Entanglement entropies of equilibrated pure states in quantum many-body systems and gravity

TL;DR

The paper develops an equilibrium approximation that turns the computation of Renyi entropies for equilibrated pure states into a universal problem governed by the equilibrium density matrix rho_eq. It recasts Renyi entropies as transition amplitudes in a replicated Hilbert space and shows how Euclidean path integrals naturally arise as the dominant contributions, with a self-consistent criterion ensuring validity in systems with large effective dimension. The framework preserves unitarity and, when applied to gravity, provides a principled derivation of replica wormholes without ensemble averaging, linking them to late-time equilibration rather than to averaging over theories. It further connects this picture to operator growth via the random void distribution, yielding a coherent picture of typicality and higher-moment structure in chaotic dynamics, and discusses subregion equilibration, causal constraints, and holographic generalizations.

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.

Figures (21)

• Figure 1: The path integrals \ref{['jh']} for \ref{['rnyi']} involve $2n$ integration contours, with those for $U$'s going forward in time and those for $U^{\dagger}$ backward in time. $\rho_0$ provides the initial conditions while the contractions implied by the traces in \ref{['rnyi']} define the final conditions for the path integrals.
• Figure 2: (a) shows the "future conditions" for each of the ${\mathcal{Z}}_n^{(A)}(\tau)$, coming from the factor in the first line of \ref{['zn_t']}, for $n=6$. In (b), we show an example of how to connect indices in the interior of the diagram for a permutation $\tau$ such that $\tau(1)=3$, and the factor of $\braket{i_{1_a} i_{1_b}| {{\mathcal{I}}}_{{\alpha}}| i'_{3_a} i'_{3_b}}$ that comes from this interior connection.
• Figure 3: Examples of planar diagrams corresponding to different choices of planar permutations $\tau$ that saturate \ref{['yeg']}.
• Figure 4: Examples of non-planar diagrams corresponding to two choices of $\tau$ that do not saturate \ref{['yeg']}.
• Figure 5: Double-line diagrams (and the corresponding polygons) obtained from the diagrams of Fig. \ref{['fig:eq']}(a) and \ref{['fig:rev']}(a).