We extend the notion of chord number in the strict large $\textit{N}$ double-scaled Sachdev-Ye-Kitaev (DSSYK) model to the corresponding finite $\textit{N}$ ETH matrix model. The chord number in the strict large $\textit{N}$ DSSYK model is known to correspond to the discrete length of the Einstein-Rosen bridge in the gravity dual, which reduces to the renormalized geodesic length in JT gravity at weak coupling. At finite $\textit{N}$, these chord number states form an over-complete basis of the non-perturbative Hilbert space, as the structure of the inner product gets significantly modified due to the Cayley-Hamilton theorem: There are infinitely many null states. In this paper, by considering “EFT for gravitational observables” or a version of “non-isometric code”, we construct a family of chord number operators at finite $\textit{N}$. While the constructed chord number operator depends on the reference chord number state, it realizes approximate $\textit{q}$-deformed oscillator algebra and reproduces semiclassical bulk geometry around the reference state. As a special case, we will show that when the reference is chosen to be the chord number zero state, the chord number operator precisely matches with the Krylov state complexity, leading to the “ramp-slope-plateau” behavior at late times, implying the formation of “gray hole”.

Japan

Masamichi Miyaji

Soichiro Mori

Kazumi Okuyama

Finite $\textit{N}$ bulk Hilbert space in ETH matrix model for double-scaled SYK. Null states, state-dependence and Krylov state complexity

The interplay between black hole interior dynamics and quantum chaos provides a crucial framework for probing quantum effects in quantum gravity. In this work, we investigate non-perturbative overlaps in Jackiw-Teitelboim (JT) gravity to uncover universal signatures of quantum chaos and quantum complexity. Taking advantage of universal spectral correlators from random matrix theory, we compute the overlaps between the thermofield double (TFD) state and two distinct classes of states: fixed-length states, which encode maximal volume slices, and time-shifted TFD states. The squared overlaps naturally define probability distributions that quantify the expectation values of gravitational observables. Central to our results is the introduction of generating functions for quantum complexity measures, such as 〈$\textit{e}$ 〉. The time evolution of these generating functions exhibits the universal slope-ramp-plateau structure, mirroring the behavior of the spectral form factor (SFF). Using generating functions, we further demonstrate that the universal time evolution of complexity for chaotic systems, which is characterized by a linear growth followed by a late-time plateau, arises from the disappearance of the linear ramp as the regularization parameter $\textit{α}$ decreases. With regard to the time-shifted TFD state, we derive a surprising result: the expectation value of the time shift, which classically grows linearly, vanishes when non-perturbative quantum corrections are incorporated. This cancellation highlights a fundamental distinction between semiclassical and quantum gravitational descriptions of the black hole interior. All our findings establish generating functions as powerful probes of quantum complexity and chaos in gravitational and quantum systems.

China

Belgium

Shan-Ming Ruan

Shono Shibuya

Kazuyoshi Yano

Non-perturbative overlaps in JT gravity: from spectral form factor to generating functions of complexity

We study the spectrum of the interior length and the horizon timeshift of a two-sided black hole by constructing non-perturbative length and timeshift operators in Jackiew-Teitelboim gravity. We first construct projection operators onto the fixed length or fixed horizon timeshift subspaces using the replica trick. We calculate the densities of state for the length and the timeshift, which are found to be finite and terminate at values of order $$ {e}^{S_0} $$ . This finiteness implies the discreteness in the spectrum of these quantities, and significant modifications in length and timeshift spectrum at order $$ {e}^{S_0} $$ . We then construct the non-perturbative length and timeshift operators, and apply them to study the time evolution of the two-sided black hole. We find that at early time, the probability distribution of the interior length and the timeshift are sharply peaked at the classical values, while after the Heisenberg time, the distribution is completely uniform over all possible values of the length and the timeshift, indicating maximal uncertainty. In particular, the probability of having the negative timeshift states, which corresponds to the white hole probability, is $\textit{O}$(1) after the Heisenberg time.

