Entanglement Wedge Cross Section Research Articles

Information theoretic measures for Lifshitz system

In this work, we have studied various mixed state information theoretic quantities for an excited state of Lifshitz spacetime in 3 + 1-dimensions. This geometry is the gravity dual to a class of 2 + 1-dimensional quantum field theories having Lifshitz symmetry. We have holographically calculated mutual information, entanglement wedge cross section, entanglement negativity and mutual complexity for strip like subsystems at the boundary. For this we have used the results of holographic entanglement entropy and complexity present in the literature. We first calculate all of these mentioned quantities for the pure state of Lifshitz spacetime. Then we have moved on to calculate all these quantities for excited state of the Lifshitz spacetime. The gravity dual of excited state of Lifshitz systems in field theory can be obtained by applying constant perturbations along the boundary direction. Further, we would like to mention that for the simplicity of calculation we are only considering results up to the first order in perturbation. The change in the obtained holographic information theoretic quantities are then related to entanglement entropy, entanglement pressure, entanglement chemical potential and charge using the stress tensor complex. These relations are analogous to the first law of entanglement thermodynamics given earlier in the literature. All the calculations are carried out for both values of dynamical scaling exponent (z) present in the Lifshitz field theory.

Entanglement islands and cutoff branes from path-integral optimization

Recently it was proposed that, the AdS/BCFT correspondence can be simulated by a holographic Weyl transformed CFT2, where the cut-off brane plays the role of the Karch-Randall (KR) brane [1]. In this paper, we focus on the Weyl transformation that optimizes the path integral computation of the reduced density matrix for a single interval in a holographic CFT2. When we take the limit that one of the endpoint of the interval goes to infinity (a half line), such a holographic Weyl transformed CFT2 matches the AdS/BCFT configuration for a BCFT with one boundary. Without taking the limit, the induced cutoff brane becomes a circle passing through the two endpoints of the interval. We assume that the cutoff brane also plays the same role as the KR brane in AdS/BCFT, hence the path-integral-optimized purification for the interval is in the island phase. This explains the appearance of negative mutual information observed in [2]. We check that, the entanglement entropy and the balanced partial entanglement entropy (BPE) calculated via the island formulas, exactly match with the RT formula and the entanglement wedge cross-section (EWCS), which are allowed to anchor on the cutoff brane.

Entanglement inside a black hole before the Page time

We investigate the evolution of entanglement within an open, strongly coupled system interacting with a heat bath as its environment, in the frameworks of both the doubly holographic model and the Sachdev-Ye-Kitaev (SYK) model. Generally, the entanglement within the system initially increases as due to internal interactions; however, it eventually dissipates into the environment. In the doubly holographic setup, we consider an end-of-the-world brane in the bulk to represent an eternal black hole coupled with its radiation and the evolution of the global thermofield double (TFD) state. For small black holes, the reflected entropy between the bipartition exhibits a ramp-plateau-slump behavior, where the plateau arises due to the phase transition of the entanglement wedge cross-section before the Page time. Similarly, the mutual information between the bipartition displays a ramp-slop-stabilizing behavior. In quantum mechanics, we consider a double copy of the SYK-plus-bath system in a global TFD state, resembling an eternal black hole interacting with an environment. The Rényi mutual information within the double-copied SYK clusters exhibits a ramp-plateau-slope-stabilizing behavior. The dynamic behaviors of the entanglement quantities observed in these two models are attributable to the competition between the internal interaction of the system and the external interaction with the baths. Our study provides a fine-grained picture of the entanglement dynamics inside black holes before their Page time.

Holographic coarse-grained states and the necessity of perfect entanglement

In the framework of the holographic principle, focusing on a central concept, conditional mutual information, we construct a class of coarse-grained states, which are intuitively connected to a family of thread configurations. These coarse-grained states characterize the entanglement structure of holographic systems at a coarse-grained level. Importantly, these coarse-grained states can be used to further reveal nontrivial requirements for the holographic entanglement structure. Specifically, we employ these coarse-grained states to probe the entanglement entropies of disconnected regions and the entanglement wedge cross section dual to the inherent correlation in a bipartite mixed state. The investigations demonstrate the necessity of perfect tensor state entanglement. Moreover, in a certain sense, our work establishes the equivalence between the holographic entanglement of purification and the holographic balanced partial entropy. We also construct a thread configuration with the multiscale entanglement renormalization ansatz (MERA) structure, reexamining the connection between the MERA structure and kinematic space. Published by the American Physical Society 2024

