General criterion for the entanglement of two indistinguishable particles

Abstract

We relate the notion of entanglement for quantum systems composed of two identical constituents to the impossibility of attributing a complete set of properties to both particles. This implies definite constraints on the mathematical form of the state vector associated with the whole system. We then analyze separately the cases of fermion and boson systems, and we show how the consideration of both the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition of the global state vector and the von Neumann entropy of the one-particle reduced density operators can supply us with a consistent criterion for detecting entanglement. In particular, the consideration of the von Neumann entropy is particularly useful in deciding whether the correlations of the considered states are simply due to the indistinguishability of the particles involved or are a genuine manifestation of the entanglement. The treatment leads to a full clarification of the subtle aspects of entanglement of two identical constituents which have been a source of embarrassment and of serious misunderstandings in the recent literature.

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

“…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.…”

The von Neumann entanglement entropy for Wigner-crystal states in one dimensional N-particle systems

“…Since the asymptotic RDM possesses a degenerate spectrum, its Schmidt decomposition fails to be unique [41]. For the sake of completeness, we derive below another form of the Schmidt decomposition of the asymptotic RDM.…”

Quantum Entanglement of Two Harmonically Trapped Dipolar Particles

“…In terms of the occupancies, we have (8) vanishes for the non-entangled points discussed above, except for the bosonic states with Sn = 2, (S B = 1). As noted in the already cited [35], the vN entropy alone is insufficient to distinguish whether the bosonic state is entangled or not, since the following may happen: S B = 1 for the state with Sn different than 2. For example, such a situation was observed in the system of bosons interacting via the short-range contact potential confined in a split trap [31].…”

“…The normalization conditions for the occupancies of the bosonic (+) and fermionic (−) states give l λ (+) l = 1, and 2 l λ (−) l = 1, where the factor 2 comes from the fact that the occupancies of the fermionic state are doubly degenerate. As discussed in [35], the state of two identical bosons is non-entangled only in two cases, i.e., when its Schmidt number (Sn) is 1 or 2, which corresponds to…”

“…As to a fermion state, it is non-entangled if, and only if, its total wavefunction can be expressed as one single determinant [35], which in turn corresponds to λ (−) i = 0.5. We will measure the amount of the entanglement via the vN entropy,…”