical Wehrl entropy is more sensitive in di- stinguishing states than the quantum von Neumann entropy, since it is dependent on the choice of pure states.However, there are properties of quantum fields (including phase decoherence or unique description of superposition states), which are not enough precisely described by the conventional Wehrl entropy.Therefore, we propose a new entropic me-
We are concerned with an information theoretic measure of uncertainty for quantum systems. Precisely, the Wehrl entropy of the phase space probability (z,m I rho Iz,m) which is known as Husimi function, where rho is a density operator and Iz,m) are coherent states attached to an Euclidean mth Landau level. We obtain the Husimi function Qmbeta of the thermal density operator rhobeta of the harmonic oscillator, which leads, by duality, to the Laguerre probability distribution of the mixed light. We discuss some basic properties of Qmbeta such as its characteristic function and its limiting logarithmic moment generating function from which we derive the rate function of the sequence of probability distributions Qmbeta, m=0,1,... For m=1,2,..., we establish an exact expression for the Wehrl entropy of the density operator rhobeta and we discuss the behavior of this entropy with respect to the temperature parameter T = 1/beta.
The Wehrl entropy is the entropy of the probability distribution in phase space corresponding to the Husimi function in terms of coherent states. We explain the significance of the Wehrl entropy in quantum information theory, and present the theory behind the Lieb conjecture, which states that, in finite dimensions, the minimum Wehrl entropy occurs for Bloch coherent states. This was proven by Lieb and Solovej in 2012. We present the theory behind coherent states, with a particular emphasis on Bloch coherent states, and give a geometrical representation of quantum states as points on a sphere. Using this representation, we identify spherical arrangements of 2–9 points that maximize the Wehrl entropy locally. We conjecture that these maxima are in fact global. Furthermore, we investigate how the maximally entangled symmetric states are related to the states corresponding to maximal Wehrl entropy.
Using the quantum phase space technique and the information-theory like the Wehrl entropy, the Husimi function and the Wehrl entropy of the quantum pure states and the corresponding mixed states in thermo field dynamics are studied. It is found that the Husimi function and the Wehrl entropy of the thermal coherent state agree with that of the corresponding mixed states. And the Wehrl entropy of thermal coherent state is not related with the displacement factor. Therefore for a quantum system, the quantum fluctuations of the observable quantities and corresponding uncertainty relation are also not related with the displacement factor in the thermal coherent state.
In this manuscript we study the Wehrl entropy of entangled oscillators. This semiclassical entropy associated with the phase-space description of quantum mechanics can be used for formulating uncertainty relations and for a quantification of entanglement. We focus on a system of two coupled oscillators described within its Segal-Bargmann space. This Hilbert space of holomorphic functions integrable with respect to a given Gaussian-like measure is particularly convenient to deal with harmonic oscillators. Indeed, the Stone-von Neumann theorem allows us to work in this space in a full correspondence with the ladder operators formalism. In addition, the Husimi pseudoprobability distribution is directly computed within the Segal-Bargmann formalism. Once we obtain the Husimi function, we analyze the Wehrl entropy and mutual information.
Abstract. Based on the recently introduced averaging procedure in phase space, a new type of entropy is defined on the von Neumann lattice. This quantity can be interpreted as a measure of uncertainty associated with simultaneous measurement of the position and momentum observables in the discrete subset of the phase space. Evaluating for a class of the coherent states, it is shown that this entropy takes a stationary value for the ground states modulo of a unit cell of the lattice in such a class. This value for the ground state depends on the ratio of the position lattice spacing and the momentum lattice spacing. It is found that its minimum is realized for the perfect square lattice, i.e., absence of squeezing. Numerical evaluation of this minimum gives 1. 386.... 1 Ever since the introduction of the Wigner distribution function [1], theory of phase-space representations [2-4] has been playing vital roles in quantum statistical mechanics, quantum chaos and signal processing. The Husimi function, which is the coherent-state representation of a wave function (or a density matrix) and is often called the Q-function, is known to be special in the sense that it is nonnegative and has a good analytical behavior in contrast to the Wigner function and the Sudarshan-Glauber P-function. Since the coherent state of the harmonic oscillator is the most classical pure state with the minimum Heisenberg uncertainty product, the Husimi Q-function is supposed to describe a semiclassical aspect of the system under consideration. However, a measure of classical nature of a normalized state, ψ, was lacking until Wehrl [5] introduced a classical entropy, which is defined by 2 W [ ψ] = −∫ ∫ d α α ψ π 2 2 α ψ ln. (1) π α in this equation is the coherent state given by α = ˆD ( α) 0 ( α ∈C), where