We study the entanglement wedge cross-section (EWCS) in holographic massive gravity theory, in which a first and second-order phase transition can occur. We find that the mixed state entanglement measures, the EWCS and mutual information (MI) can characterize the phase transitions. The EWCS and MI show exactly the opposite behavior in the critical region, which suggests that the EWCS captures distinct degrees of freedom from that of the MI. More importantly, EWCS, MI and HEE all show the same scaling behavior in the critical region. We give an analytical understanding of this phenomenon. By comparing the quantum information behavior in the thermodynamic phase transition of holographic superconductors, we analyze the relationship and difference between them, and provide two mechanisms of quantum information scaling behavior in the thermodynamic phase transition.
Generalized Entanglement Wedges and the Connected Wedge Theorem
Leinweber Institute for Theoretical Physics and Department of Physics, University of California, Berkeley（加州大学伯克利分校莱因韦伯理论物理研究所与物理系）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）
The entanglement wedge cross section in holographic picture provides a geometrical description of the entanglement for mixed states. In this paper we study the inequalities for the entanglement wedge cross section in AdS/BCFT duality. In the presence of the boundary in conformal field theory (CFT), the dual entanglement wedge cross section exhibits abundant phase structures since the extremal surface may end on the brane. We present a universal treatment which is applicable for all the possible phases such that the inequalities for the entanglement wedge cross section can be proved in an algeb…
We show that the bulk region reconstructable from a given boundary subregion --- which we term the reconstruction wedge --- can be much smaller than the entanglement wedge even when backreaction is small. We find arbitrarily large separations between the reconstruction and entanglement wedges in near-vacuum states for regions close to an entanglement phase transition, and for more general regions in states with large energy (but very low energy density). Our examples also illustrate situations for which the quantum extremal surface is macroscopically different from the Ryu-Takayanagi surface.
We study the caustics on the boundaries of entanglement wedges in the context of holography in asymptotically AdS$_3$ spacetimes. These entanglement wedges play an important role in our understanding of the emergence of bulk locality. A procedure was proposed by Sanches and Weinberg [arXiv:1703.07780] for identifying boundary operators which are local in the bulk, which also applies to certain regions that lie beyond the reach of HRT surfaces by taking advantage of the lightsheets which bound entanglement wedges. We identify the caustics which terminate these lightsheets in conical deficit and…
We establish a construction for the entanglement wedge in asymptotically flat bulk geometries for subsystems in dual $(1+1)$-dimensional Galilean conformal field theories in the context of flat space holography. In this connection we propose a definition for the bulk entanglement wedge cross section for bipartite states of such dual non relativistic conformal field theories. Utilizing our construction for the entanglement wedge cross section we compute the entanglement negativity for such bipartite states through the generalization of an earlier proposal, in the context of the usual $AdS/CFT$ …
In this paper, we find the boundary dual of the symplectic form for the bulk fields in any entanglement wedge. The key ingredient is Uhlmann holonomy, which is a notion of parallel transport of purifications of density matrices based on a maximisation of transition probabilities. Using a replica trick, we compute this holonomy for curves of reduced states in boundary subregions of holographic QFTs at large N, subject to changes of operator insertions on the boundary. It is shown that the Berry phase along Uhlmann parallel paths may be written as the integral of an abelian connection whose curv…
We relate the Riemann curvature of a holographic spacetime to an entanglement property of the dual CFT state: the Berry curvature of its modular Hamiltonians. The modular Berry connection encodes the relative bases of nearby CFT subregions while its bulk dual, restricted to the code subspace, relates the edge-mode frames of the corresponding entanglement wedges. At leading order in 1/N and for sufficiently smooth HRRT surfaces, the modular Berry connection simply sews together the orthonormal coordinate systems covering neighborhoods of HRRT surfaces. This geometric perspective on entanglement…
The problem of how the boundary encodes the bulk in AdS/CFT is still a subject of study today. One of the major issues that needs more elucidation is the problem of subregion duality; what information of the bulk a given boundary subregion encodes. Although the proof given by Dong, Harlow, and Wall states that the entanglement wedge of the bulk should be encoded in boundary subregions, no explicit procedure for reconstructing the entanglement wedge was given so far. In this paper, mode sum approach to obtaining smearing functions for a single bulk scalar is generalised to include bulk reconstr…
We study the portion of an asymptotically Anti de Sitter geometry's bulk where the metric can be reconstructed, given the areas of minimal 2-surfaces anchored to a fixed boundary subregion. We exhibit situations in which this region can reach parametrically far outside of the entanglement wedge. If the setting is furthermore holographic, so that the bulk geometry is dual to a state in a conformal field theory (CFT), these minimal 2-surface areas can be deduced from the expectation values of operators localized within the boundary subregion. This presents us with an alternative: Either the redu…
We study the problem of revealing the entanglement wedge using simple operations. We ask what operation a semiclassical observer can do to bring the entanglement wedge into causal contact with the boundary, via backreaction. In a generic perturbative class of states, we propose a unitary operation in the causal wedge whose backreaction brings all of the previously causally inaccessible `peninsula' into causal contact with the boundary. This class of cases includes entanglement wedges associated to boundary sub-regions that are unions of disjoint spherical caps, and the protocol works to first …
We consider reflected entropy in a thermofield double state perturbed by a heavy operator insertion. For sufficiently early operator insertions the dual geometry can be described by a localized shock wave geometry. We calculate the entanglement wedge cross-section in this geometry for symmetric intervals and find that it matches precisely with the CFT result for sufficiently late times. Our result exhibits a plateau before going to zero, a behaviour similar to the one observed recently in the context of global quantum quenches. We find that at high temperatures this behaviour is properly captu…
We studied the holographic entanglement entropy for a strip and sharp wedge entangling regions in momentum relaxation systems. In the case of strips, we found analytic and numerical results for the entanglement entropy and showed the effect on the minimal surface by the electric field. We also studied the entanglement entropy of wedges and confirmed that there is a linear change in the electric field. This change is proportional to the thermoelectric conductivity, $\barα$, that can be measured.
