Quantum Information

In holographic CFTs satisfying eigenstate thermalization, there is a regime where the operator product expansion can be approximated by a random tensor network. The geometry of the tensor network corresponds to a spatial slice in the holographic dual, with the tensors discretizing the radial direction. In spherically symmetric states in any dimension and more general states in 2d CFT, this leads to a holographic error-correcting code, defined in terms of OPE data, that can be systematically corrected beyond the random tensor approximation. The code is shown to be isometric for light operators outside the horizon, and non-isometric inside, as expected from general arguments about bulk reconstruction. The transition at the horizon occurs due to a subtle breakdown of the Virasoro identity block approximation in states with a complex interior.

We study entanglement entropy in (2+1)-dimensional gravity as a window into larger open questions regarding entanglement entropy in gravity. (2+1)-dimensional gravity can be rewritten as a topological field theory, which makes it a more tractable model to study. In these topological theories, there remain key questions which we seek to answer in this work, such as the questions 1) What is the entropy of the physical algebra of observables in a subregion, 2) How do we define a factorization map such that the entropy of the resulting factors agrees with this algebraic entropy, and 3) Can we use these insights to build a tensor network that exhibits non-commuting areas? We investigate non-Abelian toric codes / Levin-Wen models as a toy model for black hole entropy in Chern Simons theory. These differ from the usual model in that the stabilizers are implemented as constraints. By enforcing constraints for both Gauss' Law and the flatness of the gauge field, we obtain a choice of algebra that contains only topological operators. The desirable properties of this model are twofold: first, we produce the finiteness of black hole entropy described in previous literature while providing a natural algebraic motivation for this result. Secondly, we obtain non-commuting area operators on a toy model with the topology of a torus.

Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code's logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.

Motivated by the ground state structure of quantum models with all-to-all interactions such as mean-field quantum spin glass models and the Sachdev-Ye-Kitaev (SYK) model, we propose a tensor network architecture which can accomodate volume law entanglement and a large ground state degeneracy. We call this architecture the non-local renormalization ansatz (NoRA) because it can be viewed as a generalization of MERA, DMERA, and branching MERA networks with the constraints of spatial locality removed. We argue that the architecture is potentially expressive enough to capture the entanglement and complexity of the ground space of the SYK model, thus making it a suitable variational ansatz, but we leave a detailed study of SYK to future work. We further explore the architecture in the special case in which the tensors are random Clifford gates. Here the architecture can be viewed as the encoding map of a random stabilizer code. We introduce a family of codes inspired by the SYK model which can be chosen to have constant rate and linear distance at the cost of some high weight stabilizers. We also comment on potential similarities between this code family and the approximate code formed from the SYK ground space.

We study Generalized Free Fields (GFF) from the point of view of information measures. We begin by reviewing conformal GFF, their holographic representation, and the multiple possible assignations of algebras to a single spacetime region that arise in these theories. We will focus on manifestations of these features present in the Mutual Information (MI) of holographic GFF. First, we show that the MI can be expected to be finite even if the AdS dual space is of infinite volume. Then, we present the long-distance limit of the MI for regions with arbitrary boundaries in the light cone for the causal and entanglement wedge algebras. The pinching limit of these surfaces shows the GFF behaves as an interacting model from the MI point of view. The entanglement wedge algebra choice allows these models to ``fake'' causality, giving results consistent with their role in the description of large N models. Finally, we explore the short distance limit of the MI. Interestingly, we find that the GFF has a leading volume term rather than an area term and a logarithmic term in any dimension rather than only for even dimensions as in ordinary CFTs. We also find the dependence of some subleading terms on the conformal dimension of the GFF.

