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arXiv:2002.10272 · Universe

On Minimal Entanglement Wedge Cross Section for Holographic Entanglement Negativity

Jaydeep Kumar Basak, Vinay Malvimat, Himanshu Parihar, Boudhayan Paul, Gautam Sengupta

Published Mar 5, 2024• 25 citations
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Abstract

We demonstrate the equivalence of two different conjectures in the literature for the holographic entanglement negativity in AdS3/CFT2, modulo certain constants. These proposals involve certain algebraic sums of bulk geodesics homologous to specific combinations of subsystems, and the entanglement wedge cross section (EWCS) backreacted by a cosmic brane for the conical defect geometry in the bulk gravitational path integral. It is observed that the former conjectures reproduce the field theory replica technique results in the large central charge limit whereas the latter involves constants related to the Markov gap. In this context, we establish an alternative construction for the EWCS of a single interval in a CFT2 at a finite temperature to resolve an issue for the latter proposal involving thermal entropy elimination for holographic entanglement negativity. Our construction for the EWCS correctly reproduces the corresponding field theory results modulo the Markov gap constant in the large central charge limit.

DOI
10.3390/universe10030125

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We demonstrate the equivalence of two different conjectures in the literature for the holographic entanglement negativity in AdS3/CFT2, modulo certain constants. These proposals involve certain algebraic sums of bulk geodesics homologous to specific combinations of subsystems, and the entanglement wedge cross section (EWCS) backreacted by a cosmic brane for the conical defect geometry in the bulk gravitational path integral. It is observed that the former conjectures reproduce the field theory replica technique results in the large central charge limit whereas the latter involves constants related to the Markov gap. In this context, we establish an alternative construction for the EWCS of a single interval in a CFT2 at a finite temperature to resolve an issue for the latter proposal involving thermal entropy elimination for holographic entanglement negativity. Our construction for the EWCS correctly reproduces the corresponding field theory results modulo the Markov gap constant in the large central charge limit.
We propose a covariant holographic construction based on the extremal entanglement wedge cross section, for the entanglement negativity of bipartite states in CFT1+1s dual to non static bulk AdS3 geometries. Utilizing our proposal we obtain the holographic entanglement negativity for bipartite mixed states in CFT1+1s dual to bulk extremal and non extremal rotating BTZ black holes and the time dependent Vaidya-AdS3 geometries. Our results correctly reproduce the replica technique results modulo certain constants for the corresponding dual CFT1+1s in the large central charge limit for the bulk black hole geometries. Furthermore for all the cases our results also match upto such constants with the corresponding results from an earlier alternate covariant holographic construction in the literature. This demonstrates the equivalence modulo constants of the two constructions for the AdS3/CFT2 scenario.
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We propose a holographic entanglement negativity conjecture involving the bulk geometry, for mixed states of adjacent intervals in ( 1 + 1 ) -dimensional dual conformal field theories through the A d S / C F T correspondence. The holographic entanglement negativity is obtained from a specific algebraic sum of the geodesics anchored on respective intervals on the boundary which reduces to the holographic mutual information between them. Utilizing our conjecture we obtain the entanglement negativity of adjacent intervals in zero and finite temperature ( 1 + 1 )-dimensional holographic conformal field theories dual to the bulk A d S 3 vacuum and the Euclidean BTZ black hole respectively. Our holographic conjecture exactly reproduces the conformal field theory results obtained through the replica technique, in the large central charge limit. We briefly elucidate the corresponding issue for the A d S d + 1 / C F T d scenario.
Holographic entanglement negativity for disjoint intervals in AdS3/CFT2AdS_3/CFT_2AdS3​/CFT2​ A d S 3 / C F T 2
We advance a holographic construction for the entanglement negativity of bipartite mixed state configurations of two disjoint intervals in (1+1)(1+1)(1+1) dimensional conformal field theories ( CFT1+1CFT_{1+1}CFT1+1​ ) through the AdS3/CFT2AdS_3/CFT_2AdS3​/CFT2​ correspondence. Our construction constitutes the large central charge analysis of the entanglement negativity for mixed states under consideration and involves a specific algebraic sum of bulk space like geodesics anchored on appropriate intervals in the dual CFT1+1CFT_{1+1}CFT1+1​ . The construction is utilized to compute the holographic entanglement negativity for such mixed states in CFT1+1CFT_{1+1}CFT1+1​ s dual to bulk pure AdS3AdS_3AdS3​ geometries and BTZ black holes respectively. Our analysis exactly reproduces the universal features of corresponding replica technique results in the large central charge limit which serves as a consistency check.
We calculate logarithmic negativity, a quantum entanglement measure for mixed quantum states, in quantum error-correcting codes and find it to equal the minimal cross sectional area of the entanglement wedge in holographic codes with a quantum correction term equal to the logarithmic negativity between the bulk degrees of freedom on either side of the entanglement wedge cross section. This leads us to conjecture a holographic dual for logarithmic negativity that is related to the area of a cosmic brane with tension in the entanglement wedge plus a quantum correction term. This is closely related to (though distinct from) the holographic proposal for entanglement of purification. We check this relation for various configurations of subregions in AdS3/CFT2{\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}AdS3​/CFT2​. These are disjoint intervals at zero temperature, as well as a single interval and adjacent intervals at finite temperature. We also find this prescription to effectively characterize the thermofield double state. We discuss how a deformation of a spherical entangling region complicates calculations and speculate how to generalize to a covariant description.
Abstract We consider the minimal area of the entanglement wedge cross section (EWCS) in Einstein gravity. In the context of holography, it is proposed that this quantity is dual to different information measures, e.g., entanglement of purification, logarithmic negativity and reflected entropy. Motivated by these proposals, we examine in detail the low and high temperature corrections to this quantity and show that it obeys the area law even in the finite temperature. We also study EWCS in nonrelativistic field theories with nontrivial Lifshitz and hyperscaling violating exponents. The resultant EWCS is an increasing function of the dynamical exponent due to the enhancement of spatial correlations between subregions for larger values of z. We find that EWCS is monotonically decreasing as the hyperscaling violating exponent increases. We also obtain this quantity for an entangling region with singular boundary in a three dimensional field theory and find a universal contribution where the coefficient depends on the central charge. Finally, we verify that for higher dimensional singular regions the corresponding EWCS obeys the area law.