1. Quantum Gravity and Quantum Entanglement (lecture 1) Dmitri V. Fursaev Joint Institute for Nuclear Research Dubna, RUSSIA Talk is based on hep-th/0602134 hep-th/0606184 Dubna, July 25, 2007 Helmholtz International Summer School on Modern Mathematical Physics Dubna July 22 – 30, 2007

2. a recent review L. Amico, R. Fazio, A. Osterloch, V. Vedral, “Entanglement in Many-Body Systems”, quant-ph/0703044

3. What do the following problems have in common? finding entanglement entropy in a spin chain near a critical point finding a minimal surface in a curved space (the Plateau problem)

4. plan of the 1st lecture ● quantum entanglement (QE) and entropy (EE): general properties ● EE in QFT’s: functional integral methods ● geometrical structure of entanglement entropy ● entanglement in spin chains: 2D critical phenomena CFT’s ● (fundamental) entanglement entropy in quantum gravity ● the Plateau problem

5. Lecture 1

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

7. 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”)

8. a general definition

9. “symmetry” of EE in a pure state

10. consequence: the entropy is a function of the characteristics of the separating surface 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)

11. subadditivity of the entropy strong subadditivity equalities are applied to the von Neumann entropy and are based on the concavity property

12. effective action approach to EE in a QFT -effective action is defined on manifolds with cone-like singularities - “inverse temperature” - “partition function”

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

14. these intervals are identified Example: 2D case

15. the geometrical structure case conical singularity is located at the separating point - standard partition function

17. 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!

18. summary of calculation: find a family of manifolds corresponding to a given system compute - “geometrical” inverse temperature - partition function, have conical singularities on a co-dimension 2 hypersurface (separating surface)

19. Spectral geometry: example of calculation

20. many-body systems in higher dimensions spin lattice continuum limit A – area of a flat separation surface B which divides the system into two parts (pure quantum states!) entropy per unit area in a QFT is determined by a UV cutoff!

21. geometrical structure of the entropy edge ( L = number of edges) separating surface (of area A ) sharp corner ( C = number of corners) (method of derivation: spectral geometry) (Fursaev, hep-th/0602134) for ground state a is a cutoff C – topological term (first pointed out in D=3 by Preskill and Kitaev)

22. Ising spin chains off-critical regime at large N critical regime

23. RG-evolution of the entropy entropy does not increase under RG-flow (as a result of integration of high energy modes) IR UV is UV fixed point

24. Explanation 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

25. What is the entanglement entropy in a fundamental theory?

26. CONJECTURE (Fursaev, hep-th/0602134) - entanglement entropy per unit area for degrees of freedom of the fundamental theory in a flat space

27. arguments: ● entropy density is determined by a UV-cutoff ● entanglement entropy can be derived from the effective gravity action ● the conjecture is valid for area density of the entropy of black holes

28. BLACK HOLE THERMODYNAMICS Bekenstein-Hawking entropy - area of the horizon - measure of the loss of information about states under the horizon

29. some references: ● black hole entropy as the entropy of entanglement (Srednicki 93, Sorkin et al 86, Frolov & Novikov 93) ● iduced gravity (Sakharov 68) as a condition (Jacobson 94, Frolov, Fursaev, Zelnikov 96) ● application to de Sitter horizon (Hawking, Maldacena, Strominger 00) ● entropy of certain type black holes in string theory as the entanglement entropy in 2- and 3- qubit systems (Duff 06, Kallosh & Linde 06) ● yields the value for the fundamental entropy in flat space in terms of gravity coupling ● horizon entropy is a particular case our conjecture :

30. Open questions: ● Does the definition of a “separating surface” make sense in a quantum gravity theory (in the presence of “quantum geometry”)? ● Entanglement of gravitational degrees of freedom? ● Can the problem of UV divergences in EE be solved by the standard renormalization prescription? What are the physical constants which should be renormalized? the geometry was “frozen” till now:

31. assumption Ising model: “fundamental” dof are the spin variables on the lattice low-energies = near-critical regime low-energy theory = QFT (CFT) of fermions

32. at low energies integration over fundamental degrees of freedom is equivalent to the integration over all low energy fields, including fluctuations of the space-time metric This means that: (if the boundary of the separating surface is fixed) the geometry of the separating surface is determined by a quantum problem fluctuations of are induced by fluctuations of the space-time geometry

33. entanglement entropy in the semiclassical approximation a standard procedure

34. fix n and “average” over all possible positions of the separating surface on - entanglement entropy of quantum matter (if one goes beyond the semiclassical approximation) - pure gravitational part of entanglement entropy - some average area

35. what are the conditions on the separating surface?

36. conditions for the separating surface the separating surface is a minimal co-dimension 2 hypersurface in

37. - induced metric on the surface - normal vectors to the surface - traces of extrinsic curvatures Equations

38. NB: we worked with Euclidean version of the theory (finite temperature), stationary space-times was implied; In the Lorentzian version of the theory space-times: the surface is extremal; Hint: In non-stationary space-times the fundamental entanglement should be associated with extremal surfaces A similar conclusion in AdS/CFT context is in (Hubeny, Rangami, Takayanagi, hep-th/0705.0016)

39. Quantum corrections the UV divergences in the entropy are removed by the standard renormalization of the gravitational couplings; the result is finite and is expressed entirely in terms of low-energy variables and effective constants like G

40. a Killing vector field - a constant time hypersurface (a Riemannian manifold) is a co-dimension 1 minimal surface on a constant-time hypersurface Stationary spacetimes: simplification the statement is true for the Lorentzian theory as well !

41. variational formulae for EE - change of the entropy per unit length (for a cosmic string) - string tension -change of the entropy under the shift of a point particle -mass of the particle - shift distance

42. other approaches Jacobson : -entanglement is associated with a local causal structure of a space-time; we consider more general case; -space-like surface is arbitrary, it is considered as a local Rindler horizon for a family of accelerated observers; we: the surface is minimal (extremal), black hole horizon is a particular case; -evolution of the surface is along light rays starting at the surface; we study the evolution leaving the surface minimal (extremal).

43. the Plateau Problem (Joseph Plateau, 1801-1883) It is a problem of finding a least area surface (minimal surface) for a given boundary soap films: - the mean curvature - surface tension -pressure difference across the film - equilibrium equation

44. the Plateau Problem there are no unique solutions in general

45. the Plateau Problem simple surfaces The structure of part of a DNA double helix catenoid is a three-dimensional shape made by rotatingdimensionalshape a catenary curve (discovered by L.Euler in 1744)catenarycurve helicoid is a ruled surface, meaning that it is a trace of a lineruled surface

46. the Plateau Problem Costa’s surface (1982) other embedded surfaces

47. the Plateau Problem A minimal Klein bottle with one end Non-orientable surfaces A projective plane with three planar ends. From far away the surface looks like the three coordinate plane

48. the Plateau Problem Non-trivial topology: surfaces with hadles a surface was found by Chen and Gackstatter a singly periodic Scherk surface approaches two orthogonal planes

49. the Plateau Problem a minimal surface may be unstable against small perturbations

50. plan of the 2d lecture ● entanglement entropy in AdS/CFT: “holographic formula” ● derivation of the “holographic formula” for EE ● some examples: EE in 2D CFT’s ● conclusions