<p>We study mixed state entanglement measures in a higher dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>T</mi><mover accent="true"><mrow><mi>T</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> deformed field theory at finite temperature. The holographic dual is described by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></math> black brane geometry with a finite cutoff. We compute the entanglement wedge cross section (EWCS), proposed to be dual to entanglement of purification and holographic entanglement negativity (HEN) for striplike subsystems. The behavior of EWCS and HEN is studied across different regimes of temperature and deformation parameter. It is observed that the deformation and temperature exhibit similar effects on these two entanglement measures. Increasing the deformation leads to a decrease in the entanglement between the subsystems.</p>

India

Sanjay Pant

Taiwan

Himanshu Parihar

Mixed state entanglement in deformed field theory at finite temperature

We propose a holographic duality for the boundary Lifshitz field theory (BLFT). Similar to holographic BCFT, holographic BLFT can be consistently defined by imposing either a Neumann boundary condition (NBC) or a conformal boundary condition (CBC) on the end of the world (EOW) brane. We propose g-functions and derive g-theorem for these two types of holographic BLFT. On the field theory side, we consider BLFT whose path integral is prescribed to include also paths bouncing off the boundary. The entanglement entropy for an interval for the Lifshitz invariant ground state is computed in the saddle point approximation, and is found to agree precisely with the holographic result in both limits when the interval is very close or very far away from the boundary.

Chong-Sun Chu

Ignacio Gonzalez

Holography for boundary Lifshitz field theory

We propose a covariant holographic construction based on the extremal entanglement wedge cross section, for the entanglement negativity of bipartite states in <math altimg='si1.svg'><mi>C</mi><mi>F</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>1</mn></mrow></msub></math>s dual to non static bulk <math altimg='si2.svg'><mi>A</mi><mi>d</mi><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math> geometries. Utilizing our proposal we obtain the holographic entanglement negativity for bipartite mixed states in <math altimg='si1.svg'><mi>C</mi><mi>F</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>1</mn></mrow></msub></math>s dual to bulk extremal and non extremal rotating BTZ black holes and the time dependent Vaidya- <math altimg='si2.svg'><mi>A</mi><mi>d</mi><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math> geometries. Our results correctly reproduce the replica technique results modulo certain constants for the corresponding dual <math altimg='si1.svg'><mi>C</mi><mi>F</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>1</mn></mrow></msub></math>s in the large central charge limit for the bulk black hole geometries. Furthermore for all the cases our results also match upto such constants with the corresponding results from an earlier alternate covariant holographic construction in the literature. This demonstrates the equivalence modulo constants of the two constructions for the <math altimg='si3.svg'><mi>A</mi><mi>d</mi><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy='false'>/</mo><mi>C</mi><mi>F</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math> scenario.

Jaydeep Kumar Basak

Boudhayan Paul

Gautam Sengupta

Covariant holographic negativity from the entanglement wedge in AdS <inf>3</inf>/CFT <inf>2</inf>

By using the notion of fractional derivatives, we introduce a class of massless Lifshitz scalar field theory in (1+1)-dimension with an arbitrary anisotropy index z. The Lifshitz scale invariant ground state of the theory is constructed explicitly and takes the form of Rokhsar-Kivelson (RK). We show that there is a continuous family of ground states with degeneracy parameterized by the choice of solution to the equation of motion of an auxiliary classical system. The quantum mechanical path integral establishes a 2d/1d correspondence with the equal time correlation functions of the Lifshitz scalar field theory. We study the entanglement properties of the Lifshitz theory for arbitrary z using the path integral representation. The entanglement measures are expressed in terms of certain cross ratio functions we specify, and satisfy the c-function monotonicity theorems. We also consider the holographic description of the Lifshitz theory. In order to match with the field theory result for the entanglement entropy, we propose a z-dependent radius scale for the Lifshitz background. This relation is consistent with the z-dependent scaling symmetry respected by the Lifshitz vacuum. Furthermore, the time-like entanglement entropy is determined using holography. Our result suggests that there should exist a fundamental definition of time-like entanglement other than employing analytic continuation as performed in relativistic field theory.

