Structure of chaotic eigenstates and their entanglement entropy

Abstract

We consider a chaotic many-body system (i.e. one that satisfies the eigenstate thermalization hypothesis) that is split into two subsystems, with an interaction along their mutual boundary, and study the entanglement properties of an energy eigenstate with nonzero energy density. When the two subsystems have nearly equal volumes, we find a universal correction to the entanglement entropy that is proportional to the square root of the system's heat capacity (or a sum of capacities, if there are conserved quantities in addition to energy). This phenomenon was first noted by Vidmar and Rigol in a specific system; our analysis shows that it is generic, and expresses it in terms of thermodynamic properties of the system. Our conclusions are based on a refined version of a model of a chaotic eigenstate originally due to Deutsch, and analyzed more recently by Lu and Grover. arXiv:1906.04295v2 [cond-mat.stat-mech]

Subsystem Rényi entropy of thermal ensembles for SYK-like models

“…Here if we consider general n = 1, the result would depend on whether we assume the ensemble to be canonical or micro-canonical. Moreover, for local systems, studies show that equation(8) should be modified[59][60][61][62]. However, here we have checked that for non-local system such as SYK-like models, the correction is small.…”

Multipartite Entanglement Structure in the Eigenstate Thermalization Hypothesis

“…There has been some interest in relating ETH to the bipartite entanglement entropy [30][31][32], here we apply ETH to the quantum Fisher information F(Ô). This quantity was introduced to bound the precision of the estimation of a parameter φ, conjugated to an observableÔ using a quantum stateρ, via the so-called quantum Cramer-Rao bound ∆φ 2 ≤ 1/M F(Ô), where M is the number of independent measurements made in the protocol [33].…”

Disentangling quantum matter with measurements

“…However, a quantitative comparison with general expectations for the behavior of the CMI with the above arrangement of ABC is problematic for the following reason. Given a partition of a Hilbert space ABC where H AB is the same size as H C , a finite energy density ergodic eigenstate should have I(A : B) ∼ √ L A + L B after cancellation of the volume law terms [52,53] 4 . This represents an area law as a function of L A , while keeping L A + L B fixed (as we do in Fig.…”

“…The results of[52,53] follow from an ansatz[5,54] for the bipartition of an ergodic state in terms of a wavefunction which is a random matrix, which includes no information about the nature of the bipartition, such as locality. (Ref [52].…”

“…The conclusions depend only on the sizes of Hilbert spaces of subsystems with fixed charge. In particular,[53] generalizes the calculation to include several conserved quantities (the relevant conserved quantities here being spin, charge and energy). As a result, we expect them to apply to our non-local bipartition of an eigenstate of a local Hamiltonian.…”