Entanglement and Particle Identity: A Unifying Approach

Abstract

A novel approach to entanglement, based on the Gelfand-Naimark-Segal (GNS) construction, is introduced. It considers states as well as algebras of observables on an equal footing. The conventional approach to the emergence of mixed from pure ones based on taking partial traces is replaced by the more general notion of the restriction of a state to a subalgebra. For bipartite systems of nonidentical particles, this approach reproduces the standard results. But it also very naturally overcomes the limitations of the usual treatment of systems of identical particles. This GNS approach seems very general and can be applied for example to systems obeying para-and braid-statistics including anyons.

Quantumness of correlations in indistinguishable particles

“…Such a state is considered both entangled by some authors [12,17,25], as non entangled for others [14,15,18]. Note that such a state can actually be described by a single Slater permanent…”

“…In systems of indistinguishable particles the notion of "local measurement" will be implemented through the algebra of single-particle observables (see for example Ref. [14] for a detailed discussion), and based on this identification we shall set up an "activation protocol" for indistinguishable particles. The importance to study the correlations, particularly the entanglement, in terms of subalgebras of observables has been emphasized in [14,22,[34][35][36][37], proving to be a useful approach for such analysis.…”

“…In fact, there are a multitude of distinct approaches and an ongoing debate around the entanglement in these systems [12][13][14][15][16][17][18][19][20][21][22]. Nev-ertheless, despite the variety, the approaches consist essentially in the analysis of correlations under two different aspects: the correlations genuinely arising from the entanglement between the particles ("entanglement of particles") [14][15][16][17][18], and the correlations arising from the entanglement between the modes of the system ("entanglement of modes") [19][20][21][22]. These two notions of entanglement are complementary, and the use of one or the other depends on the particular situation under scrutiny.…”

Entanglement of indistinguishable particles as a probe for quantum phase transitions in the extended Hubbard model

“…The only non-classical correlation present in such states is the exchange, due to the antissymetrization, which does not constitute entanglement. For example, in [17] the analysis follows by using a very elegant mathematical formalism, called GNS (Gelfand-Naimark-Segal) construction, for the case of two fermions, each one with Hilbert space dimension 3 or 4, and two bosons with dimension 3; in [23,24] the authors propose a "Generalized Entanglement (GE)" measure, obtaining a simple formula for the "partial trace", and the set of fermionic unentangled states for an arbitrary number of particles; or also in [6], where a general notion of quantum correlation beyond entanglement (the quantumness of correlations) is investigated by means of an "activation protocol", which yields the same set of states with no quantumness as the above unentangled one.…”

“…In fact, there are a multitude of distinct approaches and an ongoing debate around the entanglement in these systems [6,[17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Nevertheless, despite the variety, the approaches consist essentially in the analysis of correlations under two different aspects: the correlations genuinely arising from the entanglement between the particles (entanglement of particles) [6,[17][18][19][20][21][22][23][24], and the correlations arising from the entanglement between the modes of the system (entanglement of modes) [25][26][27][28]. These two notions of entanglement are complementary, and the use of one or the other depends on the particular situation under scrutiny.…”