Cutoff brane vs the Karch-Randall brane: the fluctuating case

Recently, certain holographic Weyl transformed CFT2 is proposed to capture the main features of the AdS3/BCFT2 correspondence [1, 2]. In this paper, by adapting the Weyl transformation, we simulate a generalized AdS/BCFT set-up where the fluctuation of the Karch-Randall (KR) brane is considered. In the gravity dual of the Weyl transformed CFT, the so-called cutoff brane induced by the Weyl transformation plays the same role as the KR brane. Unlike the non-fluctuating configuration, in the 2d effective theory the additional twist operator is inserted at a different place, compared with the one inserted on the brane. Though this is well-understood in the Weyl transformed CFT set-up, it is confusing in the AdS/BCFT set-up where the effective theory is supposed to locate on the brane. This confusion indicates that the KR brane may be emergent from the boundary CFT2 via the Weyl transformations.We also calculate the balanced partial entanglement (BPE) in the fluctuating brane configurations and find it coincide with the entanglement wedge cross-section (EWCS). This is a non-trivial test for the correspondence between the BPE and the EWCS, and a non-trivial consistency check for the Weyl transformed CFT set-up.

Holographic reflected entropy and islands in interface CFTs

We investigate the reflected entropy for various mixed state configurations in the two dimensional holographic conformal field theories sharing a common interface (ICFTs). In the AdS3/ICFT2 framework, we compute the holographic reflected entropy for the required configurations in the vacuum state of the ICFT2 which is given by twice the entanglement wedge cross section (EWCS) in a spacetime involving two AdS3 geometries glued along a thin interface brane. Subsequently, we evaluate the EWCS in the bulk geometry involving eternal BTZ black strings with an AdS2 interface brane, which is dual to an ICFT2 in the thermofield double (TFD) state. We explore the system from a doubly holographic perspective and determine the island contributions to the reflected entropy in the two dimensional semi-classical description involving two CFT2s coupled to an AdS2 brane. We demonstrate that the results from the island formula match precisely with the bulk AdS3 results in the large tension limit of the interface brane. We illustrate that the phase structure of the reflected entropy is quite rich involving many novel induced island phases and demonstrate that it obeys the expected Page curve for the reflected entropy in a radiation bath coupled to the AdS2 black hole.

Reflected entropy in a BCFT on a black hole background

We obtain the reflected entropy for bipartite mixed state configurations involving two disjoint and adjacent subsystems in a two dimensional boundary conformal field theory (BCFT2) in a black hole background. The bulk dual is described by an AdS3 black string geometry truncated by a Karch-Randall brane. The entanglement wedge cross section computed for this geometry matches with the reflected entropy obtained for the BCFT2 verifying the holographic duality. In this context, we also obtain the analogues of the Page curves for the reflected entropy and investigate the behaviour of the Markov gap.

Entanglement of purification in random tensor networks

The entanglement of purification EP(A∶B) is a powerful correlation measure, but it is notoriously difficult to compute because it involves an optimization over all possible purifications. In this paper, we prove a new inequality: EP(A∶B)≥12SR(2)(A∶B), where SR(n)(A∶B) is the Renyi reflected entropy. Using this, we compute EP(A∶B) for a large class of random tensor networks at large bond dimension and show that it is equal to the entanglement wedge cross section EW(A∶B), proving a previous conjecture motivated from AdS/CFT. Published by the American Physical Society 2024

Entanglement islands read perfect-tensor entanglement

In this paper, we make use of holographic Boundary Conformal Field Theory (BCFT) to simulate the black hole information problem in the semi-classical picture. We investigate the correlation between a portion of Hawking radiation and entanglement islands by the area of an entanglement wedge cross-section. Building on the understanding of the relationship between entanglement wedge cross-sections and perfect tensor entanglement as discussed in reference [18], we make an intriguing observation: in the semi-classical picture, the positioning of an entanglement island automatically yields a pattern of perfect tensor entanglement. Furthermore, the contribution of this perfect tensor entanglement, combined with the bipartite entanglement contribution, precisely determines the area of the entanglement wedge cross-section.

Reflected entropy for communicating black holes II: Planck braneworlds

We obtain the reflected entropy for bipartite mixed state configurations of two adjacent and disjoint intervals at a finite temperature in finite-sized non-gravitating reservoirs described by CFT2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$CFT_2$$\\end{document}s each coupled to two quantum dots at their boundaries in the large central charge limit through a replica technique. These field theory results are substantiated through a holographic computation involving the entanglement wedge cross section in the dual bulk BTZ black hole geometry truncated by two Planck branes. The two Planck branes are the holographic duals of the quantum dots described by AdS2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$AdS_2$$\\end{document} slices with JT black holes. Our result reproduce the holographic duality between the reflected entropy and the bulk entanglement wedge cross section in the context of the AdS3/CFT2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$AdS_3/CFT_2$$\\end{document} correspondence. Subsequently we analyze the behaviour of the holographic Markov gap between the reflected entropy and the mutual information for different scenarios involving the subsystem sizes and time.

