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arXiv:1702.02360 · Journal of Statistical Mechanics Theory and Experiment

Entropic relations for indistinguishable quantum particles

Marius Lemm

Published Apr 9, 2024• 6 citations
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Abstract

Abstract The von Neumann entropy of a k-body-reduced density matrix γ k quantifies the entanglement between k quantum particles and the remaining ones. In this paper, we rigorously prove general properties of this entanglement entropy as a function of k; it is concave for all 1 ⩽ k ⩽ N and non-decreasing until the midpoint k ⩽ ⌊ N / 2 ⌋ . The results hold for indistinguishable quantum particles and are independent of the statistics.

DOI
10.1088/1742-5468/ad343a

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We discuss particle entanglement in systems of indistinguishable bosons and fermions, in finite Hilbert spaces, with focus on operational measures of the entanglement of particles. We show how to use von Neumann entropy, negativity, and entanglement witnesses in these cases, proving interesting relations. We obtain analytic expressions to quantify the entanglement of particles in homogeneous DDD-dimensional Hamiltonians with certain symmetries.
We discuss the entropic criterion for separability of compound quantum systems for general nonadditive entropic forms based on arbitrary concave functions f. For any separable state, the generalized entropy of the whole system is shown to be not smaller than that of the subsystems, for any choice of f, providing thus a necessary criterion for separability. Nevertheless, the criterion is not sufficient and examples of entangled states with the same property are provided. This entails, in particular, that the conjecture about the positivity of the conditional Tsallis entropy for all q, a more stringent requirement than the positivity of the conditional von Neumann entropy, is actually a necessary but not sufficient condition for separability in general. The direct relation between the entropic criterion and the largest eigenvalues of the full and reduced density operators of the system is also discussed.
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