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Revista Colombiana de Física

An Accidental Relationship Between a Relative Quantum Phase Distribution and Concurrence

Rubén Darío Guerrero, Rafael Ramón Rey González, Karen Fonseca-Romero

Published Jun 9, 2012• 0 citations
OpenAlex

Abstract

We consider some particular states of two coupled continuous systems, for which it is meaningful to use the concurrence as an entanglement quantifier, and show there is an accidental connection between the probability distribution of the relative phase of their Husimi function and their concurrence. We moreover show a quantum open system where these states can be produced. Hence, the measurement of the Husimi function allows for quantification of entanglement for these particular states.

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•Jan 1, 2011•0 citations
We consider some particular states of two coupled continuous systems, for which it is meaningful to use the concurrence as an entanglement quantifier, and show there is an accidental connection between the probability distribution of the relative phase of their Husimi function and their concurrence. We moreover show a quantum open system where these states can be produced. Hence, the measurement of the Husimi function allows for quantification of entanglement for these particular states.
For systems of two and three spins 1∕21∕21∕2 it is known that the second moment of the Husimi function can be related to entanglement properties of the corresponding states. Here, we generalize this relation to an arbitrary number of spins in a pure state. It is shown that the second moment of the Husimi function can be expressed in terms of the lengths of the concurrence vectors for all possible partitions of the NNN-spin system in two subsystems. This relation implies that the phase space distribution of an entangled state is less localized than that of a nonentangled state. As an example, the second moment of the Husimi function is analyzed for an Ising chain subject to a magnetic field perpendicular to the chain axis.
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.
I investigate quantum decoherence in a one-body density matrix of a composite particle consisting of two correlated particles. Because of a two-body correlation in the composite particle, quantum decoherence occurs in the one-body density matrix that has been reduced from the two-body density matrix. As the delocalization of the distribution of the composite particle grows, the entanglement entropy increases, and the system can be well-described by a semi-classical approximation, wherein the center position of the composite particle can be regarded as a classical coordinate. I connect the quantum decoherence in the one-body density matrix of a composite particle to the coarse graining in a phase space distribution function of a single particle and associate it with the Husimi function.
Entanglement is a nonclassical correlation of compound quantum systems. We derive Separability-Eigenvalue-Equations. The solution of these equations delivers all observables for the determination of entanglement. On the other hand, it delivers entanglement quasi-probabilities. We conclude from negativities of such distributions, that no classical statistics can describe the state under study. For the quantification of entanglement we compare different entanglement measures. We also consider different kinds of entanglement and the relation to nonclassicality.