We study and prove properties of the minimax formulation of the HRT holographic entanglement entropy formula, which involves finding the maximal-area surface on a timelike hypersurface, or time-sheet, and then minimizing over the choice of time-sheet. In this formulation, the homology condition is imposed at the level of the spacetime: the homology regions are spacetime volumes rather spatial regions. We show in particular that the smallest minimax homology region is the entanglement wedge. The minimax prescription suggests a way to construct a graph model for time-dependent states, a weighted graph on which min cuts compute HRT entropies. The existence of a graph model would imply that HRT entropies obey the same inequalities as RT entropies, in other words that the RT and HRT entropy cones coincide. Our construction of a graph model relies on the time-sheets obeying a certain “cooperating” property, which we show holds in some examples and for which we give a partial proof; however, we also find scenarios where it may fail. A video abstract is available at .

United States

Brianna Grado-White

Guglielmo Grimaldi

Matthew Headrick

Veronika Hubeny

Minimax surfaces and the holographic entropy cone

We address the question of whether holographic entropy inequalities obeyed in static states (by the RT formula) are always obeyed in time-dependent states (by the HRT formula), focusing on the case where the bulk spacetime is 2 + 1 dimensional. An affirmative answer to this question was previously claimed by Czech-Dong. We point out an error in their proof when the bulk is multiply connected. We nonetheless find strong support, of two kinds, for an affirmative answer in that case. We extend the Czech-Dong proof for simply-connected spacetimes to spacetimes with = (i.e. 2-boundary, genus-0 wormholes). Specializing to vacuum solutions, we also numerically test thousands of distinct inequalities (including all known RT inequalities for up to 6 regions) on millions of randomly chosen configurations of regions and bulk spacetimes, including three different multiply-connected topologies; we find no counterexamples. In an appendix, we prove some (dimension-independent) facts about degenerate HRT surfaces and symmetry breaking.

Testing holographic entropy inequalities in 2 + 1 dimensions

We derive several new reformulations of the Hubeny-Rangamani-Takayanagi covariant holographic entanglement entropy formula. These include: (1) a minimax formula, which involves finding a maximal-area achronal surface on a timelike hypersurface homologous to D(A) (the boundary causal domain of the region A whose entropy we are calculating) and minimizing over the hypersurface; (2) a max V-flow formula, in which we maximize the flux through D(A) of a divergenceless bulk 1-form V subject to an upper bound on its norm that is non-local in time; and (3) a min U-flow formula, in which we minimize the flux over a bulk Cauchy slice of a divergenceless timelike 1-form U subject to a lower bound on its norm that is non-local in space. The two flow formulas define convex programs and are related to each other by Lagrange duality. For each program, the optimal configurations dynamically find the HRT surface and the entanglement wedges of A and its complement. The V-flow formula is the covariant version of the Freedman-Headrick bit thread reformulation of the Ryu-Takayanagi formula. We also introduce a measure-theoretic concept of a “thread distribution”, and explain how Riemannian flows, V-flows, and U-flows can be expressed in terms of thread distributions.

Covariant bit threads

Generalizing the bit thread formalism, we use convex duality to derive dual flow programs to the bipartite and multipartite holographic entanglement of purification proposals and then prove several inequalities using these constructions. In the multipartite case we find the flows exhibit novel behavior which allows for a constrained flux on the boundary of the homology region. We show this flux can be made distinct from bi-partite terms and reflects the truly multipartite portion of the holographic entanglement of purification.

Jonathan Harper

Bit threads and holographic entanglement of purification

We explore the structure of holographic entropy relations (associated with ‘information quantities’ given by a linear combination of entanglement entropies of spatial sub-partitions of a CFT state with geometric bulk dual). Such entropy relations can be recast in multiple ways, some of which have significant advantages. Motivated by the already-noted simplification of entropy relations when recast in terms of multipartite information, we explore additional simplifications when recast in a new basis, which we dub the K-basis, constructed from perfect tensor structures. For the fundamental information quantities such a recasting is surprisingly compact, in part due to the interesting fact that entropy vectors associated to perfect tensors are in fact extreme rays in the holographic entropy cone (as well as the full quantum entropy cone). More importantly, we prove that all holographic entropy inequalities have positive coefficients when expressed in the K-basis, underlying the key advantage over the entropy basis or the multipartite information basis.

