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arXiv:2309.02506 · Physical Review A

Possibility of entanglement of purification to be less than half of the reflected entropy

Josiah Couch, Phuc Nguyen, Sarah Racz, Georgios Stratis, Yuxuan Zhang

Published Feb 20, 2024• 3 citations
arXivHTMLPDF

Abstract

In recent work, Akers et al. [arXiv:2306.06163] proved that the entanglement of purification ${E}_{p}(A:B)$ is bounded below by half of the $q$-R\'enyi reflected entropy ${S}_{R}^{(q)}(A:B)$ for all $q\ensuremath{\ge}2$, showing that ${E}_{p}(A:B)=\frac{1}{2}{S}_{R}^{(q)}(A:B)$ for a class of random tensor-network states. Naturally, the authors raise the question of whether a similar bound holds at $q=1$. 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.

DOI
10.1103/physreva.109.022426

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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≥2q\ensuremath{\ge}2q\ensuremath≥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.
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.
We prove that the von Neumann entanglement of purification is nonadditive. For two-qubit Werner states W(f)W(f)W(f) with singlet weight 1/64≤f≤1/41/64\le f\le1/41/64≤f≤1/4, we show that canonical purification is globally optimal: its entanglement, also known as reflected entropy, gives the exact entanglement of purification. In contrast, the regularized entanglement of purification lies strictly below the reflected entropy for 0<f<f∗0<f<f_\ast0<f<f∗​, where f∗≈0.017f_\ast\approx0.017f∗​≈0.017. Together, these results establish nonadditivity throughout 1/64≤f<f∗1/64\le f<f_\ast1/64≤f<f∗​. Our canonical-optimality proof generalizes the proof in [1], which established Rényi nonadditivity for 0≤α<10\leα<10≤α<1 using classical two-qubit states. We thus resolve the additivity question at α=1α=1α=1. Remarkably, nonadditivity is again witnessed by separable states and no entanglement in the underlying states is required.
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 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.
The entanglement of purification EPE_PEP​, introduced by Terhal, Horodecki, Leung, and DiVincenzo in 2002, measures the total correlations of a bipartite state by the entanglement needed to purify it; whether it is additive on tensor powers has been open since its introduction. In 2012 Chen and Winter argued "beyond reasonable doubt", from numerical evidence, that the two-qubit Werner state at singlet fraction f=1/200f = 1/200f=1/200 is a counterexample, and explicitly left a completely rigorous proof to future work. To our knowledge we give the first rigorous proof: for this state WWW, EP∞(W)≤0.9663…<97/100<EP(W)E_P^\infty(W) \le 0.9663\ldots < 97/100 < E_P(W)EP∞​(W)≤0.9663…<97/100<EP​(W), hence the entanglement of purification is strictly nonadditive at some finite tensor power. The lower bound is a finite certificate: analytic reductions send the Schmidt spectrum of every purification in the complete complex Stiefel domain to a point satisfying explicit necessary inequalities in a three-dimensional box, and an exact-rational subdivision of that box has 25,383 leaves, each of which either excludes nonphysical points or proves the entropy bound. The upper bound combines an exact symmetric-support endpoint, an explicit feasible isometry, and a convexity theorem for the regularized quantity, closed by two directed rational logarithm certificates. Every computer-assisted certificate reduces to integer comparisons; a dependency-free program verifies all of them from the supplementary artifact.