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arXiv:2401.06917 · Physical Review A

Bipartite representations and many-body entanglement of pure states of N indistinguishable particles

J. A. Cianciulli, R. Rossignoli, M. Di Tullio, N. Gigena, Federico Petrovich

Published Sep 12, 2024• 3 citations
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Abstract

We analyze a general bipartite-like representation of arbitrary pure states of N indistinguishable partcles, valid for both bosons and fermions, based on $M$- and $(N\ensuremath{-}M)$-particle states. It leads to exact $(M,N\ensuremath{-}M)$ Schmidt-like expansions of the state for any $M<N$ and is directly related to the isospectral reduced $M$- and $(N\ensuremath{-}M)$-body density matrices ${\ensuremath{\rho}}^{(M)}$ and ${\ensuremath{\rho}}^{(N\ensuremath{-}M)}$. The formalism also allows for reduced yet still exact Schmidt-like decompositions associated with blocks of these densities, in systems having a fixed fraction of the particles in some single-particle subspace. Monotonicity of the ensuing $M$-body entanglement under a certain set of quantum operations is also discussed. Illustrative examples in fermionic and bosonic systems with pairing correlations are provided, which show that in the presence of dominant eigenvalues in ${\ensuremath{\rho}}^{(M)}$, approximations based on a few terms of the pertinent Schmidt expansion can provide a reliable description of the state. The associated one- and two-body entanglement spectrum and entropies are also analyzed.

DOI
10.1103/physreva.110.032414

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We analyze a general bipartite-like representation of arbitrary pure states of N indistinguishable partcles, valid for both bosons and fermions, based on MMM- and (N\ensuremath−M)(N\ensuremath{-}M)(N\ensuremath−M)-particle states. It leads to exact (M,N\ensuremath−M)(M,N\ensuremath{-}M)(M,N\ensuremath−M) Schmidt-like expansions of the state for any M<NM<NM<N and is directly related to the isospectral reduced MMM- and (N\ensuremath−M)(N\ensuremath{-}M)(N\ensuremath−M)-body density matrices \ensuremathρ(M){\ensuremath{\rho}}^{(M)}\ensuremathρ(M) and \ensuremathρ(N\ensuremath−M){\ensuremath{\rho}}^{(N\ensuremath{-}M)}\ensuremathρ(N\ensuremath−M). The formalism also allows for reduced yet still exact Schmidt-like decompositions associated with blocks of these densities, in systems having a fixed fraction of the particles in some single-particle subspace. Monotonicity of the ensuing MMM-body entanglement under a certain set of quantum operations is also discussed. Illustrative examples in fermionic and bosonic systems with pairing correlations are provided, which show that in the presence of dominant eigenvalues in \ensuremathρ(M){\ensuremath{\rho}}^{(M)}\ensuremathρ(M), approximations based on a few terms of the pertinent Schmidt expansion can provide a reliable description of the state. The associated one- and two-body entanglement spectrum and entropies are also analyzed.
We discuss a general bipartitelike representation and Schmidt decomposition of an arbitrary pure state of NNN indistinguishable fermions, based on states of M<NM<NM<N and $(N\ensuremath{-}M)$ fermions. It is directly connected with the reduced MMM- and $(N\ensuremath{-}M)$-body density matrices (DMs), which have the same spectrum in such states. The concept of MMM-body entanglement emerges naturally in this scenario, generalizing that of one-body entanglement. Rigorous majorization relations satisfied by the normalized MMM-body DM are then derived, which imply that the associated entropy will not increase, on average, under a class of operations which have these DMs as postmeasurement states. Moreover, such entropy is an upper bound to the bipartite entanglement entropy generated by a class of operations which map the original state to a bipartite state of MMM and $N\ensuremath{-}M$ effectively distinguishable fermions. Analytic evaluation of the spectrum of MMM-body DMs in some strongly correlated fermionic states is also provided.
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.
Using an operational definition we quantify the entanglement, E(P), between two parties who share an arbitrary pure state of N indistinguishable particles. We show that E(P)< or =E(M), where E(M) is the bipartite entanglement calculated from the mode-occupation representation. Unlike E(M), E(P) is superadditive. For example, E(P)=0 for any single-particle state, but the state |1>|1>, where both modes are split between the two parties, has E(P)=1/2. We discuss how this relates to quantum correlations between particles, for both fermions and bosons.
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.
We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.
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