Wehrl Entropy Research Articles

The effect of atomic motion and two-quanta JCM on the information entropy

We study the interaction between a moving two-level atom and a single-mode field. The coupled atom–cavity system with atomic center-of-mass motion included is modeled by considering the dependence of the atomic motion along z -axis. At exact resonance between the internal atomic transition and the cavity eigenfrequency, an exact solution of the system is obtained and periodically modulated Rabi oscillations and regular translational motion are observed. We focused on the dynamics of both field Wehrl entropy and Wehrl phase distribution. The influence of the atomic motion on the evolution of von Neumann entropy and Wehrl entropy is examined. The results show that the atomic motion and the field-mode structure play important roles in the evolution of the von Neumann entropy, Wehrl entropy and Wehrl PD.

Entanglement in the Bimodal Jaynes–Cummings Model with the Two-Mode Squeezed Vacuum State

In this paper, we study the interaction between the two-level atom and a bimodal cavity field, namely, two-mode Jaynes–Cummings model when the atom and the modes are initially in the atomic superposition state and two-mode squeezed vacuum state, respectively. For this system we investigate the atomic inversion, linear entropy and atomic Wehrl entropy. We show that there is a connection between all these quantities. Also we prove that the atomic Wehrl entropy exhibits behaviors similar to those of the linear entropy and the von Neumann entropy. Moreover, we show that the bipartite exhibits periodical disentanglement and derive the explicit forms of the states of the atom and the modes at these values of the interaction times.

Fisher information, delocalization and the semiclassical description of molecular rotation

We discuss phase-space delocalization for the rigid rotor within a semiclassical context by recourse to the Husimi distributions of both the linear and the 3D-anisotropic instances. The Husimi function can be viewed in a variety of ways. In particular, we obtain it as the expectation value of the density operator in a suitable basis of coherent states. Our treatment is based upon the concomitant Fisher information measures. The pertinent Wehrl entropy is also investigated in the linear case.

Entropy of Semiclassical Measures of the Walsh-Quantized Baker’s Map

We study the baker's map and its Walsh quantization, as a toy model of a quantized chaotic system. We focus on localization properties of eigenstates, in the semiclassical regime. Simple counterexamples show that quantum unique ergodicity fails for this model. We obtain, however, lower bounds on the entropies associated with semiclassical measures, as well as on the Wehrl entropies of eigenstates. The central tool of the proofs is an "entropic uncertainty principle".

Fisher information contains all HO-quantum-statistics already at the semiclassical level

We study here the difference between quantum statistical treatments and semiclassical ones, using as the main research tool a semiclassical, shift-invariant Fisher information measure built up with Husimi distributions. Its semiclassical character notwithstanding, this measure also contains abundant information of a purely quantal nature. Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies and to discover thermodynamic-like relations that involve the degree of delocalization. Fisher-related thermal uncertainty relations are developed and the degree of purity of canonical distributions, regarded as mixed states, is connected to this Fisher measure as well.

Statistical mechanical aspects of a spin-12 entropy system

Abstract In this paper we consider some statistical properties of the entropy due to the time development of a two-level system. Analytic expression for the density matrix is obtained to investigate the influence of the mean photon number on the atomic inversion, field and atomic Wehrl entropies. The results show that the general features of both entropies are similar.

Fisher information, Wehrl entropy, and Landau diamagnetism

Using information theoretic quantities like the Wehrl entropy and Fisher's information measure we study the thermodynamics of the problem leading to Landau's diamagnetism, namely, a free spinless electron in a uniform magnetic field. We reveal a new Fisher-uncertainty relation, derived via the Cramer-Rao inequality, that involves phase space localization and energy fluctuations.

Minimum Rényi and Wehrl entropies at the output of bosonic channels

The minimum Renyi and Wehrl output entropies are found for bosonic channels in which the signal photons are either randomly displaced by a Gaussian distribution (classical-noise channel), or in which they are coupled to a thermal environment through lossy propagation (thermal-noise channel). It is shown that the Renyi output entropies of integer orders z>1 and the output Wehrl entropy are minimized when the channel input is a coherent state.

A Lower Bound for the Wehrl Entropy of Quantum Spin with Sharp High-Spin Asymptotics

A lower bound for the Wehrl entropy of a single quantum spin is derived. The high-spin asymptotics of this bound coincides with Lieb's conjecture up to, but not including, terms of first and higher order in the inverse spin quantum number. The result presented here may be seen as complementary to the verification of the conjecture in cases of lowest spin by Schupp [Commun. Math. Phys. 207 (1999), 481]. The present result for the Wehrl-entropy is obtained from interpolating a sharp norm bound that also implies a sharp lower bound for the so-called R\'enyi-Wehrl entropy with certain indices that are evenly spaced by half of the inverse spin quantum number.

