The Wehrl information entropy and its phase density, the so-called Wehrl phase distribution, are applied to describe Schr\\"odinger cat and cat-like (kitten) states. The advantages of the Wehrl phase distribution over the Wehrl entropy in a description of the superposition principle are presented. The entropic measures are compared with a conventional phase distribution from the Husimi Q-function. Compact-form formulae for the entropic measures are found for superpositions of well-separated states. Examples of Schr\\"odinger cats (including even, odd and Yurke-Stoler coherent states), as well as the cat-like states generated in Kerr medium are analyzed in detail. It is shown that, in contrast to the Wehrl entropy, the Wehrl phase distribution properly distinguishes between different superpositions of unequally-weighted states in respect to their number and phase-space configuration.
The Wehrl phase distribution is defined as a phase density of the Wehrl classical information entropy. The new measure is applied to describe the quantum phase properties of some optical fields including Fock states, coherent and squeezed states, and superposition of chaotic and coherent fields. The Wehrl phase distribution is compared with both the conventional Wehrl entropy and Husimi phase distribution (the marginal Husimi Q -function). It is shown that the Wehrl phase distribution is a good measure of the phase-space uncertainty (noise), phase locking and phase bifurcation effects. It is also demonstrated that the Wehrl phase distribution properly describes phase randomization processes, and thus can be used in a description of the quantum optical phase.
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-
The Wehrl’s entropy of the Schrödinger-cat states has been calculated numerically in this paper. In particular, we show the dependence of the Wehrl’s entropy on the two parameters of the Schrödinger-cat states, which act as, respectively, the photon number parameter and the phase difference . The Wehrl’s entropy has been found to be in the range , and its changes with respect to are not homogenized. Furthermore, we obtain both the dependence of the Wehrl’s entropy on the average photon number and the relation between the Wehrl’s entropy and the statistical entropy.