We study two disjoint universes in an entangled pure state. When only one
universe contains gravity, the path integral for the $n^{\text{th } }$ R\'enyi
entropy includes a wormhole between the $n$ copies of the gravitating universe,
leading to a standard "island formula" for entanglement entropy consistent with
unitarity of quantum information. When both universes contain gravity,
gravitational corrections to this configuration lead to a violation of
unitarity. However, the path integral is now dominated by a novel wormhole with
$2n$ boundaries connecting replica copies of both universes. The analytic
continuation of this contribution involves a quotient by $\mathbb{Z}_n$ replica
symmetry, giving a cylinder connecting the two universes. When entanglement is
large, this configuration has an effective description as a "swap wormhole", a
geometry in which the boundaries of the two universes are glued together by a
"swaperator". This description allows precise computation of a generalized
entropy-like formula for entanglement entropy. The quantum extremal surface
computing the entropy lives on the Lorentzian continuation of the cylinder/swap
wormhole, which has a connected Cauchy slice stretching between the universes
-- a realization of the ER=EPR idea. The new wormhole restores unitarity of
quantum information.

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Other Link： http://arxiv.org/pdf/2104.13383v1

We study a thermofield double type entangled state on two disjoint universes
$A$ and $B$, where one of the universes is asymptotically flat containing a
black hole. As we increase the entanglement temperature, this black hole
receives back-reaction from the stress energy tensor of the state. This results
in lengthening of the wormhole region in the black hole interior, and
decreasing of its horizon area, both of which are key features of an
evaporating black hole. We then compute the entanglement entropy on the
universe $A$ through the island formula, and argue that it naturally follows
the Page curve of an evaporating black hole in flat space. We also study the
effects of local operations in the gravitating universe with the black hole. We
find that they accelerate the evaporation of the black hole, therefore disrupt
the entanglement between two universes. Furthermore, we observe that depending
on whether the operation can be regarded as an LOCC or not, the behavior of the
entanglement entropy changes. In particular, when the operation is made neither
in the entanglement wedge of the radiation system or that of the black hole,
the transition between the island phase and the no-island phase can happen
multiple times.

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Other Link： http://arxiv.org/pdf/2104.00183v1

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

<title>A<sc>bstract</sc>
</title>We study a class of decoherence process which admits a 3 dimensional holographic bulk. Starting from a thermo-field double dual to a wormhole, we prepare another thermo-field double which plays the role of environment. By allowing the energy flow between the original and environment thermo-field double, the entanglement of the original thermo-field double eventually decoheres. We model this decoherence by four-boundary wormhole geometries, and study the time-evolution of the moduli parameters to see the change of the entanglement pattern among subsystems. A notable feature of this holographic decoherence processes is that at the end point of the processes, the correlations of the original thermo-field double are lost completely both classically and also quantum mechanically. We also discuss distinguishability between thermo-field double state and thermo mixed double state, which contains only classical correlations, and construct a code subspace toy model for that.

DOI： 10.1007/jhep03(2021)214

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Other Link： http://link.springer.com/article/10.1007/JHEP03(2021)214/fulltext.html

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

<title>A<sc>bstract</sc>
</title>We consider black holes in 2d de Sitter JT gravity coupled to a CFT, and entangled with matter in a disjoint non-gravitating universe. Tracing out the entangling matter leaves the CFT in a density matrix whose stress tensor backreacts on the de Sitter geometry, lengthening the wormhole behind the black hole horizon. Naively, the entropy of the entangling matter increases without bound as the strength of the entanglement increases, but the monogamy property predicts that this growth must level off. We compute the entropy via the replica trick, including wormholes between the replica copies of the de Sitter geometry, and find a competition between conventional field theory entanglement entropy and the surface area of extremal “islands” in the de Sitter geometry. The black hole and cosmological horizons both play a role in generating such islands in the backreacted geometry, and have the effect of stabilizing the entropy growth as required by monogamy. We first show this in a scenario in which the de Sitter spatial section has been decompactified to an interval. Then we consider the compact geometry, and argue for a novel interpretation of the island formula in the context of closed universes that recovers the Page curve. Finally, we comment on the application of our construction to the cosmological horizon in empty de Sitter space.

