Abstract In holographic duality, the entanglement entropy of a boundary region is proposed to be dual to the area of an extremal codimension-2 surface that is homologous to the boundary region, known as the Hubeny-Rangamani-Takayanagi (HRT) surface. In this paper, we study when the HRT surfaces of two boundary subregions R, A are in the same Cauchy slice. This condition is necessary for the subregion-subregion mapping to be local for both subregions and for states to have a tensor network description. To quantify this, we study the area of a surface that is homologous to A and is extremal except at possible intersections with the HRT surface of R (minimizing over all such possible surfaces), which we call the constrained area. We give a boundary proposal for an upper bound of this quantity, a bound which is saturated when the constrained surface intersects the HRT surface of R at a constant angle. This boundary quantity is the minimum entropy of region A in a modular evolved state — a state that has been evolved unitarily with the modular Hamiltonian of R. We can prove this formula in two boundary dimensions or when the modular Hamiltonian is local. This modular minimal entropy is a boundary quantity that probes bulk causality and, from this quantity, we can extract whether two HRT surfaces are in the future or past of each other. These entropies satisfy some inequalities reminiscent of strong subadditivity and can be used to remove certain corner divergences.
Abstract In holographic duality, the entanglement entropy of a boundary region is proposed to be dual to the area of an extremal codimension-2 surface that is homologous to the boundary region, known as the Hubeny-Rangamani-Takayanagi (HRT) surface. In this paper, we study when the HRT surfaces of two boundary subregions R, A are in the same Cauchy slice. This condition is necessary for the subregion-subregion mapping to be local for both subregions and for states to have a tensor network description. To quantify this, we study the area of a surface that is homologous to A and is extremal except at possible intersections with the HRT surface of R (minimizing over all such possible surfaces), which we call the constrained area. We give a boundary proposal for an upper bound of this quantity, a bound which is saturated when the constrained surface intersects the HRT surface of R at a constant angle. This boundary quantity is the minimum entropy of region A in a modular evolved state — a state that has been evolved unitarily with the modular Hamiltonian of R. We can prove this formula in two boundary dimensions or when the modular Hamiltonian is local. This modular minimal entropy is a boundary quantity that probes bulk causality and, from this quantity, we can extract whether two HRT surfaces are in the future or past of each other. These entropies satisfy some inequalities reminiscent of strong subadditivity and can be used to remove certain corner divergences.
Abstract Consider two boundary subregions A and B that lie in a common boundary Cauchy surface, and consider also the associated HRT surface γ B for B. In that context, the constrained HRT surface γ A:B can be defined as the codimension-2 bulk surface anchored to A that is obtained by a maximin construction restricted to Cauchy slices containing γ B . As a result, γ A:B is the union of two pieces, γA:BB γ A : B B and γA:BB γ A : B B ¯ lying respectively in the entanglement wedges of B and its complement B B ¯ . Unlike the area A(γA) A γ A of the HRT surface γ A , at least in the semiclassical limit, the area A(γA:B) A γ A : B of γ A:B commutes with the area A(γB) A γ B of γ B . To study the entropic interpretation of A(γA:B) A γ A : B , we analyze the Rényi entropies of subregion A in a fixed-area state of subregion B. We use the gravitational path integral to show that the n ≈ 1 Rényi entropies are then computed by minimizing A(γA) A γ A over spacetimes defined by a boost angle conjugate to A(γB) A γ B . In the case where the pieces γA:BB γ A : B B and γA:BB γ A : B B ¯ intersect at a constant boost angle, a geometric argument shows that the n ≈ 1 Rényi entropy is then given by 4GA(γA:B) A γ A : B 4 G
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.
The Ryu-Takayanagi formula equates the area of a minimal surface with the von Neumann entropy of a boundary subregion, and leaves two things about that identification open.
The first is how sharply a geometry fixes an entropy. Fannes-Audenaert answers with a Hilbert-space dimension, which diverges as the cutoff is removed however close the two states are. We replace it with the capacity of entanglement, the variance of the modular energy, whose square root grows like the square root of the entangling area where the dimensional factor grows like the regulated volume. The bound is dimension-free and saturated, and it makes the ambiguity of the entropy subextensive for any perturbation whose capacity is small compared with SvN2 times the trace norm.
The second is what the area means for a single state, since compression and dilution rates are defined only for many copies while a geometry describes one. When a single replica saddle dominates near α=1, every smooth Rényi entropy at fixed α>1 agrees with SvN to O(SvN), as do the smooth min- and max-entropies. The minimal surface therefore fixes every one-shot entropy of the state at once, with large central charge playing the role of large copy number in the asymptotic equipartition property.
As a consequence we bound how far outside the holographic entropy cone a holographic state can appear to fall, leaving estimation and certification open.
One decade ago, Ryu-Takayanagi explicitly introduced a formula that relates the entropy of a subregion in CFT system to a geometrical quantity which is called minimal surface in hyperbolic space. This formula extended to the idea of connection of gravy to quantum mechanics of gauge/gravity duality. This duality which can help us to learn a more interesting feature of each side from the other. Quantum systems obey from some constraints come from the quantum information theory. I would be interesting to find out what is the dual of this constraint in the gravitation system. Dual to the specific class of quantum theories which is called conformal field theories. One of the most significant constraint that QFTs should obey is the strong subadditivity of entanglement entropy. These constraints let the theories have bound on the energy spectrum from the below; Recently there has been the development that the combination of monotonicity of relative entropy and the strong subadditivity of entanglement entropy is equal to have a specific bound on the energy momentum tensor, called quantum null energy condition. In this thesis, we re-look to this argument by introducing the entanglement density and obtain a differential operator from the strong subadditivity and exploiting from the Markov property of the vacuum of CFT. In the next step, by using from the Ryu-Takayangi, we rewrite the strong subadditivity inequality regarding geometrical quantities. By using from the kinematic languages and intertwinement, we realize that the strong subadditivity at the boundary implies new bound on averaged energy condition which has some common feature with the quantum null energy condition statement.
The Ryu-Takayanagi conjecture provides a holographic description for the entanglement entropy for the strongly coupled holographic CFTs in the semi-classical limit. It proposes that the entanglement entropy is given by the area of the minimal homologous surface in the dual bulk. We show that the common terms of the entanglement entropy for the free massless fermions or bosons on a torus in the high-temperature expansion can be described by the sum of the signed area of extremal surfaces in the BTZ spacetime. The resulting EE and the corresponding bunch of the exgremal surfaces have preferable properties rather than those from the Ryu-Takayanagi conjecture.
We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS3/CFT2. With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are codimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval A and the points on the Ryu-Takayanagi (RT) surface EA. In the same sense an arbitrary subinterval A2 of A will correspond to a subinterval E2 of EA. This fine correspondence indicates that the length of E2 captures the contribution sA(A2) from A2 to the entanglement entropy SA, hence gives the contour function for entanglement entropy. Furthermore we propose that sA(A2) in general can be written as a simple linear combination of entanglement entropies of single intervals inside A. This proposal passes several nontrivial tests.