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arXiv:quant-ph/0206021 · Fortschritte der Physik

Entanglement and properties

G. C. Ghirardi, Luca Marinatto

Published Apr 24, 2003• 57 citations
arXivHTMLPDF

Abstract

Abstract Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents. In particular we will focus our attention to composite quantum systems composed of identical constituents, with the purpose of dealing with subtle issues, which have never been adequately discussed in the literature, originating from the true indistinguishability of the subsystems involved.

DOI
10.1002/prop.200310051

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•Jan 1, 2005•24 citations
Abstract Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents. In particular we will focus our attention to composite quantum systems composed of identical constituents, with the purpose of dealing with subtle issues, which have never been adequately discussed in the literature, originating from the true indistinguishability of the subsystems involved.
We present here an overview of our work concerning entanglement properties of composite quantum systems. The characterization of entanglement, i.e. the possibility to assert if a given quantum state is entangled with others and how much entangled it is, remains one of the most fundamental open questions in quantum information theory. We discuss our recent results related to the problem of separability and distillability for distinguishable particles, employing the tool of witness operators. Finally, we also state our results concerning quantum correlations for indistinguishable particles.
The study of entanglement in systems composed of identical particles raises interesting challenges with far-reaching implications in both, our fundamental understanding of the physics of composite quantum systems, and our capability of exploiting quantum indistinguishability as a resource in quantum information theory. Impressive theoretical and experimental advances have been made in the last decades that bring us closer to a deeper comprehension and to a better control of entanglement. Yet, when it involves composites of indistinguishable quantum systems, the very meaning of entanglement, and hence its characterization, still finds controversy and lacks a widely accepted definition. The aim of the present paper is to introduce, within an accessible and self-contained exposition, the basic ideas behind one of the approaches advanced towards the construction of a coherent definition of entanglement in systems of indistinguishable particles, with focus on fermionic systems. We also inquire whether the corresponding tools developed for studying entanglement in identical-fermion systems can be exploited when analysing correlations in distinguishable-party systems, in which the complete information of the individual parts is not available. Further, we open the discussion on the broader problem of constructing a suitable framework that accommodates entanglement in the presence of generalized statistics. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.
Introduction The first realization that the validity of the quantum superposition principle in the Hilbert space describing a composite quantum system may give rise to fundamentally new correlations between the constituent subsystems came in the landmark 1935 paper by Einstein, Podolsky, and Rosen (EPR), where it was shown how the measurement statistics of observables in certain quantum states could not be reproduced by assigning definite wavefunctions to individual subsystems. It was in response to the EPR paper that Schrödinger, in the same year, coined the term entanglement ( Verschränkung ) to acknowledge the failure of classical intuition in describing the relationship between the “parts” and the “whole” in the quantum world: Whenever one has a complete expectation catalog – a maximum total knowledge – a ψ function – for two completely separated bodies, …then one obviously has it also for the two bodies together. But the converse is not true. The best possible knowledge of a total system does not necessarily include total knowledge of all its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all. While Bell's strengthening of the original EPR-paradox setting and the subsequent experimental verification of Bell inequalities irreversibly changed the perception of entanglement from a property of counterintuitive “spookiness” to (beyond reasonable doubt) an experimental reality, the concept and implications of entanglement continue to be associated with a host of physical, mathematical, and philosophical challenges. In particular, investigation of entanglement in both its qualitative and quantitative aspects has intensified under the impetus of quantum information science (QIS).
Entanglement is nowadays considered as a key quantity for the understanding of correlations, transport properties, and phase transitions in composite quantum systems, and thus receives interest beyond the engineered applications in the focus of quantum information science. We review recent experimental and theoretical progress in the study of quantum correlations under that wider perspective, with an emphasis on rigorous definitions of the entanglement of identical particles, and on entanglement studies in atoms and molecules.
We review the main aspects of entanglement and we prove some general theorems concerning entangled states. In order to satisfy the symmetry requirements for systems containing identical constituents one is unavoidably led to consider entangled states. One could then naively conclude that the identity gives rise, by itself, to peculiar and problematic aspects of physical systems. This is by no means the case: the symmetrization or antisymmetrization of factorized states is an absolutely benign procedure and does not pose any specific conceptual problem apart from the one of the impossibility of attributing their elemetary constituents to one or another of two far-away systems. We briefly reconsider the same problem within the context of recently proposed models of dynamical reduction based on nonlinear and stochastic modifications of the standard evolution.
We review two general criteria for deciding whether a pure bipartite quantum state describing a system of two identical particles is entangled or not. The first one considers the possibility of attributing a complete set of objective properties to each particle belonging to the composed system, while the second is based both on the consideration of the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition and on the evaluation of the von Neumann entropy of the one-particle reduced statistical operators. 1