Exact relations between particle fluctuations and entanglement in Fermi gases

Abstract

We derive exact relations between the Rényi entanglement entropies and the particle number fluctuations of (connected and disjoint) spatial regions in systems of N noninteracting fermions in arbitrary dimension. We prove that the asymptotic large-N behavior of the entanglement entropies is proportional to the variance of the particle number. We also consider 1D Fermi gases with a localized impurity, where all particle cumulants contribute to the asymptotic large-N behavior of the entanglement entropies. The particle cumulant expansion turns out to be convergent for all integer-order Rényi entropies (except for the von Neumann entropy) and the first few cumulants provide already a good approximation. Since the particle cumulants are accessible to experiments, these relations may provide a measure of entanglement in these systems. The nature of the quantum correlations of manybody systems, and in particular the entanglement phenomenon, are fundamental physical issues. They have attracted much theoretical interest in the last few decades, due to the impressive progress in the experimental activity in atomic physics, quantum optics and nanoscience, which has provided a great opportunity to investigate the interplay between quantum and statistical behaviors in particle systems. The great ability in the manipulation of cold atoms in optical lattice (see, e.g., Ref.[1]) has allowed the realization of physical systems which are accurately described by theoretical models such as Hubbard and Bose-Hubbard models in different dimensions, achieving through experimental checks of the fundamental theoretical paradigma of condensed matter physics.The quantum correlations arising in the ground state of quantum many-body systems can be characterized by the expectation values of the products of local operators, such as the particle density and one-particle operators, or by their integral over a space region A, such as the particle-number correlators within A,where n(x) is the particle-density operator andcounts the number of particles in A. Quantum correlations are also characterized by the fundamental phenomenon of entanglement, which gives rise to nontrivial connections between different parts of extended quantum systems [2]. A widely accepted measure of entanglement is given by the Rényi entropies of the reduced density matrix ρ A of a subsystem A:whose limit α → 1 provides the von Neumann (vN) entropy. Local correlations and bipartite entanglement entropies provide important and complementary information of the quantum features of many-body systems, of their ground states and of their unitary evolutions, because they probe different features of the quantum dynamics. However, the entanglement entropy is a highly nonlocal quantity which is difficult to measure. Designing an experimental protocol for its measurement represents a major challenge. A recent interesting proposal considers the particle fluctuations as effective probes of many-body entanglement [3][4][5][6]. This is based on the result that, for noninteracting fermions...

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“…For systems which can be mapped to free fermions as the present one, the entanglement entropies can be related to the even cumulant V (2k) A of the particle-number distribution [80,81,82,83] V…”

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“…For systems which can be mapped to free fermions as the present one, they are intimately related to the expectation values of the correlations of local operators. Indeed, the entanglement entropies can be formally related to the even cumulant V (2k) [x,y] of the particle-number distribution [132,133,134,135] …”

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