Relative entropy of entanglement (REE) is an entanglement measure of bipartite mixed states, defined by the minimum of the relative entropy $S(ρ_{AB}|| σ_{AB})$ between a given mixed state $ρ_{AB}$ and an arbitrary separable state $σ_{AB}$. The REE is always bounded by the mutual information $I_{AB}=S(ρ_{AB} || ρ_{A}\otimes ρ_{B})$ 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 $ρ^{0}_{AB}$ on disjoint subsystems $A \cup B$ and arbitrarily separable state $σ_{AB}$ 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 a 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.
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 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 AdS metric on a time slice. On the other hand, for free scalar CFTs, we do not find a…
In this paper we study the properties of two-dimensional CFTs defined by cyclic and symmetric orbifolds of free Dirac fermions, especially by focusing on the partition function and entanglement entropy. Via the bosonization, we construct the twist operators which glue two complex planes to calculate the partition function of Z_2 orbifold CFT on a torus. We also find an expression of Z_N cyclic orbifold in terms of Hecke operators, which provides an explicit relation between the partition functions of cyclic orbifolds and those of symmetric ones. We compute the entanglement entropy and Renyi en…
We study the structure and dynamics of entanglement in CFTs and black holes. We use a local entanglement measure, the entanglement contour, which is a spatial density function for von Neumann entropy with some additional properties. The entanglement contour can be calculated in many 1+1d condensed matter systems and simple models of black hole evaporation. We calculate the entanglement contour of a state excited by a splitting quench, and find universal results for the entanglement contours of low energy non-equilibrium states in 2d CFTs. We also calculate the contour of a non-gravitational ba…
We study conformal twist field four-point functions on a $\mathbb Z_N$ orbifold. We examine in detail the case $N=3$ and analyze theories obtained by replicated $N$-times a minimal model with central charge $c<1$. A fastly convergent expansion of the twist field correlation function in terms of sphere conformal blocks with central charge $Nc$ is obtained by exploiting covering map techniques. We discuss extensive applications of the formalism to the entanglement of two disjoint intervals in CFT, in particular we propose a conformal block expansion for the partially transposed reduced density m…
In this paper, we regard the $T\bar{T}$/$J\bar{T}$-deformed CFTs as perturbation theories and calculate the first order correction of the correlation functions due to the $T\bar{T}$/$J\bar{T}$-deformation. As applications, we study the Rényi entanglement entropy of excited state in the $T\bar{T}$/$J\bar{T}$-deformed two-dimensional CFTs. We find, up to the perturbation first order of the deformation, the Rényi entanglement entropy of locally excited states will acquire a non-trivial time dependence. The excess of the Rényi entanglement entropy of locally excited state will also be dramatically…
In this work, we study the holographic entanglement entropy of two dimensional $T\bar{T}$-deformed conformal field theory. We compute the correction due to the deformation up to the leading order of the deformation parameter in the framework of New massive gravity, General minimal massive gravity, and Exotic general massive gravity. We also apply Song's prescription to obtain the entanglement entropy for deformed theories. In each cases, we find agreement between the results.
