1. Entanglement in Quantum Gravity and Space-Time Topology
Quarks-08
Sergiev Posad
Entanglement in Quantum Gravity
and Space-Time Topology
Dmitri V. Fursaev
Joint Institute for Nuclear Research
Dubna, RUSSIA
the talk is based on hep-th/ ,
hep-th/ ,
arXiv:  [hep-th]

2. entanglement has to do with quantum gravity:
quantum entanglement: states
of subsystems cannot described
independently 1 2
entanglement has to do with quantum gravity:
● 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
● entanglement entropy allows a holographic interpretation for CFT’s with AdS duals

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

4. entanglement entropy in quantum gravity
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
conditions:
● static space-times
● slices have trivial topology
● the boundary of the slice is simply connected
entropy of fundamental
d.of f. is UV finite

5. aim of the talk extension to problems with non-trivial topology:
slices which admit closed least area surfaces;

6. plan ● motivations for entanglement entropy (EE)
● problems with non-trivial topology
● tests of the suggestions

7. entanglement entropy

8. for realistic condensed matter systems the entanglement entropy is a non-trivial function of both macroscopical and microscopical parameters;
its calculation is technically involved, it does not allow an analytical treatment in general
DF: 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

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

10. finite temperature theory on an interval
Example:
finite temperature theory on an interval
these intervals
are identified

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

12. “gravitational” entanglement entropy (semiclassical approximation)
the “gravitational”entropy appears from the classical gravity action
(which is a low-energy approximation of the effective action in quantum gravity)

13. conditions for the “separating” surface
fluctuations of are induced by fluctuations of the space-time geometry
the geometry of the separating surface is determined by a quantum problem

14. the separating surface is a minimal
least area co-dimension 2
hypersurface

15. slices with non-trivial topology
the work is done with
A.I. Zelnikov
slices with non-trivial topology
slices which locally are
slices with handles 1 2 1 2
(regions where states are integrated out are dashed)

16. slices with wormhole topology

17. closed least area surfaces
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

18. slices with a single handle
we follow the principle of the least total area
suggestion: EE in quantum gravity on a slice with a handle is
are homologous to , respectively

19. slices with wormhole topology
EE in quantum gravity is:
are least area minimal hypersurfaces homologous,
respectively, to

20. observation: if the EE is
black holes: EE reproduces the Bekenstein-Hawking entropy
wormholes may be characterized by an intrinsic entropy

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

22. 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

23. 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

24. 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);
“the least area principle” can be used to generalize the entropy definition for slices with non-trivial topology;
the principle can be tested by the entropy inequalities;
BH entropy is a particular case of EE in quantum gravity;
wormholes can be characterized by an intrinsic entropy determined
by the least area surface at the throat

25. thank you for attention