Quantum Entanglement and Gravity Dmitri V. Fursaev Joint Institute for Nuclear Research and Dubna University “Gravity in three dimensions”, ESI Workshop, Vienna,

plan of the talk Part I (a review) ● general properties and examples (spin chains, 2D CFT,...) ● computation: “partition function” approach ● entanglement in CFT’s with AdS gravity duals (a holographic formula for the entropy) Part II (entanglement entropy in quantum gravity) ● suggestions and motivations ● tests ● consequences

Quantum Entanglement Quantum state of particle «1» cannot be described independently from particle «2» (even for spatial separation at long distances)

measure of entanglement - entropy of entanglement density matrix of particle «2» under integration over the states of «1» «2» is in a mixed state when information about «1» is not available S – measures the loss of information about “1” (or “2”)

definition of entanglement entropy

“symmetry” of EE in a pure state

Entanglement in many-body systems spin lattice continuum limit Entanglement entropy is an important physical quantity which helps to understand better collective effects in stringly correlated systems (both in QFT and in condensed matter)

spin chains (Ising model as an example) off-critical regime at large N critical regime

Near the critical point the Ising model is equivalent to a 2D quantum field theory with mass m proportional to At the critical point it is equivalent to a 2D CFT with 2 massless fermions each having the central charge 1/2

Behavior near the critical point and RG-interpretation IR UV is UV fixed point The entropy decreases under the evolution to IR region because the contribution of short wave length modes is ignored (increasing the mass is equivalent to decreasing the energy cutoff)

more analytical results in 2D a is a UV cutoff Calabrese, Cardy hep-th/ ground state entanglement on an interval massive case: massless case: is the length of

analytical results (continued) ground state entanglement for a system on a circle system at a finite temperature is the length of

Entropy in higher dimensions in a simple case the entropy is a fuction of the area A - in a relativistic QFT (Srednicki 93, Bombelli et al, 86) - in some fermionic condensed matter systems (Gioev & Klich 06)

geometrical structure of the entropy edge ( L = number of edges) separating surface (of area A ) sharp corner ( C = number of corners) (DF, hep-th/ ) for ground state a is a cutoff

“partition function” and effective action

replica method -effective action is defined on manifolds with cone-like singularities - “inverse temperature” - “partition function” (a path integral)

theory at a finite temperature T classical Euclidean action for a given model

these intervals are identified Example: 2D case

conical singularity is located at the separating point the geometrical structure for

effective action on a manifold with conical singularities is the gravity action (even if the manifold is locally flat) curvature at the singularity is non-trivial: derivation of entanglement entropy in a flat space has to do with gravity effects!

entanglement in CFT’s and a “holographic formula”

Holographic Formula 4d space-time manifold (asymptotic boundary of AdS) (bulk space) separating surface minimal (least area) surface in the bulk Ryu and Takayanagi, hep-th/ , entropy of entanglement is measured in terms of the area of is the gravity coupling in AdS

Holographic formula enables one to compute entanglement entropy in strongly correlated systems with the help of classical methods (the Palteau problem)

2D CFT on a circle ground state entanglement for a system on a circle is the length of c – is a central charge

gravity - AdS radius A is the length of the geodesic - UV cutoff -holographic formula - central charge minimal surface = a geodesic line

a finite temperature theory: a black hole in the bulk space Entropies are different (as they should be) because there are topologically inequivalent minimal surfaces

a simple example for higher dimensions – is IR cutoff

Motivation of the holographic formula DF, hep-th/

Low-energy approximation Partition function for the bulk gravity (for the “replicated” boundary CFT)

Boundary conditions The boundary manifold has conical singularities at the separating surface. Hence, the bulk path integral should involve manifolds with conical singularities, position of the singular surfaces in the bulk is specified by boundary conditions

- holographic entanglement entropy Semiclassical approximation

conditions for the singular surface in the bulk the separating surface is a minimal least area co-dimension 2 hypersurface