In this paper, I calculate the large $N$ limit of marginal $O(N)$ models with non-polynomial potentials in arbitrary odd dimensions $d$. This results in a new class of interacting pure conformal field theories (CFTs) in $d=3+4n$ for any $n \in \mathbb{Z}_+$. Similarly, in $d=3+4n$ I calculate the thermal entropy for all couplings on $R^{2+4n} \times S^1$ for $n=0,1,2,3$. In 2+1 dimensions I find the strong-to-weak coupling ratio of the thermal entropy to be 4/5, matching recent results, and further extend this analysis to higher odd dimensions. Next, I calculated the vacuum entanglement entrop…
We present a new method of deriving shapes of entanglement wedges directly from CFT calculations. We point out that a reduced density matrix in holographic CFTs possesses a sharp wedge structure such that inside the wedge we can distinguish two local excitations, while outside we cannot. We can determine this wedge, which we call a CFT wedge, by computing a distinguishability measure. We find that CFT wedges defined by the fidelity or Bures distance as a distinguishability measure, coincide perfectly with shadows of entanglement wedges in AdS/CFT. We confirm this agreement between CFT wedges a…
We study operator entanglement measures of the unitary evolution operators of (1+1)-dimensional conformal field theories (CFTs), aiming to uncover their scrambling and chaotic behaviors. In particular, we compute the bi-partite and tri-partite mutual information for various configurations of input and output subsystems, and as a function of time. We contrast three different CFTs: the free fermion theory, compactified free boson theory at various radii, and CFTs with holographic dual. We find that the bi-partite mutual information exhibits distinct behaviors for these different CFTs, reflecting…
In this paper, we look for signatures of quantum revivals in two-dimensional conformal field theories (2d CFTs) on a spatially compact manifold by using operator entanglement. It is believed that thermalization does not occur on spatially compact manifolds as the quantum state returns to its initial state which is a phenomenon known as quantum revival. We find that in CFTs such as the free fermion CFT, the operator mutual information exhibits quantum revival in accordance with the relativistic propagation of quasiparticles while in holographic CFTs, the operator mutual information does not exh…
We use the holographic methods to calculate the entanglement entropy for field theories modified by $T\bar{T}$ insertion. Based on the available holographic proposals, this calculation reduces to the holographic computations in AdS with finite cut-off. We perform a direct holographic calculation of the entanglement entropy for two dimensional deformed theory. By direct we mean the evaluation of the gravitational action in the bulk spacetime which has been reconstructed from a dual CFT$_2$ on $n$-sheeted Riemann surface as a finite cut-off boundary. We also apply Ryu-Takayanagi's prescription t…
We introduce a pseudo entropy extension of topological entanglement entropy called topological pseudo entropy. Various examples of the topological pseudo entropies are examined in three-dimensional Chern-Simons gauge theory with Wilson loop insertions. Partition functions with knotted Wilson loops are directly related to topological pseudo (Rényi) entropies. We also show that the pseudo entropy in a certain setup is equivalent to the interface entropy in two-dimensional conformal field theories (CFTs), and leverage the equivalence to calculate the pseudo entropies in particular examples. Furth…
Conformal Blocks in 2d Carrollian/Galilean CFTs and Excited State Entanglement Entropy
Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所引力物理与量子信息中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Department of Physics, Kyoto University（京都大学物理学系）；Perimeter Institute for Theoretical Physics（围场理论物理研究所）
The light cone OPE limit provides a significant amount of information regarding the conformal field theory (CFT), like the high-low temperature limit of the partition function. We started with the light cone bootstrap in the {\it general} CFT ${}_2$ with $c>1$. For this purpose, we needed an explicit asymptotic form of the Virasoro conformal blocks in the limit $z \to 1$, which was unknown until now. In this study, we computed it in general by studying the pole structure of the {\it fusion matrix} (or the crossing kernel). Applying this result to the light cone bootstrap, we obtained the unive…
The Ryu-Takayanagi (RT) formula has been a key ingredient in our understanding of holography. Recent work on TT deformations has also boosted our understanding of holography away from the conformal boundary of AdS. In this short note, we aim to refine some recent work demonstrating the success of the RT formula in TT deformed theories. We emphasize general arguments that justify the use of the RT formula in general holographic theories that obey a GKPW-like dictionary. In doing so, we clarify subtleties related to holographic counterterms and discuss the implications for holography in general …
Entanglement Requires Fluctuations at Conformal Interfaces
Institute for Advanced Study, Kyushu University（九州大学高等研究院）；Department of Physics, Kyushu University（九州大学物理系）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心(iTHEMS)）