On Minimal Entanglement Wedge Cross Section for Holographic Entanglement Negativity

We demonstrate the equivalence of two different conjectures in the literature for the holographic entanglement negativity in AdS3/CFT2, modulo certain constants. These proposals involve certain algebraic sums of bulk geodesics homologous to specific combinations of subsystems, and the entanglement wedge cross section (EWCS) backreacted by a cosmic brane for the conical defect geometry in the bulk gravitational path integral. It is observed that the former conjectures reproduce the field theory replica technique results in the large central charge limit whereas the latter involves constants related to the Markov gap. In this context, we establish an alternative construction for the EWCS of a single interval in a CFT2 at a finite temperature to resolve an issue for the latter proposal involving thermal entropy elimination for holographic entanglement negativity. Our construction for the EWCS correctly reproduces the corresponding field theory results modulo the Markov gap constant in the large central charge limit.

Mixed-state entanglement and transport in Einstein–Maxwell–Axion–Horndeski theory

We present a comprehensive study exploring the relationship between transport properties and measures of quantum entanglement in the Einstein–Maxwell–Axion–Horndeski theory. By using holographic duality, we study the entanglement measures, holographic entanglement entropy (HEE) and entanglement wedge cross-section (EWCS), and transport coefficients, for this model and analyze their dependence on free parameters which we classify into action parameters, observable parameters and axion factor. We find contrasting behaviors between HEE and EWCS with respect to observable parameters (charge and temperature), and the axion factor, indicating that they capture different types of quantum correlations. We also find that HEE exhibits positive correlation with both charge and thermal excitations, whereas EWCS exhibits a negative correlation with charge-related conductivities and thermal fluctuations. Furthermore, we find that the Horndeski coupling term, as the modification to standard gravity theory, does not change the qualitative behaviors of the conductivities and the entanglement measures.

Entanglement and Chaos near critical point in strongly coupled Gauge theory

We perform a holographic study of the high and low temperature behaviours of logarithmic negativity (LN) and entanglement wedge cross section (EWCS) in a large N strongly coupled thermal field theory with critical point, having a well defined gravity dual known as 1RC black hole. The bulk theory accommodates a dimensionless parameter ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}, proportional to the charge of the 1RC black hole. Holographically, ξ→2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi \\rightarrow 2$$\\end{document} limit ensures an existence of a critical point in the dual boundary theory. We show that the logarithmic negativity in low and high temperature limits enhances with increasing ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}. We analytically compute the EWCS in low and high temperature limits and find an agreement with the previously reported numerical results. We holographically explore the correlation between two identical copies of thermal field theory with critical point forming a thermofield double state (TFD) by computing the thermo mutual information (TMI). TMI shows an increasing behaviour with respect to the width of the boundary region. Furthermore, we study the chaotic behaviour of the field theory by analyzing a shock wave in the dual eternal 1 RC black hole and then estimate the degradation of TMI. The rate of such disruption of TMI slows down as the value of critical parameter ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document} takes higher values.

Ownerless island and partial entanglement entropy in island phases

In the context of partial entanglement entropy (PEE), we study the entanglement structure of the island phases realized in several 2-dimensional holographic set-ups. From a pure quantum information perspective, the entanglement islands emerge from the self-encoding property of the system, which gives us new insights on the construction of the PEE and the physical interpretation of two-point functions of twist operators in island phases. With the contributions from the entanglement islands properly taken into account, we give a generalized prescription to construct PEE and balanced partial entanglement entropy (BPE). Here the ownerless island region, which lies inside the island \text{Is}(AB)Is(AB) of A\cup BA∪B but outside \text{Is}(A)\cup \text{Is}(B)Is(A)∪Is(B), plays a crucial role. Remarkably, we find that under different assignments for the ownerless island, we get different BPEs, which exactly correspond to different saddles of the entanglement wedge cross-section (EWCS) in the entanglement wedge of A\cup BA∪B. The assignments can be settled by choosing the one that minimizes the BPE. Furthermore, under this assignment we study the PEE and give a geometric picture for the PEE in holography, which is consistent with the geometric picture in the no-island phases.