Jaydeep Basak

Adrita Chakraborty

Dimitrios Giataganas

Massless Lifshitz field theory for arbitrary z

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.

Debarshi Basu

South Korea

Vinay Malvimat

Holographic reflected entropy and islands in interface CFTs

We propose a doubly holographic version of the semiclassical island formula for the entanglement negativity in the framework of the defect AdS/BCFT correspondence where the anti–de Sitter (AdS) bulk contains a defect conformal matter theory. In this context, we propose a defect extremal surface (DES) formula for computing the entanglement negativity modified by the contribution from the defect matter theory on the end-of-the-world brane. The equivalence of the DES proposal and the semiclassical island formula for the entanglement negativity is demonstrated in <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>/</mo><msub><mrow><mi>BCFT</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math> framework. Furthermore, in the time-dependent <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>/</mo><msub><mrow><mi>BCFT</mi></mrow><mn>2</mn></msub></mrow></math> scenarios involving eternal black holes in the lower dimensional effective description, we investigate the time evolution of the entanglement negativity through the DES and the island formulas and obtain the analogs of the Page curves.

Vinayak Raj

Defect extremal surfaces for entanglement negativity

We study the entanglement entropy for time-like subsystem in two-dimensional boundary conformal field theory (BCFT) both from the field theory and holographic point of view. In field theory, we compute the time-like entanglement entropy of a pure time-like interval at zero and finite temperature using the replica technique and analytical continuation. We find that, similar to the ordinary space-like entanglement entropy in BCFT, the time-like entropy also has a bulk phase and a boundary phase which corresponds respectively to the dominance of the identity block in the bulk and boundary OPE channels. However, we find that in Lorentzian BCFT, the time-like entanglement entropy posses a third Regge phase which arises in the Regge limit of the interval, when one endpoint of the time interval approaches the light cone of the mirror image of the other endpoint. We determine the phase diagram for the time-like entanglement entropy. We find that while the time-like entropy is complex in the bulk phase and has a boundary term in the boundary phase, there is no boundary entropy in the Regge phase. Moreover, it can be real or complex depending on which side the Regge limit is approached from. On the gravity side, we obtain the holographic time-like entanglement entropy from the corresponding bulk dual geometries and find exact agreement with the field theory results. The time-like entanglement entropy may be useful to describe the entanglement of a quantum dot on a half line.

Time-like entanglement entropy in AdS/BCFT

We investigate the time evolution of reflected entropy and entanglement negativity for mixed state configurations involving two adjacent and disjoint intervals in the radiation flux of moving mirrors by utilizing the AdS/BCFT duality. These measures are computed for the required mixed state configurations by using the respective replica techniques in the large central charge limit of the BCFT 1+1. We demonstrate that the results obtained exactly agree with the corresponding holographic computations in the dual bulk AdS 3 geometry with an end of the world brane. In this context, the analogues of the Page curves for these measures are obtained for the required configurations in the radiation flux of kink and escaping mirrors which mimic the Hawking radiation from evaporating and eternal black holes respectively.

Reflected entropy and entanglement negativity for holographic moving mirrors

We investigate mixed state entanglement measures of entanglement negativity and reflected entropy for bipartite states in two dimensional conformal field theories with an anomaly through appropriate replica techniques. Furthermore we propose holographic constructions for these measures from the corresponding bulk dual geometries involving topologically massive gravity in <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math> and find exact agreement with the field theory results. In this connection we extend an earlier holographic proposal for the entanglement negativity to the bulk action with a gravitational Chern-Simons term and compute its contribution to the entanglement wedge cross section dual to the reflected entropy.

Entanglement negativity, reflected entropy, and anomalous gravitation

We obtain the entanglement negativity for various bipartite zero and finite temperature pure and mixed state configurations in a class of (<math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional Galilean conformal field theories. In this context, we establish a construction for computing the entanglement negativity for such bipartite states involving a suitable replica technique. Our construction exactly reproduces certain universal features observed for entanglement negativity of corresponding states in relativistic (<math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional conformal field theories.