Pseudoentropy in dS/CFT and Timelike Entanglement Entropy.

We study holographic entanglement entropy in dS/CFT and introduce timelike entanglement entropy in CFTs. Both of them take complex values in general and are related with each other via an analytical continuation. We argue that they are correctly understood as pseudoentropy. We find that the imaginary part of pseudoentropy implies an emergence of time in dS/CFT.

Holography in de Sitter Space via Chern-Simons Gauge Theory.

In this Letter, we propose a holographic duality for classical gravity on a three-dimensional de Sitter space. We first show that a pair of SU(2) Chern-Simons gauge theories reproduces the classical partition function of Einstein gravity on a Euclidean de Sitter space, namely S^{3}, when we take the limit where the level k approaches -2. This implies that the conformal field theory (CFT) dual of gravity on a de Sitter space at the leading semiclassical order is given by an SU(2) Wess-Zumino-Witten model in the large central charge limit k→-2. We give another evidence for this in the light of known holography for coset CFTs. We also present a higher spin gravity extension of our duality.

Pseudo-Entropy in Free Quantum Field Theories.

Pseudo-entropy is an interesting quantity with a simple gravity dual, which generalizes entanglement entropy such that it depends on both an initial and a final state. Here we reveal the basic properties of pseudo-entropy in quantum field theories by numerically calculating this quantity for a set of two-dimensional free-scalar field theories and the Ising spin chain. We extend the Gaussian method for pseudo-entropy in free-scalar theories with two parameters: mass m and dynamical exponent z. This computation finds two novel properties of pseudo-entropy which we conjecture to be universal in field theories, in addition to an area law behavior. One is a saturation behavior and the other one is nonpositivity of the difference between pseudo-entropy and averaged entanglement entropy. Moreover, our numerical results for the Ising chain imply that pseudo-entropy can play a role as a new quantum order parameter which detects whether two states are in the same quantum phase or not.

Entanglement Entropy in a Holographic Moving Mirror and the Page Curve.

We calculate the time evolution of entanglement entropy in two-dimensional conformal field theory with a moving mirror. For a setup modeling Hawking radiation, we obtain a linear growth of entanglement entropy and show that this can be interpreted as the production of entangled pairs. For the setup, which mimics black hole formation and evaporation, we find that the evolution follows the ideal Page curve. We perform these computations by constructing the gravity dual of the moving mirror model via holography. We also argue that our holographic setup provides a concrete model to derive the Page curve for black hole radiation in the strong coupling regime of gravity.

Entanglement Wedges from the Information Metric in Conformal Field Theories.

We present a new method of deriving the geometry of entanglement wedges in holography directly from conformal field theories (CFTs). We analyze an information metric called the Bures metric of reduced density matrices for locally excited states. This measures the distinguishability of states with different points excited. For a subsystem given by an interval, we precisely reproduce the expected entanglement wedge for two-dimensional holographic CFTs from the Bures metric, which turns out to be proportional to the anti-de Sitter metric on a time slice. On the other hand, for free scalar CFTs, we do not find any sharp structures like entanglement wedges. When a subsystem consists of two disconnected intervals, we manage to reproduce the expected entanglement wedge from holographic CFTs with the correct phase transitions, up to a very small error, from a quantity alternative to the Bures metric.

Entanglement of Purification in Many Body Systems and Symmetry Breaking.

We study the entanglement of purification (EOP), a measure of total correlation between two subsystems A and B, for free scalar field theory on a lattice and the transverse-field Ising model by numerical methods. In both of these models, we find that the EOP becomes a nonmonotonic function of the distance between A and B when the total number of lattice sites is small. When it is large, the EOP becomes monotonic and shows a plateaulike behavior. Moreover, we also show that the original reflection symmetry which exchanges A and B can get broken in optimally purified systems. We provide an interpretation of our results in terms of the interplay between classical and quantum correlations.

Holographic Entanglement of Purification from Conformal Field Theories.

We explore a conformal field theoretic interpretation of the holographic entanglement of purification, which is defined as the minimal area of the entanglement wedge cross section. We argue that, in AdS_{3}/CFT_{2}, the holographic entanglement of purification agrees with the entanglement entropy for a purified state, obtained from a special Weyl transformation, called path-integral optimizations. By definition, this special purified state has minimal path-integral complexity. We confirm this claim in several examples.

Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories.

We introduce a new optimization procedure for Euclidean path integrals, which compute wave functionals in conformal field theories (CFTs). We optimize the background metric in the space on which the path integration is performed. Equivalently, this is interpreted as a position-dependent UV cutoff. For two-dimensional CFT vacua, we find the optimized metric is given by that of a hyperbolic space, and we interpret this as a continuous limit of the conjectured relation between tensor networks and Anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. We confirm our procedure for excited states, the thermofield double state, the Sachdev-Ye-Kitaev model, and discuss its extension to higher-dimensional CFTs. We also show that when applied to reduced density matrices, it reproduces entanglement wedges and holographic entanglement entropy. We suggest that our optimization prescription is analogous to the estimation of computational complexity.

Continuous Multiscale Entanglement Renormalization Ansatz as Holographic Surface-State Correspondence.

We present how the surface-state correspondence, conjectured by Miyaji and Takayanagi, works in the setup of AdS(3)/CFT(2) by generalizing the formulation of a continuous multiscale entanglement renormalization group ansatz. The boundary states in conformal field theories play a crucial role in our formulation and the bulk diffeomorphism is naturally taken into account. We give an identification of bulk local operators which reproduces correct scalar field solutions on AdS(3) and bulk scalar propagators. We also calculate the information metric for a locally excited state and show that it reproduces the time slice of AdS(3).

Distance between Quantum States and Gauge-Gravity Duality.

We study a quantum information metric (or fidelity susceptibility) in conformal field theories with respect to a small perturbation by a primary operator. We argue that its gravity dual is approximately given by a volume of maximal time slice in an anti-de Sitter spacetime when the perturbation is exactly marginal. We confirm our claim in several examples.

Quantum entanglement of local operators in conformal field theories.

We introduce a series of quantities which characterize a given local operator in any conformal field theory from the viewpoint of quantum entanglement. It is defined by the increased amount of (Rényi) entanglement entropy at late time for an excited state defined by acting the local operator on the vacuum. We consider a conformal field theory on an infinite space and take the subsystem in the definition of the entanglement entropy to be its half. We calculate these quantities for a free massless scalar field theory in two, four and six dimensions. We find that these results are interpreted in terms of quantum entanglement of a finite number of states, including Einstein-Podolsky-Rosen states. They agree with a heuristic picture of propagations of entangled particles.

Thermodynamical property of entanglement entropy for excited states.

We argue that the entanglement entropy for a very small subsystem obeys a property which is analogous to the first law of thermodynamics when we excite the system. In relativistic setups, its effective temperature is proportional to the inverse of the subsystem size. This provides a universal relationship between the energy and the amount of quantum information. We derive the results using holography and confirm them in two-dimensional field theories. We will also comment on an example with negative specific heat and suggest a connection between the second law of thermodynamics and the strong subadditivity of entanglement entropy.

Holographic dual of a boundary conformal field theory.

We propose a holographic dual of a conformal field theory defined on a manifold with boundaries, i.e., boundary conformal field theory (BCFT). Our new holography, which may be called anti-de Sitter BCFT, successfully calculates the boundary entropy or g function in two-dimensional BCFTs and it agrees with the finite part of the holographic entanglement entropy. Moreover, we can naturally derive a holographic g theorem. We also analyze the holographic dual of an interval at finite temperature and show that there is a first order phase transition.

Holography and entanglement in flat spacetime.

We propose a holographic correspondence of the flat spacetime based on the behavior of the entanglement entropy and the correlation functions. The holographic dual theory turns out to be highly nonlocal. We argue that after most part of the space is traced out, the reduced density matrix gives the maximal entropy and the correlation functions become trivial. We present a toy model for this holographic dual using a nonlocal scalar field theory that reproduces the same property of the entanglement entropy. Our conjecture is consistent with the entropy of Schwarzschild black holes in asymptotically flat spacetimes.

Holographic derivation of entanglement entropy from the anti-de Sitter space/conformal field theory correspondence.

A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We argue that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal agrees perfectly with the entanglement entropy in 2D CFT when applied to AdS(3). We also compare the entropy computed in AdS(5)XS(5) with that of the free N=4 super Yang-Mills theory.