Quantum many-body systems have a rich structure in the presence of boundaries. We study the groundstates of conformal field theories (CFTs) and Lifshitz field theories in the presence of a boundary through the lens of the entanglement entropy. For a family of theories in general dimensions, we relate the universal terms in the entanglement entropy of the bulk theory with the corresponding terms for the theory with a boundary. This relation imposes a condition on certain boundary central charges. For example, in $2+1$ dimensions, we show that the corner-induced logarithmic terms of free CFTs an…
We derive a general formula for renormalized entanglement entropy in even dimensional CFTs holographically dual to Einstein gravity in one dimension higher. In order to renormalize, we adapt the Kounterterm method to asymptotically locally AdS manifolds with conical singularities. On the gravity side, the computation considers extrinsic counterterms and the use of the replica trick a la Lewkowycz-Maldacena. The boundary counterterm B_d is shown to satisfy a key property, in direct analogy to the Euler density: when evaluated on a conically singular manifold, it decomposes into a regular part p…
We use the embedding formalism to study correlation functions of a d-dimensional Euclidean CFT in the presence of a $q$ co-dimensional defect. The defect breaks the global conformal group $SO(d+1,1)$ into $SO(d-q+1,1) \times SO(q)$. We calculate all possible invariant structures that can appear in one-point, two-point and three-point correlation functions of bulk and defect operators in mixed symmetry representation. Their generalization to n-point correlation functions are also worked out. Correlation functions in the presence of a defect, in arbitrary representation of $SO(q)$, are also calc…
We study the entanglement of purification (EoP) of subsystem $A$ and B in conformal field theories (CFTs) stressing on its relation to unitary operations of disentanglement, if the auxiliary subsystem $\tilde{A}$ adjoins $A$ and $\tilde{A}\tilde{B}$ is the complement of $AB$. We estimate the amount of the disentanglement by using the holographic EoP conjecture as well as the inequality of Von Neumann entropy. Denote the state that produces the EoP by $|ψ\rangle_M$. We calculate the variance of entanglement entropy of $A\tilde{A}$ in the state $|ψ(δ)\rangle:=e^{iδH_{\tilde{A}\tilde{B}}}|ψ\rangl…
Holographic Entanglement Propagation Through Wormholes
Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所引力物理与量子信息中心）；Inamori Research Institute for Science（稻盛科学研究所）
We derive a covariant expression for the renormalized holographic entanglement entropy for Conformal Field Theories (CFTs) dual to Quadratic Curvature Gravity in arbitrary dimensions. This expression is written as the sum of the bare entanglement entropy functional obtained using standard conical defect techniques, and a counterterm defined at the boundary of the extremal surface of the functional. The latter corresponds to the cod-2 self-replicating part of the extrinsic counterterms when evaluated on the replica orbifold. This renormalization method isolates the universal terms of the hologr…
It is believed that thermalization drives the reduced density matrix of a subsystem to approach a short-range entangled operator. If the initial state is also short-range entangled, it is possible that the reduced density matrix remains low-entangled throughout thermalization; or there could exist a barrier with high operator entanglement between the initial and thermalized reduced density matrix. In this paper, we study such dynamics in three classes of models: the rational CFTs, the random unitary circuit, and the holographic CFTs, representing systems of increasing quantum chaoticity. We sh…
We study holographic entanglement entropy in dS/CFT and introduce time-like 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 pseudo entropy. We find that the imaginary part of pseudo entropy implies an emergence of time in dS/CFT.
After quantum quenches in many-body systems, finite subsystems evolve non-trivially in time, eventually approaching a stationary state. In typical situations, the reduced density matrix of a given subsystem begins and ends this endeavour as a low-entangled vector in the space of operators. This means that if its operator space entanglement initially grows (which is generically the case), it must eventually decrease, describing a barrier-shaped curve. Understanding the shape of this "entanglement barrier" is interesting for three main reasons: (i) it quantifies the dynamics of entanglement in t…
We explore a fruitful connection between the physics of conformal field theories (CFTs) in d-dimensional Minkowski spacetime and the extended gravitational thermodynamics of hyperbolic black holes in (d+1)-dimensional anti-de Sitter spacetime. The CFTs are reduced on a region bounded by a sphere. We show that Renormalization Group (RG) flows between CFTs are specific thermodynamic processes in the (p,V) plane, where the irreversibility of coarse-graining flows from the ultraviolet to the infrared corresponds to the Second Law of thermodynamics, preventing heat from flowing from low temperature…
CFT derivation of entanglement phase transition in pseudo entropy
Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学 引力物理与量子信息中心、汤川理论物理研究所）；Inamori Research Institute for Science（稻盛理科研究所）；Society of Fellows, Harvard University（哈佛大学 院士社）