In this talk, we will first present an analysis of infinitesimal null deformations for the entanglement entropy, which leads to a major simplification of the proof of the C, F and A-theorems in quantum field theory. Next, we will discuss the quantum null energy condition (QNEC) on the light-cone. Finally, we combine these tools in order to establish the irreversibility of renormalization group flows on planar d-dimensional defects, embedded in D-dimensional conformal field theories. This proof completes and unifies all known defect irreversibility theorems for defect dimensions below d=5. The F-theorem on defects (d=3) is a new result using information-theoretic methods. The geometric construction connects the proof of irreversibility with and without defects through the QNEC inequality in the bulk, and makes contact with the proof of strong subadditivity of holographic entropy taking into account quantum corrections.

We describe supersymmetric SYK models which display a scaling similarity at low temperatures, rather than the usual conformal behavior. We discuss the large N equations, which were studied previously as uncontrolled approximations to other models. We also present a picture for the physics of the model which suggest that the relevant low energy degrees of freedom are almost free. We also searched for a spin glass phase but we found no replica symmetry breaking solutions.

We show that if a massive (or charged) body is put in a quantum superposition of spatially separated states in the vicinity of any (Killing) horizon, the mere presence of the horizon will eventually destroy the coherence of the superposition in a finite time. This occurs because, in effect, the long-range fields sourced by the superposition register on the black hole horizon which forces the emission of entangling “soft gravitons/photons” through the horizon. This enables the horizon to harvest “which path” information about the superposition. We provide estimates of the decoherence time for such quantum superpositions in the presence of a black hole and cosmological horizon. Finally, we further sharpen and generalize this mechanism by recasting the gedankenexperiment in the language of (approximate) quantum error correction. This yields a complementary picture where the decoherence is due to an “eavesdropper” (Eve) in the black hole attempting to obtain "which path" information by measuring the long-range fields of the superposed body. We explicitly compute the quantum fidelity to determine the amount of information such an Eve can obtain and show, by the information-disturbance tradeoff, a direct relationship between the information gained by Eve and the decoherence of the superposition in the exterior. In particular, we show that the decoherence of the superposition corresponds to the "optimal" measurement made by Eve in the black hole interior.

We quantize JT gravity with matter on the spatial interval with two asymptotically AdS boundaries. We consider the von Neumann algebra generated by the right Hamiltonian and the gravitationally dressed matter operators on the right boundary. We prove that the commutant of this algebra is the analogously defined left boundary algebra and that both algebras are type II infinity factors. These algebras provide a precise notion of the entanglement wedge away from the semiclassical limit.

We argue that generic local subregions in semiclassical quantum gravity are associated with von Neumann algebras of type II_1, extending recent work by Chandrasekaran et.al. beyond subregions bounded by Killing horizons. The subregion algebra arises as a crossed product of the type III_1 algebra of quantum fields in the subregion by the flow generated by a gravitational constraint operator. We conjecture that this flow agrees with the vacuum modular flow sufficiently well to conclude that the resulting algebra is type II_\infty, which projects to a type II_1 algebra after imposing a positive energy condition. The entropy of semiclassical states on this algebra can be computed and shown to agree with the generalized entropy by appealing to a first law of local subregions. The existence of a maximal entropy state for the type II_1 algebra is further shown to imply a version of Jacobson’s entanglement equilibrium hypothesis. We discuss other applications of this construction to quantum gravity and holography, including the quantum extremal surface prescription and the quantum focusing conjecture.

In holography, the quantum extremal surface formula relates the entropy of a boundary state to the sum of two terms: the area term and the entropy of bulk fields inside the entanglement wedge. As the bulk effective field theory suffers from UV divergences, the second term must be regularized. It has been conjectured since the work of Susskind and Uglum that the renormalization of Newton’s constant in the area term exactly cancels the difference between different choices of regularization for bulk entropy. In this talk, I will explain how the recent developments on von Neumann algebras appearing in the large N limit of holography allow to prove this claim within the framework of holographic quantum error correction, and to reinterpret it as an instance of the ER=EPR paradigm. This talk is based on the paper arXiv:2302.01938.