Abstract
We develop a universal approximation for the Renyi entropies of a pure state at late times in a non-integrable many-body system, which macroscopically resembles an equilibrium density matrix. The resulting expressions are fully determined by properties of the associated equilibrium density matrix, and are hence independent of the details of the initial state, while also being manifestly consistent with unitary time-evolution. For equilibrated pure states in gravity systems, such as those involving black holes, this approximation gives a prescription for calculating entanglement entropies using Euclidean path integrals which is consistent with unitarity and hence can be used to address the information loss paradox of Hawking. Applied to recent models of evaporating black holes and eternal black holes coupled to baths, it provides a derivation of replica wormholes, and elucidates their mathematical and physical origins. In particular, it shows that replica wormholes can arise in a system with a fixed Hamiltonian, without the need for ensemble averages.
Highlights
Consider a quantum many-body system initially in a far-from-equilibrium pure state | 0
By applying the equilibrium approximation to these models, we provide a derivation of the replica wormholes introduced in these references, explaining how such Euclidean configurations emerge from Lorentzian time evolution at late times, and why they lead to answers that are consistent with unitarity constraints
We develop an approximation to calculate the entanglement entropies of an equilibrated pure state
Summary
Consider a quantum many-body system initially in a far-from-equilibrium pure state | 0. We develop a general approximation method for calculating Sn(A) for equilibrated pure states in systems with a fixed initial state and time-evolution operator, in the limit where the effective dimension of the Hilbert space (roughly, the dimension of the accessible part of the Hilbert space from the initial state) is large. Since the only input that goes into our approximation method is information about the equilibrium density matrix ρ(eq), it can be used to obtain universal results for the entanglement entropies of a variety of quantum many-body systems when the initial state equilibrates to a given type of ensemble. By applying the equilibrium approximation to these models, we provide a derivation of the replica wormholes introduced in these references, explaining how such Euclidean configurations emerge from Lorentzian time evolution at late times, and why they lead to answers that are consistent with unitarity constraints. V, we discuss the applicability of the equilibrium approximation to observables other than the Renyi entropies, and mention some open questions
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