Non-perturbative discrete spectrum of interior length and timeshift in two-sided black hole

The interior of a pure-state black hole is known to be reconstructed from the Petz map by collecting a sufficiently large amount of the emitted Hawking radiation. This was established based on the Euclidean replica wormhole, which comes from an ensemble averaging over gravitational theories. On the other hand, this means that the Page curve and the interior reconstruction are both ensemble averages; thus, there is a possibility of large errors. In the previous study [R. Bousso and M. Miyaji, Fluctuations in the entropy of Hawking radiation], it was shown that the entropy of the Hawking radiation has fluctuation of order <math><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>S</mi><mrow><mi>BH</mi></mrow></msub></mrow></msup></math>, thus is typical in the ensemble. In the present article, we show that the fluctuations in the information recovery are similarly suppressed by the factor <math><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>S</mi><mrow><mi>BH</mi></mrow></msub></mrow></msup></math> by studying the fluctuations of the relative entropy difference in the encoding map and the entanglement fidelity of the Petz map. This result establishes the typicality of the information recovery in the ensemble. In addition, we also compute the entanglement loss of the encoding map.

Fluctuation in the fidelity of information recovery from Hawking radiation

We use the gravitational path integral (GPI) to compute the fluctuations of the Hawking radiation entropy around the Page curve in a two-dimensional model introduced by Penington et al. Before the Page time, we find that <math><mi>δ</mi><mi>S</mi><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>S</mi></mrow></msup><mo>/</mo><msqrt><mn>2</mn></msqrt></math>, where <math><mi>S</mi></math> is the black hole entropy. This result agrees with the Haar-averaged entropy fluctuations in a bipartite system. After the Page time, we find that <math><mi>δ</mi><mi>S</mi><mo>∼</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>S</mi></mrow></msup></math>, up to a prefactor that depends logarithmically on the width of the microcanonical energy window. This is not symmetric under exchange of subsystem sizes and so does not agree with the Haar average for a subsystem of fixed Hilbert space dimension. The discrepancy can be attributed to the fact that the black hole Hilbert space dimension is not fixed by the state preparation: even in a microcanonical ensemble with a top-hat smearing function, the GPI yields an additive fluctuation in the number of black hole states. This result, and the fact that the Page curve computed by the GPI is smooth, all point towards an ensemble interpretation of the GPI.

United States

Raphael Bousso

Fluctuations in the entropy of Hawking radiation

In this paper, we propose a new gravity dual for a 2d BCFT with two conformal boundaries by introducing a defect that connects the two End-of-the-World branes. We demonstrate that the BCFT dual to this bulk model exhibits a richer lowest spectrum. The corresponding lowest energy eigenvalue can continuously interpolate between   <math> <mo>−</mo> <mfrac> <mi>πc</mi> <mrow> <mn>24</mn> <mi>Δ</mi> <mi>x</mi> </mrow> </mfrac> </math>  $$ -\frac{\pi c}{24\Delta x} $$  and 0 where ∆x is the distance between the boundaries. This range was inaccessible to the conventional AdS/BCFT model with distinct boundary conditions. We compute the holographic entanglement entropy and find that it exhibits three different phases, one of which breaks the time reflection symmetry. We also construct a wormhole saddle, analogous to a 3d replica wormhole, which connects different boundaries through the AdS bulk. This saddle is present only if the BCFT is non-unitary and is always subdominant compared to the disconnected saddle.

Chitraang Murdia

Holographic BCFT with a Defect on the End-of-the-World brane

We consider spacetime initiated by a finite-sized initial boundary as a generalization of the Hartle-Hawking no-boundary state. We study entanglement entropy of matter state prepared by such spacetime. We find that the entanglement entropy for large subregion is given either by the initial state entanglement or the entanglement island, preventing the entropy to grow arbitrarily large. Consequently, the entanglement entropy is always bounded from above by the boundary area of the island, leading to an entropy bound in terms of the island. The island I is located in the analytically continued spacetime, either at the bra or the ket part of the spacetime in Schwinger-Keldysh formalism. The entanglement entropy is given by an average of complex pseudo generalized entropy for each entanglement island. We find a necessary condition of the initial state to be consistent with the strong sub-additivity, which requires that any probe degrees of freedom are thermally entangled with the rest of the system. We then find a large parameter region where the spacetime with finite-sized initial boundary, which does not have the factorization puzzle at leading order, dominates over the Hartle-Hawking no-boundary state or the bra-ket wormhole. Due to the absence of a moment of time reflection symmetry, the island in our setup is a generalization of the entanglement wedge, called pseudo entanglement wedge. In pseudo entanglement wedge reconstruction, we consider reconstructing the bulk matter transition matrix on A ∪ I, from a fine-grained state on A. The bulk transition matrix is given by a thermofield double state with a projection by the initial state. We also provide an AdS/BCFT model by considering EOW branes with corners. We also find the exponential hardness of such reconstruction task using a generalization of Python’s lunch conjecture to pseudo generalized entropy.

