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arXiv:2211.06886 · Journal of High Energy Physics

The Markov gap in the presence of islands

Yizhou Lu, Jiong Lin

Published Mar 7, 2023• 21 citations
arXivHTMLPDF

Abstract

Abstract The Markov gap [1], namely the difference between reflected entropy and mutual information, is explicitly computed in the defect extremal surface model, JT gravity, and the generic 2d extremal black holes, in vacuum states. The phases that contain various island contributions are considered, and their existence is carefully checked. Moreover, we show explicitly how the Markov gap originates from the OPE coefficient of the boundary CFT. And, as a generalization of [1], the lower bound of the Markov gap is given by $$ \frac{C}{3} $$ C 3 log 2 times the number of EWCS boundaries on minimal surfaces. We propose a boundary way of counting the lower bound for the Markov gap, which states that the lower bound is given by $$ \frac{C}{3} $$ C 3 log 2 times the number of gaps between two boundary regions in vacuum states. We discuss the limitation and possible generalization of the boundary counting, and its relation to tripartite entanglement.

DOI
10.1007/jhep03(2023)043

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Abstract The Markov gap [1], namely the difference between reflected entropy and mutual information, is explicitly computed in the defect extremal surface model, JT gravity, and the generic 2d extremal black holes, in vacuum states. The phases that contain various island contributions are considered, and their existence is carefully checked. Moreover, we show explicitly how the Markov gap originates from the OPE coefficient of the boundary CFT. And, as a generalization of [1], the lower bound of the Markov gap is given by C3\frac{C}{3}3C​ C 3 log 2 times the number of EWCS boundaries on minimal surfaces. We propose a boundary way of counting the lower bound for the Markov gap, which states that the lower bound is given by C3\frac{C}{3}3C​ C 3 log 2 times the number of gaps between two boundary regions in vacuum states. We discuss the limitation and possible generalization of the boundary counting, and its relation to tripartite entanglement.
A bstract The reflected entropy S R ( A : B ) of a density matrix ρ AB 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- N 2 gap between S R and I . We provide an information-theoretic interpretation of this gap by observing that S R − 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 S R − I the Markov gap . We then prove that for time-symmetric states in pure AdS 3 gravity, the Markov gap is universally lower bounded by log(2) ℓ AdS / 2 G N 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 S R − 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.
We study how mixed-state correlations reorganize across island transitions in Jackiw-Teitelboim gravity coupled to a non-gravitating bath. Using the partial-dimensional-reduction realization of double holography, we compute reflected entropy for adjacent and disjoint radiation regions in the vacuum and in the thermofield-double state of an eternal black hole. Reflected entropy probes the internal partition of the entanglement wedge and therefore resolves structure that is invisible to the ordinary Page curve: its dominant saddle can change even when the entropy saddle of the union remains fixed, and it vanishes at leading order when the entanglement wedge disconnects. We show that the lower-dimensional reflected-island description agrees, to leading order in the dynamical brane fluctuation, with the higher-dimensional defect-extremal-surface picture. The finite-angle comparison further suggests a projection-covariant identification of island and brane endpoints at linear order. The resulting Page-like curves and Markov gaps provide a geometric probe of the reorganization of the purification and of non-Markovian, multipartite correlation structure during island transitions.
A bstract We consider entanglement entropies of finite spatial intervals in Minkowski radiation baths coupled to the eternal black hole in JT gravity, and the related problem involving free fermion BCFT in the thermofield double state. We show that the non-monotonic entropy evolution in the black hole problem precisely matches that of the free fermion theory in a high temperature limit, and the results have the form expected for CFTs with quasiparticle description. Both exhibit rich behaviour that involves at intermediate times, an entropy saddle with an island in the former case, and in the latter a special class of disconnected OPE channels. The quantum extremal surfaces start inside the horizon, but can emerge from and plunge back inside as time evolves, accompanied by a characteristic dip in the entropy also seen in the free fermion BCFT. Finally an entropy equilibrium is reached with a no-island saddle.
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A bstract Defect extremal surface is defined by extremizing the Ryu-Takayanagi formula corrected by the quantum defect theory. This is interesting when the AdS bulk contains a defect brane (or string). We introduce a defect extremal surface formula for reflected entropy, which is a mixed state generalization of entanglement entropy measure. Based on a decomposition procedure of an AdS bulk with a brane, we demonstrate the equivalence between defect extremal surface formula and island formula for reflected entropy in AdS 3 /BCFT 2 . We also compute the evolution of reflected entropy in evaporating black hole model and find that defect extremal surface formula agrees with island formula.