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Annals of West University of Timisoara - Physics

On The Information-Theoretical Entropy for Some Quantum Oscillators

Dušan Popov, Nicolina Pop, Simona Şimon

Published Dec 1, 2013• 0 citations
Publisher

Abstract

Abstract The information-theoretical entropy, also called the “classical” entropy, was introduced by Wehrl in terms of the Glauber coherent states (CSs) | z > , i.e. the CSs corresponding to the one-dimensional harmonic oscillator (HO-1D). In the present paper, we have focused our attention on the examination of the information-theoretical entropy, i.e. the Wehrl entropy, for both the pure and the mixed (thermal) states of some quantum oscillators.

DOI
10.1515/awutp-2015-0108

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Abstract The information-theoretical entropy, also called the “classical” entropy, was introduced by Wehrl in terms of the Glauber coherent states (CSs) | z > , i.e. the CSs corresponding to the one-dimensional harmonic oscillator (HO-1D). In the present paper, we have focused our attention on the examination of the information-theoretical entropy, i.e. the Wehrl entropy, for both the pure and the mixed (thermal) states of some quantum oscillators.
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.
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.
The Coherent states (CSs) show the correspondence between quantum oscillators and classical oscillators. It is very useful for understanding what goes on in oscillating systems, including harmonic oscillators as well as many kinds of waves. The properties of CSs for harmonic oscillator-1D were generalized to the anharmonic oscillators potential. Therefore, the Morse oscillator has aroused interest also in the construction of CSs. The properties of different nanostructures differ from those of the corresponding bulk structures. This fact leads to many applications in condensed matter, optoelectronic devices, sensors, electronic and light emitting components. The CSs formalism may play an important role in the quantum information theory, and the use of this representation is not only theoretical, but also some practical relevance by its applications. The fundamental unity of quantum information can be express through the CSs of the one-dimensional quantum oscillator. The transmission channel randomly displaces an input pure CS according to the Gaussian distribution which results in a thermal state.
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.
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