A canonical purification for the entanglement wedge cross-section

Abstract

In AdS/CFT we consider a class of bulk geometric quantities inside the entanglement wedge called reflected minimal surfaces. The areas of these surfaces are dual to the entanglement entropy associated to a canonical purification (the GNS state) that we dub the reflected entropy. From the bulk point of view, we show that half the area of the reflected minimal surface gives a reinterpretation of the notion of the entanglement wedge cross-section. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using a recently introduced approach to holographic modular flow. We also consider an explicit holographic construction of the canonical purification, introduced by Engelhardt and Wall; the reflected minimal surfaces are simply RT surfaces in this new spacetime. We contrast our results with the entanglement of purification conjecture, and finally comment on the continuum limit where we find a relation to the split property: the reflected entropy computes the von Neumann entropy of a canonical splitting type-I factor introduced by Doplicher and Longo.

Modular Nuclearity and Entanglement measures

“…Between the experts, it is a common belief that this entanglement entropy can be shown to be finite by assuming modular p-nuclearity for some 0 < p < 1. A result of this type is strongly suggested by previous works [Nar94, OT18, LX20] and shows interesting applications in AdS/CFT contexts as pointed out in [DF19], where it appeared as splitting entropy and it was used to conjecture a bound on the reflected entropy. [Nar11] linked the canonical entropy to the mutual information.…”

“…The authors of [DF19] argued on grounds of physical arguments that E C (ω) = E F ∨B ′ I (ω), and it is reasonable to expect that the results of this work can be properly strengthened. In fact, Theorem 15 implies that E F ∨B ′ I (ω) is finite if ε F ⊗B ′ →A⊗B ′ is a local operation (or finite sum thereof).…”

A universal tripartite entanglement signature of ungappable edge states

“…In this work we provide such a measure by going beyond bipartite entanglement and considering a multipartite entanglement measure h(A : B) recently referred to as the "Markov gap" [18]. This quantity was first discussed in the context of 1+1D CFTs and holography [18][19][20][21]. After defining a procedure which eliminates the nonuniversal short-distance contribution, we argue that the remainder takes the universal value…”

“…For a one-dimensional system, we choose A, B, C to be adjacent intervals. It has been shown [19,21] that the ground state of a gapped system has h = 0 and the ground state of a gapless system has h = h CFT ≡ c 3 log 2 , where c is the central charge of the CFT. Now we consider a two-dimensional lattice with the tripartition given in Fig.…”

Reflected Entropy in Double Holography

“…It is expected that this identification captures both classical and quantum correlations between two disjoint subregions. Meanwhile, a similar concept called holographic reflected entropy, which describes the entanglement involving the canonical purification of mixed states, has also been related to the EWCS [9]. EWCS, as a good measure of mixed state entanglement, has been widely studied in recent literature [9][10][11][12][13][14][15][16][17][18][19][20].…”