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arXiv:2112.06840 · Physical Review A

ℏ→0 limit of the entanglement entropy

Giuseppe Mussardo, Jacopo Viti

Published Mar 2, 2022• 9 citations
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Abstract

It may be naively assumed that certain entanglement measures, like the entanglement entropy of a pure state, approach zero in the classical limit. Here, however, the authors show that the limit is both well-behaved and nonzero: for a bipartite system of N particles, it coincides with the Shannon entropy of N bits. These results have been demonstrated previously in the context of quantum field theory, but here the authors use fully analytic methods to provide a general physical interpretation.

DOI
10.1103/physreva.105.032404

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It may be naively assumed that certain entanglement measures, like the entanglement entropy of a pure state, approach zero in the classical limit. Here, however, the authors show that the limit is both well-behaved and nonzero: for a bipartite system of N particles, it coincides with the Shannon entropy of N bits. These results have been demonstrated previously in the context of quantum field theory, but here the authors use fully analytic methods to provide a general physical interpretation.
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To this day, von Neumann definition of entropy remains the most popular measure of quantum entanglement. Much of the literature on entanglement entropy, particularly in the context of field theory, has focused on isolating the UV divergences. Zero-mode divergences of the entanglement entropy are less studied in this context, and apart from being easier to isolate, they offer an interesting insight into the physics of the system. To gain a better understanding of the system in this limit, we develop the free particle approximation of harmonic oscillator, with which we investigate the properties of entropy divergence in continuous bi-partite quantum systems such as the coupled harmonic oscillators and the hydrogen atom. We also show zero-mode divergence of the entropy of environment-induced entanglement in a tri-partite oscillator system. We discuss the implications of our result for field theory and IR structure of gravity.
We analyze numerically the dynamical generation of quantum entanglement in a system of two interacting particles, started in a coherent separable state, for decreasing values of ℏ. As ℏ→0 the entanglement entropy, computed at any finite time, converges to a finite nonzero value. The limit law that rules the time dependence of entropy is well reproduced by purely classical computations. Its general features can be explained by simple classical arguments, which expose the different ways entanglement is generated in systems that are classically chaotic or regular.
The paper is an elaborate comparative assessment of the most relevant entropy-based quantities to be applied in quantifying quantum entanglement in pure, mixed, bipartite, and multipartite quantum systems. The study fills the existing gap in the literature in which most of the previous literature examines individual measures, but a lot of knowledge is lacking on what these measures are regarding their strengths, weaknesses, and the operational usefulness against each other. The study can prove the behaviour of each measure under noise, decoherence, dimensional scaling, and variation of state-type by analysing von Neumann entropy, the Tsallis entropy, entropy of entanglement of formation, concurrence and logarithmic negativity. Numerical studies with pure two-qubit states, Werner states, GHZ states, W states and mixed random states demonstrate specific sensitivity properties and computational limitations which ensure that no single measure can be a universal quantifier. The paper goes on to say that entropy-only measures do not work in mixed regimes because of a dominance in classical correlations. The results have several important implications to quantum communication, quantum simulation, and quantum computing in the NISQ-era, where the choice of measurement has a direct impact on benchmarking, error-correction scheme, and protocol performance based on entanglement. The paper ends with a recommendation of contextual, purpose-based scheme of selecting entanglement measures and suggests a further study on scalable benchmarking, hybrid quantifiers, and operational performance-based entanglement measurements. These observations reinforce the theoretical premise of entanglement studies and enable the advancement of more effective quantum technologies in the field of practice.
We establish relations between tripartite pure state entanglement and additivity properties of the bipartite relative entropy of entanglement. Our results pertain to the asymptotic limit of local manipulations on a large number of copies of the state. We show that additivity of the relative entropy would imply that there are at least two inequivalent types of asymptotic tripartite entanglement. The methods used include the application of some useful lemmas that enable us to analytically calculate the relative entropy for some classes of bipartite states.