Reflected entropy in random tensor networks

Abstract

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 Page Curve for Reflected Entropy

“…Evidence for this conjecture is provided in Ref. [18], while here we simply utilize the conjecture and obtain physically sensible results.…”

“…Note: This paper provides a complementary analysis to Ref. [18] where we analyze the reflected entropy phase transition in random tensor networks.…”

“…The microcanonical case is identical to a random state and is discussed in detail in Ref. [18], 5 so our primary focus here will be on the canonical ensemble. For this case, we can use an integral representation for the n-boundary partition function Z n given by…”

Entanglement negativity, reflected entropy, and anomalous gravitation

“…Furthermore in a related development another mixed state correlation measure termed reflected entropy was introduced and computed for bipartite states in CFT 2 through another replica technique described in [20]. In a recent communication [21] this was further explored in the context of random tensor networks to include novel non-perturbative effects in the Rényi reflected entropy spectrum.…”

“…In this connection it should also be noted that following the gravitational path integral techniques developed in [12], a holographic duality between the reflected entropy and the minimal EWCS was established in [20]. Note that the minimal cross section of the entanglement wedge 2 (EWCS) has been proposed as putative dual of several quantum information measures, for example the entanglement of purification [49,50], the reflected entropy [20,21] and the balanced partial entanglement [51]. We should also mention here that an alternative 1 For applications of these holographic proposals to the black hole information loss problem, see for example [36], where analogues of the Page curve for the entanglement negativity were obtained.…”