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Helmholtz International Summer School on Modern Mathematical Physics Dubna July 22 – 30, 2007. Quantum Gravity and Quantum Entanglement (lecture 1). Dmitri V. Fursaev Joint Institute for Nuclear Research Dubna, RUSSIA. Talk is based on hep-th/0602134
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Helmholtz International Summer SchoolonModern Mathematical Physics Dubna July 22 – 30, 2007 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
a recent review L. Amico, R. Fazio, A. Osterloch, V. Vedral, “Entanglement in Many-Body Systems”, quant-ph/0703044
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)
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
Quantum Entanglement Quantum state of particle «1» cannot be described independently fromparticle «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”)
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)
subadditivity of the entropy strong subadditivity equalities are applied to the von Neumann entropy and are based on the concavity property
effective action approach to EE in a QFT - “partition function” • effective action is defined on manifolds • with cone-like singularities - “inverse temperature”
theory at a finite temperature T classical Euclidean action for a given model
Example: 2D case these intervals are identified
the geometrical structure - standard partition function case conical singularity is located at the separating point
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!
summary of calculation: find a family of manifolds corresponding to a given system have conical singularities on a co-dimension 2 hypersurface (separating surface) - partition function, compute - “geometrical” inverse temperature
many-body systems in higher dimensions spin lattice continuum limit A – area of a flat separation surfaceB which divides the system into two parts(pure quantum states!) entropy per unit area in a QFT is determined by a UV cutoff!
geometrical structure of the entropy (method of derivation: spectral geometry) edge (L = number of edges) separating surface (of area A) sharp corner (C = number of corners) (Fursaev, hep-th/0602134) for ground state a is a cutoff C – topological term (first pointed out in D=3 by Preskill and Kitaev)
Ising spin chains off-critical regime at large N critical regime
RG-evolution of the entropy UV is UV fixed point IR IR entropy does not increase under RG-flow (as a result of integration of high energy modes)
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
CONJECTURE(Fursaev, hep-th/0602134) • entanglement entropy per unit areafor degrees of freedom of the fundamental theory in a flat space
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
BLACK HOLE THERMODYNAMICS Bekenstein-Hawking entropy - area of the horizon • measure of the loss of information about states under • the horizon
some references: ● black hole entropy as the entropy of entanglement (Srednicki 93, Sorkin et al86, 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) our conjecture : ● yields the value for the fundamental entropy in flat space in terms of gravity coupling ● horizon entropy is a particular case
Open questions: the geometry was “frozen” till now: ● 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?
assumption Ising model: “fundamental” dof are the spin variables on the lattice low-energies = near-critical regime low-energy theory = QFT (CFT) of fermions
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
entanglement entropy in the semiclassical approximation a standard procedure
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
conditions for the separating surface the separating surface is a minimal co-dimension 2 hypersurface in
Equations - induced metric on the surface - normal vectors to the surface - traces of extrinsic curvatures
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)
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
Stationary spacetimes: simplification a Killing vector field - a constant time hypersurface (a Riemannian manifold) is a co-dimension 1 minimal surface on a constant-time hypersurface the statement is true for the Lorentzian theory as well !
variational formulae for EE • change of the entropy under • the shift of a point particle • mass of the particle • shift distance - change of the entropy per unit length (for a cosmic string) - string tension
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).
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: - equilibrium equation - the mean curvature - surface tension -pressure difference across the film
the Plateau Problem there are no unique solutions in general
the Plateau Problem simple surfaces catenoid is a three-dimensionalshape made by rotating a catenarycurve (discovered by L.Euler in 1744) helicoid is a ruled surface, meaning that it is a trace of a line The structure of part of a DNA double helix
the Plateau Problem other embedded surfaces Costa’s surface (1982)
the Plateau Problem Non-orientable surfaces A projective plane with three planar ends. From far away the surface looks like the three coordinate plane A minimal Klein bottle with one end
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
the Plateau Problem a minimal surface may be unstable against small perturbations
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