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arXiv:1303.0688 · Physical Review Letters

Entanglement and Particle Identity: A Unifying Approach

A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz, Andrés Fernando Reyes-Lega

Published Feb 22, 2013• 115 citations
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Abstract

It has been known for some years that entanglement entropy obtained from partial trace does not provide the correct entanglement measure when applied to systems of identical particles. Several criteria have been proposed that have the drawback of being different according to whether one is dealing with fermions, bosons, or distinguishable particles. In this Letter, we give a precise and mathematically natural answer to this problem. Our approach is based on the use of the more general idea of the restriction of states to subalgebras. It leads to a novel approach to entanglement, which is suitable to be used in general quantum systems and especially in systems of identical particles. This settles some recent controversy regarding entanglement for identical particles. The prospects for applications of our criteria are wide ranging, from spin chains in condensed matter to entropy of black holes.

DOI
10.1103/physrevlett.110.080503

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It has been known for some years that entanglement entropy obtained from partial trace does not provide the correct entanglement measure when applied to systems of identical particles. Several criteria have been proposed that have the drawback of being different according to whether one is dealing with fermions, bosons, or distinguishable particles. In this Letter, we give a precise and mathematically natural answer to this problem. Our approach is based on the use of the more general idea of the restriction of states to subalgebras. It leads to a novel approach to entanglement, which is suitable to be used in general quantum systems and especially in systems of identical particles. This settles some recent controversy regarding entanglement for identical particles. The prospects for applications of our criteria are wide ranging, from spin chains in condensed matter to entropy of black holes.
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'.
Quantum entanglement of identical particles is essential in quantum information theory. Yet, its correct determination remains an open issue hindering the general understanding and exploitation of many-particle systems. Operator-based methods have been developed that attempt to overcome the issue. Here we introduce a state-based method which, as second quantization, does not label identical particles and presents conceptual and technical advances compared to the previous ones. It establishes the quantitative role played by arbitrary wave function overlaps, local measurements and particle nature (bosons or fermions) in assessing entanglement by notions commonly used in quantum information theory for distinguishable particles, like partial trace. Our approach furthermore shows that bringing identical particles into the same spatial location functions as an entangling gate, providing fundamental theoretical support to recent experimental observations with ultracold atoms. These results pave the way to set and interpret experiments for utilizing quantum correlations in realistic scenarios where overlap of particles can count, as in Bose-Einstein condensates, quantum dots and biological molecular aggregates.
The concept of entanglement in systems where the particles are indistinguishable has been the subject of much recent interest and controversy. In this paper we study the notion of entanglement of particles introduced by Wiseman and Vaccaro [Phys. Rev. Lett. 91, 097902 (2003)] in several specific physical systems, including some that occur in condensed-matter physics. The entanglement of particles is relevant when the identical particles are itinerant and so not distinguished by their position as in spin models. We show that entanglement of particles can behave differently than other approaches that have been used previously, such as entanglement of modes (occupation-number entanglement) and the entanglement in the two-spin reduced density matrix. We argue that the entanglement of particles is what could actually be measured in most experimental scenarios and thus its physical significance is clear. This suggests that entanglement of particles may be useful in connecting theoretical and experimental studies of entanglement in condensed-matter systems.
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
The study of the entanglement properties of systems of N fermions has attracted considerable interest during the last few years. Various separability criteria for pure states of N identical fermions have been recently discussed but, except for the case of two-fermions systems, these criteria are difficult to implement and are of limited value from the practical point of view. Here we advance simple necessary and sufficient separability criteria for pure states of N identical fermions. We found that to be identified as separable, a state has to comply with one single identity involving either the purity or the von Neumann entropy of the single-particle reduced density matrix. These criteria, based on the verification of only one identity, are drastically simpler than the criteria discussed in the recent literature. We also derive two inequalities verified, respectively, by the purity and the entropy of the single-particle, reduced density matrix, which lead to natural entanglement measures for N -fermion pure states. Our present considerations are related to some classical results from the Hartree-Fock theory, which are here discussed from a different point of view in order to clarify some important points concerning the separability of fermionic pure states.
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 analogue of the Schmidt decomposition and on the evaluation of the von Neumann entropy of the one-particle reduced statistical operators.