New features of the atomic Wehrl entropy and its density in multi-quanta two-level system

This contribution describes new features of the atomic Wehrl entropy in a multi-quanta two-level system in the presence of the Kerr medium. A definition of the atomic Wehrl entropy is presented for this system, based on the atomic Q-function as information measurement about the atomic phase space. The influence of the nonlinear interaction of the Kerr medium, Stark shift, one and two-photon processes and the detuning parameters on the properties of the atomic Wehrl entropy is examined. The atomic Wehrl entropy gives equivalent results to the von Neumann entropy.

Heisenberg-Fisher thermal uncertainty measure.

We establish a connection among (i) the so-called Wehrl entropy, (ii) Fisher's information measure I(beta), and (iii) the canonical ensemble entropy for the one-dimensional quantum harmonic oscillator (HO). We show that the contribution of the excited HO spectrum to the mean thermal energy is given by I(beta), while the pertinent canonical partition function is essentially given by another Fisher measure: the so-called shift invariant one. Our findings should be of interest in view of the fact that it has been shown that the Legendre transform structure of thermodynamics can be replicated without any change if one replaces the Boltzmann-Gibbs-Shannon entropy by Fisher's information measure [Phys. Rev. E 60, 48 (1999)]]. Fisher-related uncertainty relations are also advanced, together with a Fisher version of thermodynamics' third law.

Escort Husimi distributions, Fisher information and nonextensivity

We evaluate generalized information measures constructed with Husimi distributions and connect them with the Wehrl entropy, on the one hand, and with thermal uncertainty relations, on the other one. The concept of escort distribution plays a central role in such a study. A new interpretation concerning the meaning of the nonextensivity index q is thereby provided. A physical lower bound for q is also established, together with a “state equation” for q that transforms the escort-Cramer–Rao bound into a thermal uncertainty relation.

Escort Husimi distributions, Fisher information and nonextensivity

We evaluate generalized information measures constructed with Husimi distributions and connect them with the Wehrl entropy, on the one hand, and with thermal uncertainty relations, on the other one. The concept of escort distribution plays a central role in such a study. A new interpretation concerning the meaning of the nonextensivity index q is thereby provided. A physical lower bound for q is also established, together with a “state equation” for q that transforms the escort-Cramer–Rao bound into a thermal uncertainty relation.

Wehrl entropy, Lieb conjecture, and entanglement monotones

We propose to quantify the entanglement of pure states of $N\ifmmode\times\else\texttimes\fi{}N$ bipartite quantum systems by defining its Husimi distribution with respect to $\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(N)$ coherent states. The Wehrl entropy is minimal if and only if the analyzed pure state is separable. The excess of the Wehrl entropy is shown to be equal to the subentropy of the mixed state obtained by partial trace of the bipartite pure state. This quantity, as well as the generalized (R\'enyi) subentropies, are proved to be Schur concave, so they are entanglement monotones and may be used as alternative measures of entanglement.

Proof of the generalized Lieb-Wehrl conjecture for integer indices larger than one

Gnutzmann and Zyczkowski have proposed the Renyi-Wehrl entropy as a generalization of the Wehrl entropy, and conjectured that its minimum is obtained for coherent states. We prove this conjecture for the Renyi index q=2,3,... in the cases of compact semisimple Lie groups. A general formula for the minimum value is given.

Rényi-Wehrl entropies as measures of localization in phase space

We generalize the concept of the Wehrl entropy of quantum states which gives a basis-independent measure of their localization in phase space. We discuss the minimal values and the typical values of these R{enyi-Wehrl entropies for pure states for spin systems. According to Lieb's conjecture the minimal values are provided by the spin coherent states. Though Lieb's conjecture remains unproven, we give new proofs of partial results that may be generalized for other systems. We also investigate random pure states and calculate the mean Renyi-Wehrl entropies averaged over the natural measure in the space of pure quantum states.

Wehrl's entropy and a measure of intermode correlations in phase space

The properties of the entangled pure states in phase space are analysed using the classical entropy introduced by Wehrl. The general entropy inequalities for non-entangled pure states are derived, which are violated by any entangled state. To measure the strength of intermode correlations in phase space the parameters related to these inequalities are proposed. As an example we study the correlations between amplitudes and phases for two-mode Fock states. We find that the amplitudes as well as phases of different modes are correlated. It is also shown that the degree of the intermode correlation strongly depends on the photon number difference in two-mode Fock states.

Wehrl information entropy and phase distributions of Schrödinger cat and cat-like states

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.

Localization of eigenstates and mean Wehrl entropy

Dynamics of a periodically time-dependent quantum system is reflected in the features of the eigenstates of the Floquet operator. Of the special importance, are their localization properties quantitatively characterized by the eigenvector entropy, the inverse participation ratio or the eigenvector statistics. Since these quantities depend on the choice of the eigenbasis, we suggest to use the overcomplete basis of coherent states, uniquely determined by the classical phase space. In this way, we define the mean Wehrl entropy of eigenvectors of the Floquet operator and demonstrate that this quantity is useful to describe the quantum chaotic systems.

Classical information entropy and phase distributions of optical fields

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.