Temple He

Holographic entropy relations repackaged

As a probe of circuit complexity in holographic field theories, we study sub-system analogues based on the entanglement wedge of the bulk quantities appearing in the “complexity = volume” and “complexity = action” conjectures. We calculate these quantities for one exterior region of an eternal static neutral or charged black hole in general dimensions, dual to a thermal state on one boundary with or without chemical potential respectively, as well as for a shock wave geometry. We then define several analogues of circuit complexity for mixed states, and use tensor networks to gain intuition about them. In the action approach, we find two possible cases depending on an ambiguity in the definition of the action associated with a counterterm. In one case, there is a promising qualitative match between the holographic action and what we call the purification complexity, the minimum number of gates required to prepare an arbitrary purification of the given mixed state. In the other case, the match is to what we call the basis complexity, the minimum number of gates required to prepare the given mixed state starting from a minimal complexity state with the same eigenvalue spectrum. One way to fix this ambiguity is to choose an action definition such that UV divergent part is positive, in which case the best match to the action result is the basis complexity. In contrast, the holographic volume does not appear to match any of our definitions of mixed-state complexity.

Cesar Agón

Brian Swingle

Subsystem complexity and holography

We generalize holographic bit threads to bulk theories with a gravitational action containing higher-curvature terms. Bit threads are a reformulation of holographic entanglement entropy, where the entropy is given by the maximum number of threads emanating from a boundary region into the bulk. We show that the addition of higher-curvature terms adds corrections to the bit thread thickness that depend on the local geometry and thread orientation. Two different methods are given: determination of the density bound by requiring the maximum number of threads through a given surface to reproduce the entanglement entropy functional on that surface, and application of Lagrange dualization. The results of the two methods are applied to Gauss-Bonnet gravity as the simplest non-trivial example.

Andrew Rolph

Bit threads in higher-curvature gravity

We compute the contribution of the vacuum Virasoro representation to the genus-two partition function of an arbitrary CFT with central charge c &gt; 1. This is the perturbative pure gravity partition function in three dimensions. We employ a sewing construction, in which the partition function is expressed as a sum of sphere four-point functions of Virasoro vacuum descendants. For this purpose, we develop techniques to efficiently compute correlation functions of holomorphic operators, which by crossing sym-metry are determined exactly by a finite number of OPE coefficients; this is an analytic implementation of the conformal bootstrap. Expanding the results in 1 /c , corresponding to the semiclassical bulk gravity expansion, we find that — unlike at genus one — the result does not truncate at finite loop order. Our results also allow us to extend earlier work on multiple-interval Rényi entropies and on the partition function in the separating degeneration limit.

Canada

Alexander Maloney

Eric Perlmutter

Ida Zadeh

Rényi entropies, the analytic bootstrap, and 3D quantum gravity at higher genus

We identify conditions for the entanglement entropy as a function of spatial region to be compatible with causality in an arbitrary relativistic quantum field theory. We then prove that the covariant holographic entanglement entropy prescription (which relates entanglement entropy of a given spatial region on the boundary to the area of a certain extremal surface in the bulk) obeys these conditions, as long as the bulk obeys the null energy condition. While necessary for the validity of the prescription, this consistency requirement is quite nontrivial from the bulk standpoint, and therefore provides important additional evidence for the prescription. In the process, we introduce a codimension-zero bulk region, named the entanglement wedge, naturally associated with the given boundary spatial region. We propose that the entanglement wedge is the most natural bulk region corresponding to the boundary reduced density matrix.

United Kingdom

Albion Lawrence

Mukund Rangamani

Causality &amp; holographic entanglement entropy

Recently it has been shown that the Bekenstein-Hawking entropy formula evaluated on certain closed surfaces in the bulk of a holographic spacetime has an interpretation as the differential entropy of a particular family of intervals (or strips) in the boundary theory [1, 2]. We first extend this construction to bulk surfaces which vary in time. We then give a general proof of the equality between the gravitational entropy and the differential entropy. This proof applies to a broad class of holographic backgrounds possessing a generalized planar symmetry and to certain classes of higher-curvature theories of gravity. To apply this theorem, one can begin with a bulk surface and determine the appropriate family of boundary intervals by considering extremal surfaces tangent to the given surface in the bulk. Alternatively, one can begin with a family of boundary intervals; as we show, the differential entropy then equals the gravitational entropy of a bulk surface that emerges from the intersection of the neighboring entanglement wedges, in a continuum limit.