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.
The Wehrl′s Entropy and Shannon′s entropy for the non-classical optical fields with negative P representation are studied in this paper. The results show that the Wehrl′s entropy of the states decreases along with the growth of the state parameter and the average photon numbers and tend to be constant. While the Shannon′s entropy increases along with the increase of state parameter and the average photon numbers. It is also found that the Wehrl′s entropy of the non-classical optical fields with negative P representation decreases while the total noise increases and the Shannon′s entropy decreases along with the decrease of observable fluctuation.
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 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.
Abstract The Wehrl entropy of a quantum state is the Shannon entropy of its coherent-state distribution function, and remains non-zero even for pure states. We investigate the relationship between this entropy and the many-particle quantum entanglement, for N spin-1/2 particles. Explicitly, we numerically calculate the Wehrl entropy of various N-particle ( 2 ⩽ N ⩽ 20 ) entangled pure states, with respect to the SU(2) ⊗ N coherent states. Our results show that for the large-N ( N ≳ 10 ) systems the Wehrl entropy of the highly chaotic entangled states (e.g. 2 − N / 2 ∑ s 1 , s 2 , … , s N =↑ , ↓ | s 1 , s 2 , … , s N ⟩ e − i ϕ s 1 , s 2 , … , s N , with ϕ s 1 , s 2 , … , s N being random angles) are substantially larger than that of the very regular entangled states (e.g. the Greenberger–Horne–Zeilinger state). Therefore, the Wehrl entropy can reflect the complexity of the quantum entanglement of many-body pure states, as proposed by Sugita (2003 J. Phys. A: Math. Gen. 36 9081). In particular, the Wehrl entropy per particle (WEPP) can be used as a quantitative description of this entanglement complexity. Unlike other quantities used to evaluate this complexity (e.g. the degree of entanglement between a subsystem and the other particles), the WEPP does not necessitate the division of the total system into two subsystems. We further demonstrate that many-body pure entangled states can be classified into three types, based on the behavior of the WEPP in the limit N → ∞ : states approaching that of a maximally mixed state