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arXiv:2211.16045 · Journal of High Energy Physics

Fun with replicas: tripartitions in tensor networks and gravity

Geoff Penington, Michael Walter, Freek Witteveen

Published May 2, 2023• 35 citations
arXivHTMLPDF

Abstract

A bstract We analyse a simple correlation measure for tripartite pure states that we call G ( A : B : C ). The quantity is symmetric with respect to the subsystems A , B , C , invariant under local unitaries, and is bounded from above by log d A d B . For random tensor network states, we prove that G ( A : B : C ) is equal to the size of the minimal tripartition of the tensor network, i.e., the logarithmic bond dimension of the smallest cut that partitions the network into three components with A , B , and C . We argue that for holographic states with a fixed spatial geometry, G ( A : B : C ) is similarly computed by the minimal area tripartition. For general holographic states, G ( A : B : C ) is determined by the minimal area tripartition in a backreacted geometry, but a smoothed version is equal to the minimal tripartition in an unbackreacted geometry at leading order. We briefly discuss a natural family of quantities G n ( A : B : C ) for integer n ≥ 2 that generalize G = G 2 . In holography, the computation of G n ( A : B : C ) for n > 2 spontaneously breaks part of a ℤ n × ℤ n replica symmetry. This prevents any naive application of the Lewkowycz-Maldacena trick in a hypothetical analytic continuation to n = 1.

