We extensively explore the connections between timelike entanglement and non-Hermitian density matrices in quantum many-body systems. We classify setups where we encounter non-Hermitian density matrices into two types: one is due to causal influences under unitary evolutions, and the other is due to nonunitary evolutions in non-Hermitian systems. We provide various examples of these setups including interacting harmonic oscillators, two dimensional conformal field theories, and holographic dualities. In them, we compute the timelike entanglement entropy and imagitivity, which measures how much density matrices are non-Hermitian. In both two classes, typical holographic examples are given by traversable AdS wormholes. We explain how causal influences in a wormhole dual to a pair of non-Hermitian quantum systems is possible even without interactions between them. We argue that to realize a traversable wormhole we need not only ordinary quantum entanglement but also timelike entanglement.

Japan

Jonathan Harper

Taishi Kawamoto

Ryota Maeda

Nanami Nakamura

Tadashi Takayanagi

Non-Hermitian density matrices from timelike entanglement and wormholes

We study the inverse problem of holographic entanglement entropy in AdS using a data-driven generative model. Training data consist of randomly generated geometries and their holographic entanglement entropies using the Ryu-Takayanagi formula. After training, the Transformer reconstructs the blackening function within our metric ansatz from previously unseen inputs. The Transformer achieves accurate reconstructions on smooth black hole geometries and extrapolates to horizonless backgrounds. We describe the architecture and data generation process, and we quantify accuracy on both $\textit{f}$($\textit{z}$) and the reconstructed $\textit{S}$($\textit{ℓ}$). Code and evaluation scripts are available at the provided repository.

South Korea

Sejin Kim

Learning the inverse Ryu-Takayanagi formula with transformers

In this paper, we demonstrate that the first law of holographic pseudo-entropy, which is a non-Hermitian generalization of entanglement entropy in a two-dimensional conformal field theory (CFT), is equivalent to the perturbative Einstein equation in three-dimensional de Sitter (dS) space, assuming the dS/CFT correspondence. Our analysis reveals that the geodesic that accurately satisfies the first law of holographic pseudo-entropy consists of a timelike curve and a curve whose coordinates are complex. We also demonstrate that infinitesimal changes to the pseudo entropy satisfy a Klein-Gordon equation in two-dimensional de Sitter space. These imply the emergence of a time coordinate from a Euclidean CFT in dS/CFT.

Kosei Fujiki

Michitaka Kohara

Kotaro Shinmyo

Yu-ki Suzuki

Entropic interpretation of Einstein equation in dS/CFT

We model the Hayden-Preskill (HP) information recovery protocol in 2d CFTs via local joining quenches. Euclidean path integrals with slits prepare the HP subsystems: the message $\textit{M}$, its reference $\textit{N}$, the Page-time black hole $\textit{B}$, the early radiation $\textit{E}$, and the late radiation $\textit{R}$; the remaining black hole after emitting $\textit{R}$ is denoted as $\textit{B}$ . The single-slit geometry provides an analytically tractable toy model, while the bounded-slit geometry more closely captures the HP setup. In the free Dirac fermion 2d CFT, the mutual information $\textit{I}$($\textit{N}$ : $\textit{B}$ ) shows quasi-particle dynamics with partial or full revivals, whereas that in holographic 2d CFTs, which are expected to be maximally chaotic, exhibits sharp transitions: in the bounded-slit case, when the size of the late radiation becomes comparable to that of the reference $\textit{N}$, $\textit{I}$($\textit{N}$ : $\textit{B}$ ) vanishes at late time, otherwise it remains finite. This contrast between free CFTs and holographic CFTs gives a clear characterization of the HP recovery threshold.

China

Weibo Mao

Hayden-Preskill model via local quenches

We consider the evolution of entanglement entropy in a two-dimensional conformal field theory with a holographic dual. Specifically, we are interested in a class of excited states produced by a combination of pure-state (local operator) and mixed-state local quenches. We employ a method that allows us to determine the full time evolution analytically. While a single insertion of a local operator gives rise to a logarithmic time profile of entanglement entropy relative to the vacuum, we find that this growth is heavily suppressed in the presence of a mixed-state quench, reducing it to a time-independent constant bump. The degree of suppression depends on the relative position of the quenches as well as the ratio of regularization parameters associated with the quenches. This work sheds light on the interesting properties of gravitational scattering involving black holes.