Island for gravitationally prepared state and pseudo entanglement wedge

We study overlaps between two regularized boundary states in conformal field theories. Regularized boundary states are dual to end of the world branes in an AdS black hole via the AdS/BCFT. Thus they can be regarded as microstates of a single sided black hole. Owing to the open-closed duality, such an overlap between two different regularized boundary states is exponentially suppressed as   <math> <mfenced> <mrow> <mfenced> <msub> <mi>ψ</mi> <mi>a</mi> </msub> </mfenced> <msub> <mi>ψ</mi> <mi>b</mi> </msub> </mrow> </mfenced> <mo>∼</mo> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>O</mi> <mfenced> <msubsup> <mi>h</mi> <mi>ab</mi> <mfenced> <mo>min</mo> </mfenced> </msubsup> </mfenced> </mrow> </msup> </math>  $$ \left\langle \left.{\psi}_a\right|{\psi}_b\right\rangle \sim {e}^{-O\left({h}_{ab}^{\left(\min \right)}\right)} $$ , where   <math> <msubsup> <mi>h</mi> <mi>ab</mi> <mfenced> <mo>min</mo> </mfenced> </msubsup> </math>  $$ {h}_{ab}^{\left(\min \right)} $$  is the lowest energy of open strings which connect two different boundaries a and b. Our gravity dual analysis leads to   <math> <msubsup> <mi>h</mi> <mi>ab</mi> <mfenced> <mo>min</mo> </mfenced> </msubsup> </math>  $$ {h}_{ab}^{\left(\min \right)} $$  = c/24 for a pure AdS3 gravity. This shows that a holographic boundary state is a random vector among all left-right symmetric states, whose number is given by a square root of the number of all black hole microstates. We also perform a similar computation in higher dimensions, and find that   <math> <msubsup> <mi>h</mi> <mi>ab</mi> <mfenced> <mo>min</mo> </mfenced> </msubsup> </math>  $$ {h}_{ab}^{\left(\min \right)} $$  depends on the tensions of the branes. In our analysis of holographic boundary states, the off diagonal elements of the inner products can be computed directly from on-shell gravity actions, as opposed to earlier calculations of inner products of microstates in two dimensional gravity.

Tadashi Takayanagi

Tomonori Ugajin

Spectrum of end of the world branes in holographic BCFTs

Recent work has demonstrated the need to include contributions from entanglement islands when computing the entanglement entropy in quantum field theory states coupled to regions of semiclassical gravity. We propose a new formula for the reflected entropy that includes additional contributions from such islands. We derive this formula from the gravitational path integral by finding additional saddles that include generalized replica wormholes. We also demonstrate that our covariant formula satisfies all the inequalities required of the reflected entropy. We use this formula in various examples that demonstrate its relevance in illustrating the structure of multipartite entanglement that are invisible to the entropies.

Venkatesa Chandrasekaran

Pratik Rath

Including contributions from entanglement islands to the reflected entropy

We study entanglement entropy after a double local quench in two-dimensional conformal field theories (CFTs). In the holographic CFT, such a state with double excitation is dual to an anti–de Sitter space with two massive particles injected from the boundary. We show that the growth after the double local excitations in pure CFT is universal and is given by the sum of two local quenches with an additional negative term. This negative contribution can be interpreted naturally in holography as being due to the attractive force of gravity. On the CFT side, this evaluation of the entanglement entropy is accomplished by a special limit of six-point functions, where we employ the fusion matrix approach for multipoint conformal blocks developed by the authors [J. High Energy Phys. 08 (2019) 063].