TY - JOUR

T1 - Typical entanglement entropy in the presence of a center

T2 - Page curve and its variance

AU - Bianchi, Eugenio

AU - Donà, Pietro

N1 - Publisher Copyright:
© 2019 authors. Published by the American Physical Society.

PY - 2019/11/18

Y1 - 2019/11/18

N2 - 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.

AB - 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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U2 - 10.1103/PhysRevD.100.105010

DO - 10.1103/PhysRevD.100.105010

M3 - Article

AN - SCOPUS:85075318492

VL - 100

JO - Physical Review D

JF - Physical Review D

SN - 2470-0010

IS - 10

M1 - 105010

ER -