Abstract
The reflected entropy SR(A : B) of a density matrix ρAB is a bipartite correlation measure lower-bounded by the quantum mutual information I(A : B). In holographic states satisfying the quantum extremal surface formula, where the reflected entropy is related to the area of the entanglement wedge cross-section, there is often an order-N2 gap between SR and I. We provide an information-theoretic interpretation of this gap by observing that SR− I is related to the fidelity of a particular Markov recovery problem that is impossible in any state whose entanglement wedge cross-section has a nonempty boundary; for this reason, we call the quantity SR− I the Markov gap. We then prove that for time-symmetric states in pure AdS3 gravity, the Markov gap is universally lower bounded by log(2)ℓAdS/2GN times the number of endpoints of the cross-section. We provide evidence that this lower bound continues to hold in the presence of bulk matter, and comment on how it might generalize above three bulk dimensions. Finally, we explore the Markov recovery problem controlling SR− I using fixed area states. This analysis involves deriving a formula for the quantum fidelity — in fact, for all the sandwiched Rényi relative entropies — between fixed area states with one versus two fixed areas, which may be of independent interest. We discuss, throughout the paper, connections to the general theory of multipartite entanglement in holography.
Highlights
1. the (a) A boundary state canonical purification ρ whose bulk dual is a Rindler wedge √ρ, formed by pasting W(ρ) to its in AdS3. (b) The initial CPT conjugate along the quantum extremal surface that forms the spatial boundary of W(ρ)
We provide an information-theoretic interpretation of this gap by observing that SR − I is related to the fidelity of a particular Markov recovery problem that is impossible in any state whose entanglement wedge cross-section has a nonempty boundary; for this reason, we call the quantity SR − I the Markov gap
Our goal in this paper was threefold: (i) we established a connection between the quantity SR(A : B) − I(A : B) and Markov recovery processes in canonical purifications, arguing in the process that boundaries in the entanglement wedge cross-section of a holographic state require SR − I to be nonzero at order 1/GN ; (ii) we proved inequality (1.12) for time-symmetric states in AdS3 gravity, establishing a quantitative bound on the Markov gap in such states, and (iii) we explored a fixed area toy model of the Markov recovery process to see how information inequalities partially reproduce the geometric bound on the Markov gap given in equation (1.11)
Summary
As mentioned in the introduction, the quantity SR(A : B) − I(A : B) can be related to the fidelity of a particular Markov recovery process on the canonical purification of ρAB. In subsection 2.1, we review and explain the results from quantum information theory needed to understand this claim. We define Markov recovery maps and the quantum fidelity of states, and give a refinement of the inequality SR − I ≥ 0 in terms of the fidelity of a Markov recovery map. In subsection 2.2, we give a holographic interpretation of the refined inequality, and explain why the Markov gap must be nonzero at order 1/GN whenever the entanglement wedge cross-section of ρAB has a nontrivial boundary.
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