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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
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
Lecture 1
Quantum Entanglement
Quantum state of particle «1» cannot be described independently from
particle «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”)
a general definition
 ( A, a | B, b)
1 ( A | B)    ( A, a | B, a),
a
 2 (a | b)    ( A, a | A, b),
A
A
1  Tr2  ,  2  Tr1  ,
a
S1  Tr1 1 ln 1 , S2  Tr2 2 ln  2
e H / T

Tr e  H / T
 S1  S2
“symmetry” of EE in a pure state
 
   C Aa A a
aA
1 ( A | B)   C Aa C  Ba  1  CC 
a
 2 (a | b)   C AaC  Ab ,   2  C T C 
A
if d   Ce ,  2 e   e
S1  S 2
(  0) 
1 d    d 
consequence: the entropy is a function of
the characteristics of the separating
surface
S1  S2  f ( A)
in a simple case the entropy
is a fuction of the area A
S
S
A
A ln A
- in a relativistic QFT (Srednicki 93, Bombelli et al, 86)
- in some fermionic condensed matter
systems (Gioev & Klich 06)
subadditivity of the entropy
| S1  S2 |  S  S1  S2 , S  Tr  ln 
S1  S2  S1
2
strong subadditivity
1 2
S1  S2  S1 2  S1
2
equalities are applied to the von Neumann entropy
and are based on the concavity property
effective action approach to EE in a QFT

S (T )   lim n1 Tr1 1n   lim  2     1 ln Z (  , T )
n
Z ( , T )
- “partition function”
1  Tr2 
ln Z (  , T ) -effective action is defined on manifolds
with cone-like singularities
  2 n
- “inverse temperature”
theory at a finite temperature T

e H / T
1
{1},{2 }  { '1},{ '2 } 
N
I [ ] 
{ '1 },{ '2 }

 
{ 1 },{
[ D ] e  I [ ]
2}
classical Euclidean action for a given model
1  Tr2 
{ '1 },{2 }
1
{1} 1 { '1}   d 2  [ D ] e  I [ ]
N
{1 },{2 }
Example: 2D case
{ '1}
{2 }
1
2
  1/T
these intervals are identified
1
2
{1}
{2 }
 0
the geometrical structure
Z (   2 , T ) 
Z (T ) - standard
partition function
Tr1 
3
1
case
n3
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:
R  2(2   )
(2)
( B)
derivation of entanglement entropy in a
flat space has to do with gravity effects!
summary of calculation:
1)
find a family of manifolds
given system
corresponding to a
have conical singularities on a co-dimension 2 hypersurface
(separating surface)
2)
compute
  2 n
Z1 (  , T )
- partition function,
- “geometrical” inverse temperature
3) S1 (T )  lim  2     1 ln Z1 (  , T )
Spectral geometry: example of calculation
L   2
 tL
Tr (e )
1
2

A

A
t

A
t
 ...
1
2
D/2  0
(4 t )
  2  
A1  

 vol ( B )
3   2 

1 dt
 tL
 tm 2
    Tr (e )e
2 2 t
1  1
2
2 2

 m ln m   A1
2 
2
32  

S
1
48
 1
2
2 2
 2  m ln m   vol ( B )


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!
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)
for ground state
S
A L
  C ln a
2
a
a
(Fursaev,
hep-th/0602134)
a is a cutoff
C – topological term (first pointed out in D=3 by Preskill
and Kitaev)
Ising spin chains
N
H   ( KX  KX1   KZ )
K 1
off-critical regime at large N
 1
1
S ( N ,  )   log 2 |   1|
6
critical regime
 1
1
N
S ( N ,  )  log 2
6
2
|   1| 1
RG-evolution of the entropy
UV
  1 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 |   1|
c
S   ln ma
6
At the critical point it is equivalent to a 2D CFT with 2 massless
fermions each having the central charge 1/2
c L
S  ln
6 a
What is the entanglement entropy
in a fundamental theory?
CONJECTURE
(Fursaev, hep-th/0602134)
(d  4)
- entanglement entropy per unit area for 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
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)
our conjecture :
● yields the value for the fundamental entropy in flat space in
terms of gravity coupling
● horizon entropy is a particular case
the geometry was “frozen” till now:
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?
assumption

fundamental
dof
...

...
low energy
dof
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
B
This means that:
(if the boundary B of the
separating surface is fixed)
1
B
2
the geometry of the separating
surface B is determined by a
quantum problem
fluctuations of B are induced by
fluctuations of the space-time geometry
entanglement entropy in the semiclassical approximation
a standard procedure
Z (T )   [ Dg  ][ D ] e  I [ g , ] ,
1
I[g ]  
16 G
Z (T )
1
M n R gd x  8 G
F (T )   ln Z (T )
4
I [ g ,  ],

M n
I [ g ,  ]  I [ g ]  I matter [ g ,  ],
K hd 3 y ,
Z ( , T )
 I (  , g , )
e

B

Mn
fix n and “average” over all possible positions
of the separating surface on
R gd 4 x   R regular  2(2   ) A( B),
I (  , g ,  )  I regular (  , g ,  )  (2   )
S   lim n1

Tr1 1n   lim  2     1 ln Z (  , T )  S g  S m ,
n
- entanglement entropy of quantum matter (if
Sm
Sg 
A( B )
A( B )
,
8 G
one goes beyond the semiclassical approximation)
A( B )
4G
- pure gravitational part of entanglement entropy
- some average area
what are the conditions on the separating
surface?
conditions for the separating surface
Z ( , T )
e
 I (  , g , )
e
B
e
I
regular
(  , g , )
e
 (   2 )
A( B )
4G
,
B
A( B )
 (   2 )
4G
e

A( B )
4G
,
B
 A( B )  0,  A( B )  0
2
the separating surface is a minimal
co-dimension 2 hypersurface in
Equations
X   X ,i X , j   0

 ij
n ,




ij
- induced metric on the surface
p
- normal vectors to the surface
n   p  1, np  0,
2
2
kn      n  0,
k p     p  0.

- 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
S  S g  Sq
A
Sg 
4G
Sq
Sq  S
div
q
S
fin
q
the UV divergences in the entropy are
removed by the standard renormalization of the
gravitational couplings;
S
div
q
A
2
A
A
div
 Sq 
4Gbare
4Gren
the result is finite and is expressed entirely in
terms of low-energy variables and effective
constants like G
Stationary spacetimes: simplification
 A   d 2 y   ij X ,i X ,j X  ;  0
B

 X 

 ;


 t

 ;
a Killing vector field
0

t
- a constant time hypersurface (a Riemannian manifold)
B
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
 S   M 
-change of the entropy under
the shift of a point particle
M
-mass of the particle

- shift distance
 S  
- 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:
k  h( p1  p2 )
k
- equilibrium equation
- the mean curvature
h 1 - surface tension
p1  p2 -pressure difference across the film
the Plateau Problem
there are no unique solutions in general
the Plateau Problem
simple surfaces
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 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 singly periodic Scherk surface
approaches two orthogonal planes
a surface was found by Chen and
Gackstatter
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