DOI
10.1007/jhep05(2023)008

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A bstract We analyse a simple correlation measure for tripartite pure states that we call G ( A : B : C ). The quantity is symmetric with respect to the subsystems A , B , C , invariant under local unitaries, and is bounded from above by log d A d B . For random tensor network states, we prove that G ( A : B : C ) is equal to the size of the minimal tripartition of the tensor network, i.e., the logarithmic bond dimension of the smallest cut that partitions the network into three components with A , B , and C . We argue that for holographic states with a fixed spatial geometry, G ( A : B : C ) is similarly computed by the minimal area tripartition. For general holographic states, G ( A : B : C ) is determined by the minimal area tripartition in a backreacted geometry, but a smoothed version is equal to the minimal tripartition in an unbackreacted geometry at leading order. We briefly discuss a natural family of quantities G n ( A : B : C ) for integer n ≥ 2 that generalize G = G 2 . In holography, the computation of G n ( A : B : C ) for n > 2 spontaneously breaks part of a ℤ n × ℤ n replica symmetry. This prevents any naive application of the Lewkowycz-Maldacena trick in a hypothetical analytic continuation to n = 1.
A bstract For general random tensor network states at large bond dimension, we prove that the integer Rényi reflected entropies (away from phase transitions) are determined by minimal triway cuts through the network. This generalizes the minimal cut description of bipartite entanglement for these states. A natural extrapolation away from integer Rényi parameters, suggested by the triway cut problem, implies the holographic conjecture S R = 2EW, where S R is the reflected entropy and EW is the entanglement wedge cross-section. Minimal triway cuts can be formulated as integer programs which cannot be relaxed to find a dual maximal flow/bit-thread description. This sheds light on the gap between the existence of tripartite entanglement in holographic states and the bipartite entanglement structure motivated by bit-threads. In particular, we prove that the Markov gap that measures tripartite entanglement is lower bounded by the integrality gap of the integer program that computes the triway cut.
A bstract The entanglement negativity E \mathcal{E} E E ( A : B ) is a useful measure of quantum entanglement in bipartite mixed states. In random tensor networks (RTNs), which are related to fixed-area states, it was found in ref. [1] that the dominant saddles computing the even Rényi negativity E(2k) {\mathcal{E}}^{(2k)} E(2k) E 2 k generically break the ℤ 2 k replica symmetry. This calls into question previous calculations of holographic negativity using 2D CFT techniques that assumed ℤ 2 k replica symmetry and proposed that the negativity was related to the entanglement wedge cross section. In this paper, we resolve this issue by showing that in general holographic states, the saddles computing E(2k) {\mathcal{E}}^{(2k)} E(2k) E 2 k indeed break the ℤ 2 k replica symmetry. Our argument involves an identity relating E(2k) {\mathcal{E}}^{(2k)} E(2k) E 2 k to the k -th Rényi entropy on subregion AB ∗ in the doubled state ∣ρAB⟩AA∗BB∗ {\left.|{\rho}_{AB}\right\rangle}_{A{A}^{\ast }{BB}^{\ast }} ∣ρAB​⟩AA∗BB∗​ ρ AB A A ∗ BB ∗ , from which we see that the ℤ 2 k replica symmetry is broken down to ℤ k . For k < 1, which includes the case of E \mathcal{E} E E ( A : B ) at k = 1/2, we use a modified cosmic brane proposal to derive a new holographic prescription for E(2k) {\mathcal{E}}^{(2k)} E(2k) E 2 k and show that it is given by a new saddle with multiple cosmic branes anchored to subregions A and B in the original state. Using our prescription, we reproduce known results for the PSSY model and show that our saddle dominates over previously proposed CFT calculations near k = 1. Moreover, we argue that the ℤ 2 k symmetric configurations previously proposed are not gravitational saddles, unlike our proposal. Finally, we contrast holographic calculations with those arising from RTNs with non-maximally entangled links, demonstrating that the qualitative form of backreaction in such RTNs is different from that in gravity.
We propose a holographic dual of the relative entropy of entanglement for a boundary tripartition A:B:CA:B:CA:B:C. Our proposal is that, at the leading order, \begin{align} E_R(ρ_{AB}) = \frac{1}{4G_N}\Big[\mathrm{Area}(Γ_{\min}) - \mathrm{Area}(γ_{AB})\Big] \notag \end{align} where Γmin⁡Γ_{\min}Γmin​ is the minimal bulk tripartition surface and γABγ_{AB}γAB​ is the minimal surface homologous to ABABAB. For a general random tensor network, we prove the corresponding upper bound by constructing a separable state associated with Γmin⁡Γ_{\min}Γmin​ and evaluating its relative entropy. We also establish the matching lower bound rigorously for networks of one, two, and three Haar random tensors by bounding the maximal tripartite product-state overlap. In appropriate geometries, the minimal tripartition surface can form a nontrivial tri-junction resembling the Mercedes logo. The proof proceeds by sequential optimization over candidate product states, which has a natural geometric interpretation as a local search for the minimal tripartition surface.
A bstract Since the work of Ryu and Takayanagi, deep connections between quantum entanglement and spacetime geometry have been revealed. The negative eigenvalues of the partial transpose of a bipartite density operator is a useful diagnostic of entanglement. In this paper, we discuss the properties of the associated entanglement negativity and its Rényi generalizations in holographic duality. We first review the definition of the Rényi negativities, which contain the familiar logarithmic negativity as a special case. We then study these quantities in the random tensor network model and rigorously derive their large bond dimension asymptotics. Finally, we study entanglement negativity in holographic theories with a gravity dual, where we find that Rényi negativities are often dominated by bulk solutions that break the replica symmetry. From these replica symmetry breaking solutions, we derive general expressions for Rényi negativities and their special limits including the logarithmic negativity. In fixed-area states, these general expressions simplify dramatically and agree precisely with our results in the random tensor network model. This provides a concrete setting for further studying the implications of replica symmetry breaking in holography.
A bstract In this paper, we analyze the question of replica symmetry in the bulk for multi-partite entanglement measures in the vacuum state of two dimensional holographic CFTs. We first define a class of multi-partite local unitary invariants, multi-invariants , with a given replica symmetry that acts freely and transitively on the replicas. We look for a subclass of measures such that the dual bulk geometry also preserves replica symmetry. We obtain the most general solution to this problem if we require the bulk to preserve replica symmetry for general configurations of the regions. Orbifolding the bulk solution with the replica symmetry gives us a bulk geometry with a network of conical singularities. Our approach makes it clear that there are infinitely many infinitely large families of multi-invariants such that each family evaluates identically on the holographic state. Geometrically, these are equalities involving volumes of handlebodies, possibly of different genus, at particular points in the moduli space. In certain cases, we check our bulk computation with an explicit calculation in CFT. Finally we comment on the generalization to higher dimension.
For general random tensor network states at large bond dimension, we prove that the integer Rényi reflected entropies (away from phase transitions) are determined by minimal triway cuts through the network. This generalizes the minimal cut description of bipartite entanglement for these states. A natural extrapolation away from integer Rényi parameters, suggested by the triway cut problem, implies the holographic conjecture SR=2EWS_R=2EWSR​=2EW, where SRS_RSR​ is the reflected entropy and EWEWEW is the entanglement wedge cross-section. Minimal triway cuts can be formulated as integer programs which cannot be relaxed to find a dual maximal flow/bit-thread description. This sheds light on the gap between the existence of tripartite entanglement in holographic states and the bipartite entanglement structure motivated by bit-threads. In particular, we prove that the Markov gap that measures tripartite entanglement is lower bounded by the integrality gap of the integer program that computes the triway cut.