Quantum Entanglement and Gravity

1. “Gravity in three dimensions”, ESI Workshop, Vienna, 24.04.09 Quantum Entanglement and Gravity Dmitri V. Fursaev Joint Institute for Nuclear Research and Dubna University

2. 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

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

4. measure of entanglement • entropy ofentanglement 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”)

5. definition of entanglement entropy

6. “symmetry” of EE in a pure state

7. 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)

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

9. 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

10. Behavior near the critical pointand RG-interpretation UV is UV fixed point IR IR 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)

11. more analytical results in 2D ground state entanglement on an interval Calabrese, Cardy hep-th/0405152 is the length of a is a UV cutoff massive case: massless case:

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

13. 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)

14. geometrical structure of the entropy edge (L = number of edges) separating surface (of area A) sharp corner (C = number of corners) for ground state a is a cutoff (DF, hep-th/0602134)

15. “partition function” and effective action

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

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

18. Example: 2D case these intervals are identified

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

20. 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!

21. entanglement in CFT’s and a “holographic formula”

22. Holographic Formula Ryu and Takayanagi, hep-th/0603001, 0605073 (bulk space) minimal (least area) surface in the bulk 4d space-time manifold (asymptotic boundary of AdS) separating surface is measured in terms of the area of entropy of entanglement is the gravity coupling in AdS

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

24. 2D CFT on a circle is the length of ground state entanglement for a system on a circle c – is a central charge

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

26. 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

27. a simple example for higher dimensions 2 – is IR cutoff 1 2

28. Motivation of the holographic formula DF, hep-th/0606184

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

30. 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

31. Semiclassical approximation - holographic entanglement entropy

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

33. Part IIentanglement entropyin quantum gravity

34. 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

35. quantum gravity theory Can one define an entanglement entropy, S(B), of fundamental degrees of freedom spatially separated by a surface B? How canthe fluctuations of the geometry be taken into account? the hypothesis ● 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

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

37. 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

38. 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

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

40. 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

41. tests

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

43. strong subadditivity: c d c d 1 2 f f b a a b generalization in the presence of closed least area surfaces is straightforward

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

45. variational formulae

46. 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

47. simple variational formulae

48. 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

49. For extension to non-static spherically symmetric wormholes and ideas of wormhole thermodynamics see S. Hayward 0903.5438 [gr-qc]; P. Martin-Moruno and P. Gonzalez-Diaz 0904.0099 [gr-qc]

50. 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?....)