Mixed-state entanglement for AdS Born-Infeld theory

We study the mixed-state entanglement for AdS Born-Infeld (BI) theory. We calculate the mixed-state entanglement and investigate the relationship between it and the system parameters. We find that the holographic entanglement entropy (HEE) and mutual information (MI) exhibit monotonically increasing and decreasing behavior with BI factor b. However, the entanglement wedge cross-section (EWCS) exhibits a very rich set of phenomena about system parameters. EWCS always increases with b when b is small and then monotonically decreases with b. These behaviors suggest that increasing the BI factor, which is essentially enhancing the coupling between the background geometry and the transport properties can always enhance the EWCS. The coupling between the entanglement and the transport behaviors has also been studied in condensed matter theories and is important to construct a stable quantum circuit. We also provide analytical understanding of the above phenomenon. Furthermore, we have tested two additional BI-like models and find the universality of these results, suggesting the crucial role of the BI term in governing the interplay between nonlinear electromagnetic effects and entanglement.

Entanglement wedge cross section growth during thermalization

Motivated by exploring the thermalization process in relativistic and non-relativistic holographic field theories after a non-local quench, we investigate some features in the time evolution of the entanglement wedge cross section (EWCS). This quantity is a possible holographic dual to some non-local information measures such as entanglement of purification. In particular, we focus on the time dependence of EWCS during black hole formation in D+2 dimensional AdS spacetime as well as geometries with Lifshitz and hyperscaling violating exponents. A combination of analytic and numerical results for large symmetric strip shaped boundary subregions shows that the scaling of EWCS at early times only depends on the Lifshitz exponent. In addition, this early growth regime is followed by a linear growth regime whose velocity depends on the dimensions of spacetime, the Lifshitz exponent, and the hyperscaling parameter. This velocity is the same as the entanglement velocity and for nontrivial dynamical exponent depends on the temperature of the final equilibrium state.

The mixed-state entanglement in holographic p-wave superconductor model

In this paper, we investigate the mixed-state entanglement in a model of p-wave superconductivity phase transition using holographic methods. We calculate several entanglement measures, including holographic entanglement entropy (HEE), mutual information (MI), and entanglement wedge cross-section (EWCS). Our results show that these measures display critical behavior at the phase transition points, with the EWCS exhibiting opposite temperature behavior compared to the HEE. Furthermore, we explore the behavior of thermodynamics and holographic quantum information at the zeroth-order phase transition point and find that it is opposite to that observed in the first-order phase transition. Additionally, we find that the critical exponents of all entanglement measures are twice those of the condensate. Our findings also suggest that the EWCS is a more sensitive indicator of the critical behavior of phase transitions than the HEE. Lastly, we uncover a universal inequality in the growth rates of EWCS and MI near critical points in thermal phase transitions, such as p-wave and s-wave superconductivity, suggesting that MI captures more information than EWCS when a phase transition first occurs.

More on entanglement properties of $$Lif_4^{(2)}\times {S}^1\times S^5$$ spacetime with string excitations

The $$Lif_{4}^{(2)} \times S^1 \times S^5$$ spacetime is an exact solution of $$F1-D2-D8$$ configuration in type IIA supergravity and can accommodate charged excitations of the fundamental string. By gauge/gravity duality, it is related to an excited state of a non-relativistic QFT with anisotropic Lifshitz scaling symmetry. We study mutual and tripartite information and entanglement wedge cross section in bulk gravity for boundary subsystems that are disjoint strips of very narrow width. Our work helps understand the nature of entanglement in the QFT excited state, which is in general a mixed one.

Mixed state information theoretic measures in boosted black brane

In this paper, we study various mixed state information theoretic quantities for a boosted black brane geometry. We have considered two setups, namely, a strip-like subsystem taken parallel and perpendicular to the direction of the boost. The quantities we calculate are the entanglement wedge cross-section, mutual information, entanglement negativity, and purification complexity. In the subsequent analysis, we have incorporated the thin-strip approximation and computed the leading order change (over pure AdS) in the concerned information theoretic quantities due to the boost parameter. We also show the relation of these quantities computed holographically to the energy and pressure of the boundary field theory. We then proceed to calculate the asymmetry ratios of these quantities, and observe that they are independent of the subsystem size. Finally, we proceed to study an interesting limit of the boosted black brane geometry, which is the so called AdS wave geometry. We once again compute all the mixed state information theoretic quantities for this geometry.

Reflected entropy for communicating black holes. Part I. Karch-Randall braneworlds

We obtain the reflected entropy for bipartite mixed state configurations of two adjacent and disjoint intervals at a finite temperature in BCFT2s with two distinct boundaries through a replica technique in the large central charge limit. Subsequently these field theory results are reproduced from bulk computations involving the entanglement wedge cross section in the dual BTZ black hole geometry truncated by two Karch-Randall branes. Our result confirms the holographic duality between the reflected entropy and the bulk entanglement wedge cross section in the context of the AdS3/BCFT2 scenario. We further investigate the critical issue of the holographic Markov gap between the reflected entropy and the mutual information for these configurations from the bulk braneworld geometry and study its variation with subsystem sizes and time.