arXiv:2204.08290

Effective reflected entropy and entanglement negativity for general 2D eternal black holes

Jiong Lin, Yizhou Lu

Published Apr 18, 2022• 3 citations

Abstract

Both reflected entropy and entanglement negativity provide valid measures of entanglement between subsystems of a mixed state. For general 2D eternal black holes coupled with CFT matters in large $c$ limit, we perform the replica-trick computation and find that both effective Renyi reflected entropy and effective entanglement negativity can be expressed in terms of the combination of modified backreacting cosmic branes in $\mathrm{AdS}_3$ bulk. We then develop a holographic scheme to calculate effective reflected entropy and entanglement negativity for general 2D eternal black holes coupled with CFT matters in large $c$ limit. Using the scheme, we check the consistency condition of the island formulae for entanglement negativity and reflected entropy. We find that the combinations of modified backreacting cosmic branes in $\mathrm{AdS}_3$ bulk from the two island proposals of entanglement negativity exactly match with each other. Finally, we study the saturation of the reflected entropy inequality.

DOI: 10.48550/arxiv.2204.08290

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arXiv:2204.08290

Effective reflected entropy and entanglement negativity for general 2D eternal black holes

Jiong Lin, Yizhou Lu

Published Apr 18, 2022• 3 citations

arXiv HTML PDF

Abstract

Both reflected entropy and entanglement negativity provide valid measures of entanglement between subsystems of a mixed state. For general 2D eternal black holes coupled with CFT matters in large

c

limit, we perform the replica-trick computation and find that both effective Renyi reflected entropy and effective entanglement negativity can be expressed in terms of the combination of modified backreacting cosmic branes in

\mathrm{AdS}_3

bulk. We then develop a holographic scheme to calculate effective reflected entropy and entanglement negativity for general 2D eternal black holes coupled with CFT matters in large

c

limit. Using the scheme, we check the consistency condition of the island formulae for entanglement negativity and reflected entropy. We find that the combinations of modified backreacting cosmic branes in

\mathrm{AdS}_3

bulk from the two island proposals of entanglement negativity exactly match with each other. Finally, we study the saturation of the reflected entropy inequality.

DOI: 10.48550/arxiv.2204.08290

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Xi Dong, Jonah Kudler-Flam, Pratik Rath

A bstract The entanglement negativity $\mathcal{E}$ E ( A : B ) is a useful measure of quantum entanglement in bipartite mixed states. In random tensor networks (RTNs), which are related to fixed-area states, it was found in ref. [1] that the dominant saddles computing the even Rényi negativity ${\mathcal{E}}^{(2k)}$ E 2 k generically break the ℤ 2 k replica symmetry. This calls into question previous calculations of holographic negativity using 2D CFT techniques that assumed ℤ 2 k replica symmetry and proposed that the negativity was related to the entanglement wedge cross section. In this paper, we resolve this issue by showing that in general holographic states, the saddles computing ${\mathcal{E}}^{(2k)}$ E 2 k indeed break the ℤ 2 k replica symmetry. Our argument involves an identity relating ${\mathcal{E}}^{(2k)}$ E 2 k to the k -th Rényi entropy on subregion AB ∗ in the doubled state ${\left.|{\rho}_{AB}\right\rangle}_{A{A}^{\ast }{BB}^{\ast }}$ ρ AB A A ∗ BB ∗ , from which we see that the ℤ 2 k replica symmetry is broken down to ℤ k . For k < 1, which includes the case of $\mathcal{E}$ E ( A : B ) at k = 1/2, we use a modified cosmic brane proposal to derive a new holographic prescription for ${\mathcal{E}}^{(2k)}$ E 2 k and show that it is given by a new saddle with multiple cosmic branes anchored to subregions A and B in the original state. Using our prescription, we reproduce known results for the PSSY model and show that our saddle dominates over previously proposed CFT calculations near k = 1. Moreover, we argue that the ℤ 2 k symmetric configurations previously proposed are not gravitational saddles, unlike our proposal. Finally, we contrast holographic calculations with those arising from RTNs with non-maximally entangled links, demonstrating that the qualitative form of backreaction in such RTNs is different from that in gravity.A bstract The entanglement negativity $$ \mathcal{E} $$ E ( A : B ) is a useful measure of quantum entanglement in bipartite mixed states. In random tensor networks (RTNs), which are related to fixed-area states, it was found in ref. [1] that the dominant saddles computing the even Rényi negativity $$ {\mathcal{E}}^{(2k)} $$ E 2 k generically break the ℤ 2 k replica symmetry. This calls into question previous calculations of holographic negativity using 2D CFT techniques that assumed ℤ 2 k replica symmetry and proposed that the negativity was related to the entanglement wedge cross section. In this paper, we resolve this issue by showing that in general holographic states, the saddles computing $$ {\mathcal{E}}^{(2k)} $$ E 2 k indeed break the ℤ 2 k replica symmetry. Our argument involves an identity relating $$ {\mathcal{E}}^{(2k)} $$ E 2 k to the k -th Rényi entropy on subregion AB ∗ in the doubled state $$ {\left.|{\rho}_{AB}\right\rangle}_{A{A}^{\ast }{BB}^{\ast }} $$ ρ AB A A ∗ BB ∗ , from which we see that the ℤ 2 k replica symmetry is broken down to ℤ k . For k < 1, which includes the case of $$ \mathcal{E} $$ E ( A : B ) at k = 1/2, we use a modified cosmic brane proposal to derive a new holographic prescription for $$ {\mathcal{E}}^{(2k)} $$ E 2 k and show that it is given by a new saddle with multiple cosmic branes anchored to subregions A and B in the original state. Using our prescription, we reproduce known results for the PSSY model and show that our saddle dominates over previously proposed CFT calculations near k = 1. Moreover, we argue that the ℤ 2 k symmetric configurations previously proposed are not gravitational saddles, unlike our proposal. Finally, we contrast holographic calculations with those arising from RTNs with non-maximally entangled links, demonstrating that the qualitative form of backreaction in such RTNs is different from that in gravity.