Part II entanglement entropy in quantum gravity

entanglement has to do with quantum gravity: ● entanglement entropy allows a holographic interpretation for CFT’s with AdS duals ● possible source of the entropy of a black hole (states inside and outside the horizon); ● d=4 supersymmetric BH’s are equivalent to 2, 3,… qubit systems

● S(B) is a macroscopical quantity (like thermodynamical entropy); ● S(B) can be computed without knowledge of a microscopical content of the theory (for an ordinary quantum system it can’t) ● the definition of the entropy is possible for surfaces B of a certain type quantum gravity theory Can one define an entanglement entropy, S(B), of fundamental degrees of freedom spatially separated by a surface B? How can the fluctuations of the geometry be taken into account? the hypothesis

Suggestion (DF, 06,07): EE in quantum gravity between degrees of freedom separated by a surface B is conditions: ● static space-times ● slices have trivial topology ● the boundary of the slice is simply connected B is a least area minimal hypersurface in a constant-time slice 1 2 the system is determind by a set of boundary conditions; subsets, “1” and “2”, in the bulk are specified by the division of the boundary

a Killing symmetry + orthogonality of the Killing field to constant-time slices: a hypersurface minimal in a constant time slice is minimal in the entire space-time a “proof” of the entropy formula is the same as the motivation of the “holographic formula” Higher-dimensional (AdS) bulk -> physical space-time AdS boundary -> boundary of the physical space

Slices with wormhole topology (black holes, wormholes) on topological grounds, on a space-time slice which locally is there are closed least area surfaces example: for stationary black holes the cross-section of the black hole horizon with a constant-time hypersurface is a minimal surface: there are contributions from closed least area surfaces to the entanglement

EE in quantum gravity is: are least area minimal hypersurfaces homologous, respectively, to slices with wormhole topology we follow the principle of the least total area

consequences: if the EE is for black holes one reproduces the Bekenstein-Hawking formula wormholes may be characterized by an intrinsic entropy associated to the area of he mouth Entropy of a wormhole: analogous conclusion (S. Hayward, P. Martin-Moruno and P. Gonzalez-Diaz) is based on variational formulae

tests

Araki-Lieb inequality strong subadditivity property equalities are applied to the von Neumann entropy and are based on the concavity property inequalities for the von Neumann entropy

strong subadditivity: ab c d f a b cd f 12 generalization in the presence of closed least area surfaces is straightforward

entire system is in a mixed state because the states on the other part of the throat are unobervable Araki-Lieb inequality, case of slices with a wormhole topology

variational formulae

for realistic condensed matter systems the entanglement entropy is a non-trivial function of both macroscopical and microscopical parameters; entanglement entropy in a quantum gravity theory can be measured solely in terms of macroscopical (low-energy) parameters without the knowledge of a microscopical content of the theory

simple variational formulae

variational formula for a wormhole - position of the w.h. mouth (a marginal sphere) - a Misner-Sharp energy (in static case) stress-energy tensor of the matter on the mouth - a surface gravity

For extension to non-static spherically symmetric wormholes and ideas of wormhole thermodynamics see S. Hayward [gr-qc]; P. Martin-Moruno and P. Gonzalez-Diaz [gr-qc]

conclusions and future questions there is a deep connection between quantum entanglement and gravity which goes beyond the black hole physics; entanglement entropy in quantum gravity may be a pure macroscopical quantity, information about microscopical structure of the fundamental theory is not needed (analogy with thermodynamical entropy) entanglement entropy is given by the “Bekenstein-Hawking” formula in terms of the area of a co-dimensiin 2 hypersurface ; black hole entropy is a particular case; entropy formula passes tests based on inequalities; wormholes may possess an intrinsic entropy; variational formulae for a wormhole might imply thermodynamical interpretation (microscopical derivation?, Cardy formula?....)

Extension of the formula for entanglement entropy to non-static space times? minimal surfaces on constant time sections