DOI： 10.1007/jhep02(2021)072

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

<title>A<sc>bstract</sc>
</title>We consider a black hole in three dimensional AdS space entangled with an auxiliary radiation system. We model the microstates of the black hole in terms of a field theory living on an end of the world brane behind the horizon, and allow this field theory to itself have a holographic dual geometry. This geometry is also a black hole since entanglement of the microstates with the radiation leaves them in a mixed state. This “inception black hole” can be purified by entanglement through a wormhole with an auxiliary system which is naturally identified with the external radiation, giving a realization of the ER=EPR scenario. In this context, we propose an extension of the Ryu-Takayanagi (RT) formula, in which extremal surfaces computing entanglement entropy are allowed to pass through the brane into its dual geometry. This new rule reproduces the Page curve for evaporating black holes, consistently with the recently proposed “island formula”. We then separate the radiation system into pieces. Our extended RT rule shows that the entanglement wedge of the union of radiation subsystems covers the black hole interior at late times, but the union of entanglement wedges of the subsystems may not. This result points to a secret sharing scheme in Hawking radiation wherein reconstruction of certain regions in the interior is impossible with any subsystem of the radiation, but possible with all of it.

DOI： 10.1007/jhep01(2021)177

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Other Link： https://link.springer.com/article/10.1007/JHEP01(2021)177/fulltext.html

We holographically compute the R\'enyi relative divergence $D_{\alpha}
(\rho_{+} || \rho_{-})$ between two density matrices $\rho_{+}, \; \rho_{-}$
prepared by path integrals with constant background fields $\lambda_{\pm}$
coupled to a marginal operator in JT gravity. Our calculation is non
perturbative in the difference between two sources $ \lambda_{+} -\lambda_{-}$.
When this difference is large, the bulk geometry becomes a black hole with the
maximal temperature allowed by the R\'enyi index $\alpha$. In this limit, we
find an analytic expression of the R\'enyi relative divergence, which is given
by the on shell action of the back reacted black hole plus the contribution
coming from the discontinuous change of the background field.

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Other Link： http://arxiv.org/pdf/2011.05539v1

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

<title>A<sc>bstract</sc>
</title>We extend a 2d topological model of the gravitational path integral to include sums over spin structure, corresponding to Neveu-Schwarz (NS) or Ramond (R) boundary conditions for fermions. This path integral corresponds to a correlator of boundary creation operators on a non-trivial baby universe Hilbert space, and vanishes when the number of R boundaries is odd. This vanishing implies a non-factorization of the correlator, which necessitates a dual interpretation of the bulk path integral in terms of a product of partition functions (associated to NS boundaries) and Witten indices (associated to R boundaries), averaged over an ensemble of theories with varying Hilbert space dimension and different numbers of bosonic and fermionic states. We also consider a model with End-of-the-World (EOW) branes, for which the dual ensemble then includes a sum over randomly chosen fermionic and bosonic states. We propose two modifications of the bulk path integral which restore an interpretation in a single dual theory: (i) a geometric prescription where we add extra boundaries with a sum over their spin structures, and (ii) an algebraic prescription involving “spacetime D-branes”. We extend our ideas to Jackiw-Teitelboim gravity, and propose a dual description of a single unitary theory with spin structure in a system with eigenbranes.

DOI： 10.1007/jhep09(2020)192

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We use the replica method to compute the entanglement entropy of a universe
without gravity entangled in a thermofield-double-like state with a disjoint
gravitating universe. Including wormholes between replicas of the latter gives
an entropy functional which includes an "island" on the gravitating universe.
We solve the back-reaction equations when the cosmological constant is negative
to show that this island coincides with a causal shadow region that is created
by the entanglement in the gravitating geometry. At high entanglement
temperatures, the island contribution to the entropy functional leads to a
bound on entanglement entropy, analogous to the Page behavior of evaporating
black holes. We demonstrate that the entanglement wedge of the non-gravitating
universe grows with the entanglement temperature until, eventually, the
gravitating universe can be entirely reconstructed from the non-gravitating
one.

DOI： 10.1007/JHEP02(2021)136

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Other Link： http://arxiv.org/pdf/2008.05274v2

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/jhep06(2020)053

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

Relative entropy of entanglement (REE) is an entanglement measure of bipartite mixed states, defined by the minimum of the relative entropy S((AB) ||sigma(AB) ) between a given mixed state (AB) and an arbitrary separable state sigma(AB) . The REE is always bounded by the mutual information I-AB = S((AB) ||(A) circle times (B)) because the latter measures not only quantum entanglement but also classical correlations. In this paper we address the question of to what extent REE can be small compared to the mutual information in conformal field theories (CFTs). For this purpose, we perturbatively compute the relative entropy between the vacuum reduced density matrix AB0 on disjoint subsystems A B and arbitrarily separable state sigma(AB) in the limit where two subsystems A and B are well separated, then minimize the relative entropy with respect to the separable states. We argue that the result highly depends on the spectrum of CFT on the subsystems. When we have a few low energy spectrum of operators as in the case where the subsystems consist of finite number of spins in spin chain models, the REE is considerably smaller than the mutual information. However in general our perturbative scheme breaks down, and the REE can be as large as the mutual information.

