Living on the edge: a non-perturbative resolution to the negativity of bulk entropies

TL;DR

The paper analyzes a non-perturbative entanglement-negativity puzzle in two-sided black holes caused by many matter insertions behind the horizon. It resolves the paradox by summing over all gravitational topologies and by exploiting a dual matrix-model description, with the Airy regime providing a universal link between gravity and pure JT thermodynamics. Beyond the Airy limit, one- and two-eigenvalue instantons in the GUE matrix model restore positivity, and a random tensor network approach corroborates the bulk picture by connecting Wick contractions to wormhole geometries. The work also extends the analysis to non-SUSY JT gravity, higher dimensions, and generic Δ via edge-universality and non-perturbative saddles, offering a cohesive framework for entropy positivity in holography.

Abstract

Lin, Maldacena, Rozenberg, and Shan (LMRS) presented a new information paradox in black hole physics by noticing that the entanglement and Rényi entropies in a two-sided black hole can become negative when the geometry contains a very large number of matter excitations behind the black hole horizon. While originally this puzzle was presented in the context of BPS two-sided black holes in two-dimensional supergravity, the negativity in fact persists for more general two-sided black holes in the presence of a large number of matter excitations. Since the entanglement and Rényi entropies in ordinary quantum systems cannot be negative, resolving this puzzle is a necessary step towards understanding the quantum mechanical description of black holes. In this paper, we explain how to address the entanglement negativity puzzle, both in the original setting discussed by LMRS and in more general non-supersymmetric settings, by summing over all non-perturbative contributions to the gravitational path integral. We then interpret this result from the point of view of a dual matrix integral, which we use to extend our analysis beyond the regime of validity of the genus re-summation performed in the gravitational path integral. In this regime, positivity is rescued by new saddles of the matrix integral, a one-eigenvalue instanton and a two-eigenvalue instanton. Finally, we formulate a similar puzzle and its resolution using random tensor network techniques.

Figures (12)

• Figure 1: Comparing semi-quenched, annealed and quenched second Rényi-2 entropies, zoomed in near the Airy edge $k=\alpha e^{2S_{\text{BH}}/3}$. Semi-quenched and annealed Rényis are obtained analytically, whereas the quenched Rényi is obtained numerically. Inset: We zoom in on the quenched and semi-quenched entropies at large $k$ and plot them on a log-log scale. Notice the different behavior: the quenched entropy has an $\alpha^{-1}$ power-law decay (dashed blue line) which can be obtained analytically (see Appendix \ref{['app:quenched']}), whereas the semi-quenched entropy falls off exponentially. Nevertheless, as expected (and unlike the annealed entropy), our calculation shows that both entropies remain positive.
• Figure 2: Comparing $(\overline{\operatorname{tr}\rho})^2$ (first term in the denominator of Eq. \ref{['eq:renyi-2']}) and $\overline{$trρ$^2}_{\text{conn.}}$ (second term in the denominator of Eq. \ref{['eq:renyi-2']}). Here $\alpha\equiv ke^{-2S_\text{BPS}/3}$ and we set $E_0=0$ for simplicity. For small $\alpha$, the disconnected geometry $(\overline{\operatorname{tr}\rho})^2$ dominates. At large $\alpha$, the connected geometry $\overline{$trρ$^2}_{\text{conn.}}$ dominates.
• Figure 3: The random tensor network preparing the state \ref{['eq:TNstate']} dual to a two-sided black hole with a long wormhole. The structure of the network provides a bulk interpretation of the state. The different tensors can be regarded as insertions of particles of different flavor (emphasized by the different colors) supporting the long wormhole. The red dots represent $k+1$ degenerate extremal surfaces.
• Figure 4: The annealed (yellow, Eq. \ref{['eq:TNannealedres']}) and semi-quenched (blue, Eq. \ref{['eq:TNSQres']}) Rényi-2 entropies obtained from our RTN calculation as a function of $\alpha=k/D$. For a large number of insertions, the annealed entropy becomes negative, even when including all-genus contributions. The semi-quenched entropy stays positive and vanishes for $k\to\infty$ thanks to connected wormhole contributions.
• Figure 5: Top left: For $q\to 0$ (we plot here $q=0.0001$), the $q$-deformed Gaussian spectral density reduces to the Wigner semicircle. Top right: Increasing $q$ (we plot here $q=0.6$), the $q$-deformed Gaussian (in blue) is roughly a Gaussian for small $x$, but it retains a square-root edge at $x_{\text{edge}}=2/\sqrt{1-q}$, shown in the inset. $\bar{\sigma}_\text{Gauss}$ in Eq. \ref{['eq:gaussspectrum']} is plotted in yellow for reference. Bottom: As we take $q\to 1$ (we plot here $q=0.99)$, the spectrum becomes unbounded and the $q$-deformed Gaussian approaches the Gaussian spectrum \ref{['eq:gaussspectrum']} (the $q$-deformed Gaussian is plotted in blue, indistinguishable from $\bar{\sigma}_\text{Gauss}$ in yellow).