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Typical entanglement entropy in the presence of a center: Page curve and its variance

Eugenio Bianchi and Pietro Donà
Phys. Rev. D 100, 105010 – Published 18 November 2019

Abstract

In a quantum system in a pure state, a subsystem generally has a nonzero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? We present a method to compute the exact formula of the momenta of the probability P(SA)dSA that a subsystem has entanglement entropy SA. The method applies to subsystems defined by a subalgebra of observables with a center. In the case of a trivial center, we reobtain the well-known result for the average entropy and the formula for the variance. In the presence of a nontrivial center, the Hilbert space does not have a tensor product structure and the well-known formula does not apply. We present the exact formula for the average entanglement entropy and its variance in the presence of a center. We show that for large systems the variance is small, ΔSA/SA1, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems where a center arises. In particular, for a system of noninteracting spins in a magnetic field and for a free quantum field, we show how the thermal entropy arises as the typical entanglement entropy of energy eigenstates.

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  • Received 1 June 2019

DOI:https://doi.org/10.1103/PhysRevD.100.105010

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Gravitation, Cosmology & AstrophysicsStatistical Physics & ThermodynamicsQuantum Information, Science & Technology

Authors & Affiliations

Eugenio Bianchi and Pietro Donà

  • Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, Pennsylvania 16802, USA and Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA

Article Text

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Issue

Vol. 100, Iss. 10 — 15 November 2019

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