DOI： 10.1007/jhep10(2018)166

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DOI： 10.1007/jhep07(2018)002

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/jhep01(2018)012

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/jhep10(2017)184

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Publishing type：Research paper (scientific journal)   Publisher：IOP Publishing  

DOI： 10.1088/1742-5468/aa85c1

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Other Link： http://stacks.iop.org/1742-5468/2017/i=9/a=093104?key=crossref.fbdd76e38be70ea8b2541a39620c4fb9

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/jhep04(2017)156

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/jhep02(2017)060

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/jhep07(2016)114

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Time-dependent entanglement entropy (EE) is computed for a single interval in
two-dimensional conformal theories from a quenched initial state in the
presence of spatial boundaries. The EE is found to be periodic in time with
periodicity equal to the system size $L$. For large enough $L$, the EE shows a
rise to a thermal value (characterized by a temperature $1/\beta$ determined by
the initial state), followed by periodic returns to the original value. This
works irrespective of whether the conformal field theory (CFT) is rational or
irrational. For conformal field theories with a holographic dual, the large $c$
limit plays an essential role in ensuring that the EE computed from the CFT is
universal (independent of the details of the CFT and of boundary conditions)
and is exactly matched by the holographic EE. The dual geometry is computed and
it interpolates between a BTZ black hole at large $L$ and global AdS at large
$\beta$.

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Other Link： http://arxiv.org/pdf/1604.07830v1

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/jhep07(2015)080

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Other Link： http://link.springer.com/content/pdf/10.1007/JHEP07(2015)080.pdf

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/jhep12(2014)048

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We propose a holographic realization of quantum quenches in two dimensional
conformal field theories. In particular, we discuss time evolutions of
holographic entanglement entropy in these backgrounds and compare them with CFT
results. The key ingredient of the construction is an introduction of a
spacetime boundary into bulk geometries, which is the gravity counterpart of a
boundary state in the dual CFT. We consider several examples, including local
quenches and an inhomogeneous quench which is dual to fusion of two black
string into the third one.

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Other Link： http://arxiv.org/pdf/1311.2562v1

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/jhep05(2013)160

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Other Link： http://link.springer.com/article/10.1007/JHEP05(2013)160/fulltext.html

Publishing type：Research paper (scientific journal)   Publisher：American Physical Society (APS)  

DOI： 10.1103/physrevlett.110.091602

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Other Link： http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.110.091602/fulltext

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DOI： 10.1007/jhep06(2012)066

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

In this paper, we study a holographic dual of a confined Fermi liquid state by putting a charged fluid of fermions in the AdS soliton geometry. This can be regarded as a confined analogue of electron stars. Depending on the parameters such as the mass and charge of the bulk fermion field, we found three different phase structures when we change the values of total charge density at zero temperature. In one of the three cases, our confined solution (called soliton star) is always stable and this solution approaches to the electron star away from the tip. In both the second and third case, we find a confinement/deconfinement phase transition. Moreover, in the third one, there is a strong indication that the soliton star decays into an inhomogeneous solution. We also analyze the probe fermion equations (in the WKB approximation) in the background of this soliton star geometry to confirm the presence of many Fermi-surfaces in the system.

DOI： 10.1007/jhep02(2012)137

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

We argue that Landau-Fermi liquids do not have any gravity duals in the purely classical limit. We employ the logarithmic behavior of entanglement entropy to characterize the existence of Fermi surfaces. By imposing the null energy condition, we show that the specific heat always behaves anomalously. We also present a classical gravity dual which has the expected behavior of the entanglement entropy and specific heat for non-Fermi liquids.

DOI： 10.1007/jhep01(2012)125

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Other Link： http://link.springer.com/article/10.1007/JHEP01(2012)125/fulltext.html

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/jhep04(2011)115

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Other Link： http://link.springer.com/article/10.1007/JHEP04(2011)115/fulltext.html

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

We argue that the entanglement entropy offers us a useful coarse-grained entropy in time-dependent AdS/CFT. We show that the total von-Neumann entropy remains vanishing even when a black hole is created in a gravity dual, being consistent with the fact that its corresponding CFT is described by a time-dependent pure state. We analytically calculate the time evolution of entanglement entropy for a free Dirac fermion on a circle following a quantum quench. This is interpreted as a toy holographic dual of black hole creations and annihilations. It is manifestly free from the black hole information problem.

DOI： 10.1007/jhep11(2010)054

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Other Link： http://link.springer.com/article/10.1007/JHEP11(2010)054/fulltext.html