We study the entanglement wedge cross-section (EWCS) in holographic Aether gravity theory, a gravity theory with Lorentz symmetry violation while keeping the general covariance intact. We find that only a limited parameter space is allowed to obtain a black brane with positive Hawking temperature. Subject to these allowed parameter regions, we find that the EWCS could exhibit non-monotonic behaviors with system parameters. Meanwhile, the holographic entanglement entropy (HEE), and the corresponding mutual information (MI), can only exhibit monotonic behaviors. These phenomena suggest that the …
Starting from the entanglement wedge of a multipartite mixed state we describe a purification procedure which involves the gluing of several copies. The resulting geometry has non-trivial topology and a single oriented boundary for each original boundary region. In the purified geometry the original multipartite entanglement wedge cross section is mapped to a minimal surface of a particular non-trivial homology class. In contrast each original bipartite entanglement wedge cross section is mapped to the minimal wormhole throat around each boundary. Using the bit thread formalism we show how max…
We study the mixed state entanglement properties in two holographic axion models by examining the behavior of the entanglement wedge minimum cross section (EWCS), and comparing it with the holographic entanglement entropy (HEE) and mutual information (MI). We find that the behavior of HEE, MI and EWCS with Hawking temperature is monotonic, while the behavior with the axion parameter $k$ is more rich, which depends on the size of the configuration and the values of the other two parameters. Interestingly, the EWCS monotonically increases with the coupling constant $κ$ between the axion field an…
We introduce a simple geometrical construction similar to covariant holographic entanglement entropy but with the addition of a new term proportional to boundary region volume. This new procedure has properties strongly resembling classical Shannon entropy of probability distributions rather than von Neumann entropy, so we call the quantity holographic Shannon entropy. The holographic Shannon entropy of a region $A$ is divergent in AdS/CFT, but upon regulation, it appears to be equal to the outer entropy of the entanglement wedge of $A$ with the entanglement wedge of the complement of $A$ held…
We calculate logarithmic negativity, a quantum entanglement measure for mixed quantum states, in quantum error-correcting codes and find it to equal the minimal cross sectional area of the entanglement wedge in holographic codes with a quantum correction term equal to the logarithmic negativity between the bulk degrees of freedom on either side of the entanglement wedge cross section. This leads us to conjecture a holographic dual for logarithmic negativity that is related to the area of a cosmic brane with tension in the entanglement wedge plus a quantum correction term. This is closely relat…
The entanglement wedge cross section (EWCS) is numerically investigated statically and dynamically in a five-dimension AdS-Vaidya spacetime with Gauss-Bonnet (GB) corrections, focusing on two identical rectangular strips on the boundary. In the static case, EWCS increases as the GB coupling constant $α$ increases and disentangles at small separation between two strips for smaller $α$. For the dynamic case, such a monotonic relationship between EWCS and $α$ holds but the two strips no longer disentangle monotonically as in the static case. In the early thermal quenching stage, the disentangleme…
We consider the reconstruction of a Lifshitz spacetime from three perspectives: differential entropy (or "hole-ography"), causal wedges and entanglement wedges. We find that not all time-varying bulk curves in vacuum Lifshitz can be reconstructed via the differential entropy approach, adding a caveat to the general analysis of \cite{Headrick:2014eia}. We show that the causal wedge for Lifshitz spacetimes degenerates, while the entanglement wedge requires the additional consideration of a set of boundary-emanating light-sheets. The need to include bulk surfaces with no clear field theory interp…
We generalize the constructions for the multipartite reflected entropy in order to construct spacetimes capable of representing multipartite entanglement wedge cross sections of differing party number as Ryu-Takayanagi surfaces on a single replicated geometry. We devise a general algorithm for such constructions for arbitrary party number and demonstrate how such methods can be used to derive novel inequalities constraining mulipartite entanglement wedge cross sections.