Kazuki Doi

Entanglement suppression due to black hole scattering

We explore the reconstruction of bulk local states in three-dimensional flat spacetime (Flat ) using states from two-dimensional Carrollian conformal field theories (CCFT ), proposed as dual field theories in one lower dimension. For massive scalar-type bulk excitations, reconstruction is achieved through states in the induced representation. This method successfully reproduces the bulk massive scalar spectrum and the bulk-to-bulk propagator. Additionally, we identify a new flat limit from AdS and dS spacetimes, further validating our proposal for scalar reconstruction in Flat /CCFT .

Peng-Xiang Hao

Shunta Takahashi

Bulk reconstruction of scalar excitations in Flat

We study the Ryu-Takayanagi (RT) surfaces associated with timelike subregions in static spacetimes with a horizon. It is possible to find the analytical continuation of the RT surfaces that can extend into the horizon, allowing us to probe the interior of the black hole. The horizon typically divides the RT surface into two distinct parts. We demonstrate that the area of the surface inside the horizon can be reconstructed from the contributions of the surfaces outside the horizon, along with additional RT surfaces for spacelike subregions that are causally related to the timelike subregions. This result provides a concrete realization of black hole complementarity at the level of classical metric, where the spacetime in the black hole interior can be reconstructed from the degrees of freedom outside the horizon.

Wu-zhong Guo

Jin Xu

Duality of Ryu-Takayanagi surfaces inside and outside the horizon

In this paper, we study a new class of AdS/BCFT setups, where the world-volumes of end-of-the-world branes (EOW branes) are given by flat spaces, to explore flat space holography from an AdS bulk. We show that they provide gravity duals of CFTs in the presence of null boundaries. Our holographic calculations lead to many new predictions on entanglement entropy, correlation functions and partition functions for CFTs with null boundaries. By considering a bulk region between two EOW branes, we present an AdS/BCFT explanation that the flat space gravity is dual to a Carrollian CFT (CCFT), including the swing surface calculation of entanglement entropy.

Naoki Ogawa

Takahiro Waki

Flat space holography via AdS/BCFT

We study a class of quantum states involving multiple entangled CFTs in AdS /CFT , associated with multi-boundary black hole geometries, and demonstrate that the Ryu-Takayanagi (RT) formula for entanglement entropy can be derived using only boundary CFT data. Approximating the OPE coefficients by their Gaussian moments within the 2D large-$\textit{c}$ CFT ensemble, we show that both the norm of the states and the entanglement entropies associated with various bipartitions — reproducing the expected bulk dual results — can be computed purely from the CFT. All $\textit{macroscopic geometric}$ structures arising from gravitational saddles emerge entirely from the universal statistical moments of the $\textit{microscopic algebraic}$ CFT data, revealing a statistical-mechanical mechanism underlying semiclassical gravity. We establish a precise correspondence between the CFT norm, the Liouville partition function with ZZ boundary conditions, and the exact gravitational path integral over 3D multi-boundary black hole geometries. For entanglement entropy, each RT phase arises from a distinct leading-order Gaussian contraction, with phase transitions — analogous to replica wormholes — emerging naturally from varying dominant statistical patterns in the CFT ensemble. Our derivation elucidates how the general mechanism behind holographic entropy, namely a boundary replica direction that elongates and becomes contractible in the bulk dual, is encoded explicitly in the statistical structure of the CFT data.

United States

Ning Bao

Hao Geng

Yikun Jiang

Ryu-Takayanagi formula for multi-boundary black holes from 2D large-$\textit{c}$ CFT ensemble

Holographic duality describes gravitational theories in terms of quantum many-body systems. In holography, quantum information theory provides a crucial tool that directly connects microscopic structures of these systems to the geometries of gravitational spacetimes. One manifestation is that the entanglement entropy in quantum many-body systems can be calculated from the area of an extremal surface in the corresponding gravitational spacetime. This implies that a gravitational spacetime can emerge from an enormous number of entangled qubits. In this Essay, I will discuss open problems in this area of research, considering recent developments and outlining future prospects towards a complete understanding of quantum gravity. The first step in this direction is to understand what kind of quantum circuits each holographic spacetime corresponds to, drawing on recent developments in quantum complexity theories and studying concrete examples of holography in string theory. Next, we should extend the concept of holography to general spacetimes, e.g., those spacetimes which appear in realistic cosmologies, by utilizing the connections between quantum information and holography. To address the fundamental question of how time emerges, I will propose the concepts of pseudoentropy and timelike entanglement as a useful tool in our exploration. Part of a series of Essays in Physical Review Letters which concisely present author visions for the future of their field.