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arXiv:2306.06163 · Physical review. D/Physical review. D.

Entanglement of purification in random tensor networks

Chris Akers, Thomas Faulkner, Simon Lin, Pratik Rath

Published May 6, 2024• 8 citations
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Abstract

The entanglement of purification E P ( A ∶ B ) is a powerful correlation measure, but it is notoriously difficult to compute because it involves an optimization over all possible purifications. In this paper, we prove a new inequality: E P ( A ∶ B ) ≥ 1 2 S R ( 2 ) ( A ∶ B ) , where S R ( n ) ( A ∶ B ) is the Renyi reflected entropy. Using this, we compute E P ( A ∶ B ) for a large class of random tensor networks at large bond dimension and show that it is equal to the entanglement wedge cross section E W ( A ∶ B ) , proving a previous conjecture motivated from AdS/CFT. Published by the American Physical Society 2024

DOI
10.1103/physrevd.109.l101902

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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.

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The entanglement of purification E P ( A ∶ B ) is a powerful correlation measure, but it is notoriously difficult to compute because it involves an optimization over all possible purifications. In this paper, we prove a new inequality: E P ( A ∶ B ) ≥ 1 2 S R ( 2 ) ( A ∶ B ) , where S R ( n ) ( A ∶ B ) is the Renyi reflected entropy. Using this, we compute E P ( A ∶ B ) for a large class of random tensor networks at large bond dimension and show that it is equal to the entanglement wedge cross section E W ( A ∶ B ) , proving a previous conjecture motivated from AdS/CFT. Published by the American Physical Society 2024
A bstract In the AdS/CFT context, the entanglement of purification (EoP, denoted as E P ) of CFT is conjectured to be dual to the entanglement wedge cross section (EWCS) in bulk. However, another quantity called reflected entropy S R is also supposed to be dual to two times the EWCS. A natural question is whether they are the same in holographic CFTs even though they are different in general. Previous studies have shown E P ≥ 12SR(n) \frac{1}{2}{S}_R^{(n)} 21​SR(n)​ 1 2 S R n , n ≥ 2 for random tensor networks. In this paper, we study this inequality beyond n ≥ 2, and we focus on the range 0 < n < 2. However, the calculations of EoP are notoriously difficult in general. Thus, our calculations mainly focus on the free scalar theory which is close to the holographic CFTs. We generalized the previous strategy for EoP in [1] to the Rényi case. And we have also presented two methods for Rényi reflected entropy, one is using correlators, the other one is Gaussian wavefunction ansatz. Our calculations show that the inequality still holds for 0 < n < 2, and it may give us some insights into the equivalence of EoP and half reflected entropy in holographic CFTs. As byproducts of our research, we have also demonstrated the positivity of the Rényi Markov gap and the monotonicity of the Rényi reflected entropy in the free scalar theory.
A bstract We introduce a new information-theoretic measure of multipartite correlations Δ P , by generalizing the entanglement of purification to multipartite states. We provide proofs of its various properties, focusing on several entropic inequalities, in generic quantum systems. In particular, it turns out that the multipartite entanglement of purification gives an upper bound on multipartite mutual information, which is a generalization of quantum mutual information in the spirit of relative entropy. After that, motivated by a tensor network description of the AdS/CFT correspondence, we conjecture a holographic dual of multipartite entanglement of purification Δ W , as a sum of minimal areas of codimension-2 surfaces which divide the entanglement wedge into multi-pieces. We prove that this geometrical quantity satisfies all properties we proved for the multipartite entanglement of purification. These agreements strongly support the Δ P = Δ W conjecture. We also show that the multipartite entanglement of purification is larger than multipartite squashed entanglement, which is a promising measure of multipartite quantum entanglement. We discuss potential saturation of multipartite squashed entanglement onto multipartite mutual information in holographic CFTs and its applications.
A bstract In holographic theories, the reflected entropy has been shown to be dual to the area of the entanglement wedge cross section. We study the same problem in random tensor networks demonstrating an equivalent duality. For a single random tensor we analyze the important non-perturbative effects that smooth out the discontinuity in the reflected entropy across the Page phase transition. By summing over all such effects, we obtain the reflected entanglement spectrum analytically, which agrees well with numerical studies. This motivates a prescription for the analytic continuation required in computing the reflected entropy and its Rényi generalization which resolves an order of limits issue previously identified in the literature. We apply this prescription to hyperbolic tensor networks and find answers consistent with holographic expectations. In particular, the random tensor network has the same non-trivial tripartite entanglement structure expected from holographic states. We furthermore show that the reflected Rényi spectrum is not flat, in sharp contrast to the usual Rényi spectrum of these networks. We argue that the various distinct contributions to the reflected entanglement spectrum can be organized into approximate superselection sectors. We interpret this as resulting from an effective description of the canonically purified state as a superposition of distinct tensor network states. Each network is constructed by doubling and gluing various candidate entanglement wedges of the original network. The superselection sectors are labelled by the different cross-sectional areas of these candidate entanglement wedges.
In recent work, Akers et al. [arXiv:2306.06163] proved that the entanglement of purification Ep(A:B){E}_{p}(A:B)Ep​(A:B) is bounded below by half of the qqq-R\'enyi reflected entropy SR(q)(A:B){S}_{R}^{(q)}(A:B)SR(q)​(A:B) for all $q\ensuremath{\ge}2$, showing that Ep(A:B)=12SR(q)(A:B){E}_{p}(A:B)=\frac{1}{2}{S}_{R}^{(q)}(A:B)Ep​(A:B)=21​SR(q)​(A:B) for a class of random tensor-network states. Naturally, the authors raise the question of whether a similar bound holds at q=1q=1q=1. Our work answers that question in the negative by finding explicit counterexamples, which we arrive at through numerical optimization. Nevertheless, this result does not preclude the possibility that restricted sets of states, such as conformal field theory states with semiclassical gravity duals, could obey the bound in question.
Purification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of many degrees of freedom, one natural purification is the one with minimal entanglement. Here we study the entropy of the minimally entangled purification, called the entanglement of purification, in three model systems: an Ising spin chain, conformal field theories holographically dual to Einstein gravity, and random stabilizer tensor networks. We conjecture values for the entanglement of purification in all these models, and we support our conjectures with a variety of numerical and analytical results. We find that such minimally entangled purifications have a number of applications, from enhancing entanglement-based tensor network methods for describing mixed states to elucidating novel aspects of the emergence of geometry from entanglement in the AdS/CFT correspondence.
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.