Effective dynamics and quantum information in de Sitter wedge holography
University of Science and Technology of China（中国科学技术大学）；Peng Huanwu Center for Fundamental Theory（彭桓武理论研究中心）；Saha Institute of Nuclear Physics（萨哈核物理研究所）；Homi Bhabha National Institute（霍米·巴巴国立学院）
We develop further the codim-2 future-past extremal surfaces stretching between the future and past boundaries in de Sitter space, discussed in previous work. We first make more elaborate the construction of such surfaces anchored at more general subregions of the future boundary, and stretching to equivalent subregions at the past boundary. These top-bottom symmetric future-past extremal surfaces cannot penetrate beyond a certain limiting surface in the Northern/Southern diamond regions: the boundary subregions become the whole boundary for this limiting surface. For multiple disjoint subregi…
We examine general features of causal wedges in asymptotically AdS spacetimes and show that in a wide variety of cases they have non-trivial topology. We also prove some general results regarding minimal area surfaces on the causal wedge boundary and thereby derive constraints on the causal holographic information. We go on to demonstrate that certain properties of the causal wedge impact significantly on features of extremal surfaces which are relevant for computation of holographic entanglement entropy.
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Case Western Reserve University（凯斯西储大学）；Inamori Research Institute for Science（稻盛科学研究所）
There is some tension between two well-known ideas in holography. On the one hand, subregion duality asserts that the reduced density matrix associated with a limited region of the boundary theory is dual to a correspondingly limited region in the bulk, known as the entanglement wedge. On the other hand, correlators that in the boundary theory can be computed solely with that density matrix are calculated in the bulk via the GKPW or BDHM prescriptions, which require input from beyond the entanglement wedge. We show that this tension is resolved by recognizing that the reduced state is only ful…
Subregion duality in AdS/CFT implies certain constraints on the geometry: entanglement wedges must contain causal wedges, and nested boundary regions must have nested entanglement wedges. We elucidate the logical connections between these statements and the Quantum Focussing Conjecture, Quantum Null Energy Condition, Boundary Causality Condition, and Averaged Null Energy Condition. Our analysis does not rely on the classical limit of bulk physics, but instead works to all orders in $G\hbar \sim 1/N$. This constitutes a nontrivial check on the consistency of subregion duality, entanglement wedg…
We define a new information theoretic quantity called odd entanglement entropy (OEE) which enables us to compute the entanglement wedge cross section in holographic CFTs. The entanglement wedge cross section has been introduced as a minimal cross section of the entanglement wedge, a natural generalization of the Ryu-Takayanagi surface. By using the replica trick, we explicitly compute the OEE for two-dimensional holographic CFT (AdS${}_3$ and planar BTZ blackhole) and see agreement with the entanglement wedge cross section. We conjecture this relation will hold in general dimensions.
Holographic relationships between entanglement entropy on the boundary of a spacetime and the area of minimal surfaces in the bulk provide an important entry in the bulk/boundary dictionary. While constructing the necessary causal and entanglement wedges is well understood in asymptotically AdS spacetimes, less is known about the equivalent constructions in spacetimes with different asymptotics. In particular, recent attempts to construct entanglement and causal wedges for asymptotically Lifshitz solutions in relativistic gravitational theories have proven problematic. We note a simple observa…
We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse…
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…
In AdS/CFT, there can exist local 2-to-2 bulk scattering processes even when local scattering is not possible on the boundary; these have previously been studied in connection with boundary correlation functions. We show that boundary regions associated with these scattering configurations must have $O(1/G_N)$ mutual information, and hence a connected entanglement wedge. One of us previously argued for this statement from the boundary theory using operational tools in quantum information theory. We improve that argument to make it robust to small errors and provide a proof in the bulk using fo…
In the holographic correspondence, subregion duality posits that knowledge of the mixed state of a finite spacelike region of the boundary theory allows full reconstruction of a specific region of the bulk, known as the entanglement wedge. This statement has been proven for local bulk operators. In this paper, specializing first for simplicity to a Rindler wedge of AdS$_3$, we find that generic curves within the wedge are in fact not fully reconstructible with entanglement entropies in the corresponding boundary region, even after using the most general variant of hole-ography, which was recen…
We review classical results on holographic entanglement entropy utilizing the Hamilton-Jacobi approach. Possibility of using the entanglement entropy as a probe of confinement is shortly discussed in the context of lattice data.
We argue that holographic CFT states require a large amount of tripartite entanglement, in contrast to the conjecture that their entanglement is mostly bipartite. Our evidence is that this mostly-bipartite conjecture is in sharp conflict with two well-supported conjectures about the entanglement wedge cross section surface $E_W$. If $E_W$ is related to either the CFT's reflected entropy or its entanglement of purification, then those quantities can differ from the mutual information at $\mathcal{O}(\frac{1}{G_N})$. We prove that this implies holographic CFT states must have $\mathcal{O}(\frac{…
We study the consequences of Entanglement Wedge Nesting for CFTs with holographic duals. The CFT is formulated on an arbitrary curved background, and we include the effects of curvature-squared couplings in the bulk. In this setup we find necessary and sufficient conditions for Entanglement Wedge Nesting to imply the Quantum Null Energy Condition in $d\leq 5$, extending its earlier holographic proofs. We also show that the Quantum Focusing Conjecture yields the Quantum Null Energy Condition as its nongravitational limit under these same conditions.
