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Quantum Gravity and Quantum Entanglement (lecture 1)
Dmitri V. Fursaev
Joint Institute for Nuclear ResearchDubna, RUSSIA
Talk is based on hep-th/0602134 hep-th/0606184 Dubna, July 25, 2007
Helmholtz International Summer School onModern Mathematical PhysicsDubna July 22 – 30, 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
Lecture 1
Quantum Entanglement
Quantum state of particle «1» cannot be described independently from particle «2» (even for spatial separation at long distances)
1 2 1 2
1| (| | | | )
2
measure of entanglement
2 2 2
2 1
( ln )
(| |)
S Tr
Tr
- 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 availableS – measures the loss of information about “1” (or “2”)
a general definition
A a
1
2
1 2 2 1
1 1 1 1 2 2 2 2
/
1 2/
( , | , )
( | ) ( , | , ),
( | ) ( , | , ),
, ,
ln , ln
a
A
H T
H T
A a B b
A B A a B a
a b A a A b
Tr Tr
S Tr S Tr
eS S
Tr e
“symmetry” of EE in a pure state
1 1
2 2
2 1
1 2
( | )
( | ) ,
, ( 0)
AaaA
Aa Baa
TAa Ab
A
C A a
A B C C CC
a b C C C C
if d Ce e e d d
S S
consequence: the entropy is a function of the characteristics of the separating
surface
1 2 ( )S S f A
in a simple case the entropyis a fuction of the area A
ln
S A
S A A
- in a relativistic QFT (Srednicki 93, Bombelli et al, 86)
- in some fermionic condensed matter systems (Gioev & Klich 06)
subadditivity of the entropy
1 2 1 2
1 2 1 2
| | , lnS S S S S S Tr
S S S
1 2
strong subadditivity
1 2 1 2 1 2S S S S
equalities are applied to the von Neumann entropyand are based on the concavity property
effective action approach to EE in a QFT
-effective action is defined on manifolds with cone-like singularities
- “inverse temperature”
1 1 1 2
1 2
( ) lim lim 1 ln ( , )
( , )
ln ( , )
2
nnS T Tr Z T
n
Z T Tr
Z T
n
- “partition function”
theory at a finite temperature T
1 2
1 2
1 2
1 2
/
{ ' },{ ' }[ ]
1 2 1 2
{ },{ }
1 2
{ ' },{ }[ ]
1 1 1 2
{ },{ }
1{ },{ } { ' },{ ' } [ ]
[ ]
1{ } { ' } [ ]
H T
I
I
e
D eN
I
Tr
d D eN
classical Euclidean action for a given model
1{ ' }
1{ }
2{ }
2{ }0
1/T
1
1
2
2these intervals are identified
Example: 2D case
the geometrical structure
3n
31 1Tr
case
conical singularity is located at the separating point
( 2 , )
( )
Z T
Z T
- standardpartition function
effective action on a manifold with conical singularities is the gravity action (even if the manifold is locally flat)
(2)2(2 ) ( )R B
curvature at the singularity is non-trivial:
derivation of entanglement entropy in a flat space has to do with gravity effects!
summary of calculation:
1
1 2 1
1)
2) ( , )
2
3) ( ) lim 1 ln ( , )
Z T
n
S T Z T
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)
Spectral geometry: example of calculation
2
2
2
20 1 2/ 2
1
2 2 212 2
2 2 22
1( ) ...
(4 )
2( )
3 2
1( )
2
1 1ln
32
1 1ln ( )
48
tLD
tL tm
L
Tr e A At A tt
A vol B
dtTr e et
m m A
S m m vol B
many-body systems in higher dimensions
a
spin lattice continuum limit
2
AS
a A – area of a flat separation surface 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
2ln
A LS C a
a a
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 statea is a cutoff
C – topological term (first pointed out in D=3 by Preskill and Kitaev)
Ising spin chains
11
( )N
X X ZK K K
K
H
2
1( , ) log
6 2
NS N
2
1( , ) log | 1|
6S N
1 | 1| 1 off-critical regime at large N
critical regime 1
RG-evolution of the entropy
entropy does not increase under RG-flow (as a result of integration of high energy modes)
IR IR
UV
1 is UV fixed point
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
| 1|
ln6
ln6
cS ma
c LS
a
What is the entanglement entropy in a fundamental theory?
CONJECTURE(Fursaev, hep-th/0602134)
3
4FUNDN
cs
G
FUNDs - entanglement entropy per unit area for degrees of freedom of the fundamental theory in a flat space
( 4)d
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
3
4BH H
N
AS c
G
HA
Bekenstein-Hawking entropy
- area of the horizon
BHS - measure of the loss of information about states underthe horizon
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 :
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:
assumption
... ...fundamental low energydof dof
Ising model:
“fundamental” dof are the spin variables on the lattice
low-energies = near-critical regime
low-energy theory = QFT (CFT) of fermions
B
1
B
2
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
B
B
Bfluctuations of are induced by fluctuations of the space-time geometry
entanglement entropy in the semiclassical approximation
[ , ]
4 3
( ) [ ][ ] , ( ) [ , ] [ ] [ , ],
1 1[ ] ,
16 8
( ) ln ( ) [ , ],
n n
I gmatter
M M
Z T Dg D e Z T I g I g I g
I g R gd x K hd yG G
F T Z T I g
a standard procedure
( , , )
4
1 1 1 2
( , )
2(2 ) ( ),
( )( , , ) ( , , ) (2 ) ,
8
lim lim 1 ln ( , ) ,
( )
4
( )
n
I g
B
regular
M
regular
nn g m
m
g
Z T e
R gd x R A B
A BI g I g
G
S Tr Z T S Sn
S
A BS
G
A B
fix n and “average” over all possible positionsof 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
what are the conditions on the separating surface?
conditions for the separating surface
( )( 2 )( , , ) ( , , ) 4
( )( )( 2 ) 44
2
( , ) ,
,
( ) 0, ( ) 0
regularA B
I g I g G
B B
A BA BGG
B
Z T e e e
e e
A B A B
the separating surface is a minimalco-dimension 2 hypersurface in
, ,
2 2
0
,
1, 0,
0,
0.
iji j
ij
n
p
X X X
n p
n p np
k n
k p
- induced metric on the surface
- normal vectors to the surface
- traces of extrinsic curvatures
Equations
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
2
4
4 4
g q
g
q
div finq q q
divq
divq
bare ren
S S S
AS
GS
S S S
AS
A AS
G G
the UV divergences in the entropy areremoved by the standard renormalization of thegravitational couplings;
the result is finite and is expressed entirely in terms of low-energy variables and effective constants like G
2, , ;
; ;
0
0
iji j
B
t
A d y X X X
X
t
B
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 !
variational formulae for EE
S M
M
S
- 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
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:1 2
1
1 2
( )k h p p
k
h
p p
- the mean curvature
- surface tension
-pressure difference across the film
- equilibrium equation
the Plateau Problem there are no unique solutions in general
the Plateau Problem simple surfaces
The structure of part of a DNA double helix
catenoid is a three-dimensional shape made by rotating a catenary curve (discovered by L.Euler in 1744) helicoid is a ruled surface, meaning that it is a trace of a line
the Plateau Problem
Costa’s surface (1982)
other embedded surfaces
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
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