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arXiv:1605.08606

Wehrl entropies attached to Euclidean Landau levels

Zouhaı̈r Mouayn, Hamidou Kassogué, P. Kayupe Kikodio, Imade Fahd Eddine Fatani

Published May 27, 2016• 0 citations
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Abstract

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.

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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.
We are concerned with a class of generalized coherent states (GCS) attached to Euclidean Landau levels (or to [Formula: see text]-true-polyanalytic spaces), which can be obtained by displacing a Gaussian–Hermite function as an admissible (or window) function. Precisely, we evaluate the Wehrl entropy for a density operator representing a projector on a Fock state (pure states) and we give an upper bound for this entropy. We also establish an exact formula of this entropy for the heat operator (mixed states) associated with the harmonic oscillator. In this case, the behavior of the entropy with respect to the temperature parameter shows the dependence of its minimum to the Landau level or equivalently to the window function by means of which the GCS involved in the Wehrl entropy were constructed.
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.
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.
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.
In this work, the definition of the density operator on quantum states in Hilbert spaces and some of its aspects relevant in thermodynamics and information-theoretical entropy calculations are given. In this framework, a physical model describing an electron in a magnetic field is investigated. The so-called exotic Landau problem in noncommutative plane is also considered. Then, a model related to the fractional quantum Hall effect is revisited. Thanks to the completeness relations verified by the coherent states (CS) in these models, the thermodynamics is discussed by using the diagonal PPP-representation of thedensity operator. Specifically, the Q-Husimi distribution and the Wehrl entropy are determined.
The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.