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Mir Afrasiar, Jaydeep Kumar Basak, Ashish Chandra +1 more

Abstract We obtain the reflected entropy for bipartite mixed state configurations of two adjacent and disjoint intervals at a finite temperature in finite-sized non-gravitating reservoirs described by $CFT_2$ C F T 2 s each coupled to two quantum dots at their boundaries in the large central charge limit through a replica technique. These field theory results are substantiated through a holographic computation involving the entanglement wedge cross section in the dual bulk BTZ black hole geometry truncated by two Planck branes. The two Planck branes are the holographic duals of the quantum dots described by $AdS_2$ A d S 2 slices with JT black holes. Our result reproduce the holographic duality between the reflected entropy and the bulk entanglement wedge cross section in the context of the $AdS_3/CFT_2$ A d S 3 / C F T 2 correspondence. Subsequently we analyze the behaviour of the holographic Markov gap between the reflected entropy and the mutual information for different scenarios involving the subsystem sizes and time.Abstract We obtain the reflected entropy for bipartite mixed state configurations of two adjacent and disjoint intervals at a finite temperature in finite-sized non-gravitating reservoirs described by $$CFT_2$$ C F T 2 s each coupled to two quantum dots at their boundaries in the large central charge limit through a replica technique. These field theory results are substantiated through a holographic computation involving the entanglement wedge cross section in the dual bulk BTZ black hole geometry truncated by two Planck branes. The two Planck branes are the holographic duals of the quantum dots described by $$AdS_2$$ A d S 2 slices with JT black holes. Our result reproduce the holographic duality between the reflected entropy and the bulk entanglement wedge cross section in the context of the $$AdS_3/CFT_2$$ A d S 3 / C F T 2 correspondence. Subsequently we analyze the behaviour of the holographic Markov gap between the reflected entropy and the mutual information for different scenarios involving the subsystem sizes and time.

arXiv:2302.12810The European Physical Journal CMar 9, 20246 citations

Both reflected entropy and entanglement negativity provide valid measures of entanglement between subsystems of a mixed state. For general 2D eternal black holes coupled with CFT matters in large

c

\mathrm{AdS}_3

bulk. We then develop a holographic scheme to calculate effective reflected entropy and entanglement negativity for general 2D eternal black holes coupled with CFT matters in large

c

\mathrm{AdS}_3

bulk from the two island proposals of entanglement negativity exactly match with each other. Finally, we study the saturation of the reflected entropy inequality.

A bstract The entanglement negativity

\mathcal{E}

E ( A : B ) is a useful measure of quantum entanglement in bipartite mixed states. In random tensor networks (RTNs), which are related to fixed-area states, it was found in ref. [1] that the dominant saddles computing the even Rényi negativity

{\mathcal{E}}^{(2k)}

E 2 k generically break the ℤ 2 k replica symmetry. This calls into question previous calculations of holographic negativity using 2D CFT techniques that assumed ℤ 2 k replica symmetry and proposed that the negativity was related to the entanglement wedge cross section. In this paper, we resolve this issue by showing that in general holographic states, the saddles computing

{\mathcal{E}}^{(2k)}

E 2 k indeed break the ℤ 2 k replica symmetry. Our argument involves an identity relating

{\mathcal{E}}^{(2k)}

E 2 k to the k -th Rényi entropy on subregion AB ∗ in the doubled state

{\left.|{\rho}_{AB}\right\rangle}_{A{A}^{\ast }{BB}^{\ast }}

ρ AB A A ∗ BB ∗ , from which we see that the ℤ 2 k replica symmetry is broken down to ℤ k . For k < 1, which includes the case of

\mathcal{E}

E ( A : B ) at k = 1/2, we use a modified cosmic brane proposal to derive a new holographic prescription for

{\mathcal{E}}^{(2k)}

E 2 k and show that it is given by a new saddle with multiple cosmic branes anchored to subregions A and B in the original state. Using our prescription, we reproduce known results for the PSSY model and show that our saddle dominates over previously proposed CFT calculations near k = 1. Moreover, we argue that the ℤ 2 k symmetric configurations previously proposed are not gravitational saddles, unlike our proposal. Finally, we contrast holographic calculations with those arising from RTNs with non-maximally entangled links, demonstrating that the qualitative form of backreaction in such RTNs is different from that in gravity.

CFT_2

C F T 2 s each coupled to two quantum dots at their boundaries in the large central charge limit through a replica technique. These field theory results are substantiated through a holographic computation involving the entanglement wedge cross section in the dual bulk BTZ black hole geometry truncated by two Planck branes. The two Planck branes are the holographic duals of the quantum dots described by

AdS_2

A d S 2 slices with JT black holes. Our result reproduce the holographic duality between the reflected entropy and the bulk entanglement wedge cross section in the context of the

AdS_3/CFT_2

A d S 3 / C F T 2 correspondence. Subsequently we analyze the behaviour of the holographic Markov gap between the reflected entropy and the mutual information for different scenarios involving the subsystem sizes and time.