Transcript entanglement entropy in quantum gravity

“Gravity in three dimensions”,
ESI Workshop, Vienna, 24.04.09
Quantum Entanglement and Gravity
Dmitri V. Fursaev
Joint Institute for Nuclear Research
and
Dubna University
plan of the talk
Part I (a review)
● general properties and examples (spin chains, 2D CFT, ...)
● computation: “partition function” approach
● entanglement in CFT’s with AdS gravity duals (a holographic formula
for the entropy)
Part II (entanglement entropy in quantum gravity)
● suggestions and motivations
● tests
● consequences
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”)
definition of entanglement entropy
 ( A, a | B, b)
1 ( A | B)    ( A, a | B, a),
a
 2 (a | b)    ( A, a | A, b),
A
1  Tr2  ,  2  Tr1  ,
A
a
S1  Tr1 1 ln 1 , S 2  Tr2  2 ln  2
e H / T

Tr e  H / T
 S1  S 2
“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 Aa C  Ab ,   2  C T C 
A
if d   Ce ,  2 e   e
S1  S 2
(  0) 
1 d    d 
Entanglement in many-body systems
spin lattice
continuum limit
Entanglement entropy is an important physical quantity which helps
to understand better collective effects in stringly correlated systems
(both in QFT and in condensed matter)
spin chains
(Ising model as an example)
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
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
Behavior near the critical point
and RG-interpretation
UV
  1 is UV fixed point
IR
IR
The entropy decreases under the evolution to IR region because the
contribution of short wave length modes is ignored (increasing the
mass |   1| is equivalent to decreasing the energy cutoff)
more analytical results in 2D
ground state entanglement
on an interval
L
Calabrese, Cardy
hep-th/0405152
L1 is the length of
massive case:
c
1
S  ln
6 ma
a is a UV cutoff
1/ m
massless case:
 L1 
c  L
S  ln 
sin
  2g
6 a
L 
c L1
S  ln
6 a
L1
analytical results (continued)
 L1 
c  L
S  ln 
sin

3 a
L 
L1 is the length of
ground state entanglement for a
system on a circle
 L1 
c  
S  ln 
sinh
3 a
 
system at a finite temperature
T  1/ 
Entropy in higher dimensions
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)
geometrical structure of the entropy
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
(DF, hep-th/0602134)
a is a cutoff
“partition function”
and effective action
replica method

S (T )   lim n1 Tr2  2 n   lim  2     1 ln Z (  , T )
n
Z ( , T )
- “partition function” (a path integral)
ln Z (  , T )
-effective action is defined on manifolds
with cone-like singularities
  2 n
- “inverse temperature”
 2  Tr1 
theory at a finite temperature T

e H / T
1
{ 2 },{1}  { '2 },{ '1} 
N
I [ ] 
{ '2 },{ '1 }



{
[ D ] e  I [ ]
2 },{ 1 }
classical Euclidean action for a given model
 2  Tr1 
{ '2 },{1 }
1
{ 2 }  2 { '2 }   d1  [ D ] e  I [ ]
N
{ 2 },{1 }
Example: 2D case
{1}
{ '2 }
1
2
  1/ T
these intervals
are identified
1
2
{1}
{2 }
 0
the geometrical structure for Tr2 2
conical singularity is located at the separating point
3
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!
entanglement in CFT’s and
a “holographic formula”
Holographic Formula
Ryu and Takayanagi,
hep-th/0603001, 0605073
AdSd 1(bulk space)
B
minimal (least area)
surface in the bulk
B  B
4d space-time manifold (asymptotic
boundary of AdS)
entropy of entanglement
G( d 1)
B separating surface
A( B)
is measured in terms of the area of B
S
4G( d 1)
is the gravity coupling in AdS
Holographic formula enables one to compute entanglement entropy
in strongly correlated systems with the help of classical
methods (the Palteau problem)
2D CFT on a circle
L1 is the length of
 L1 
c  L
S  ln 
sin

3 a
L 
ground state entanglement for a
system on a circle
c – is a central charge
ds 2  l 2  d  2  cosh 2  dt 2  sinh 2  d 2 
l
- AdS radius
gravity
minimal surface =
a geodesic line
2 L1
1 
ds 2CFT  ds 2   0
L
A
2
2   L1 
A is the length of the
cosh  1  2sinh 0 sin 

l
geodesic
 L 
L
0
- UV cutoff
e 
a
A
c  0
  L1  
S
 ln  e sin 

-holographic formula
4G3 3 
L


3l
c
2G3
- central charge
a finite temperature theory:
a black hole in the bulk space
A( B1 ) A( B2 )
S1 

 S2
4G3
4G3
c  1

S1  ln 
sinh  TL1 
3   Ta

Entropies are different (as they should be) because there are topologically
inequivalent minimal surfaces
a simple example for higher dimensions
2
l
ds52  2 (dz 2  ds42 )
z
2
A
1
l3
A
2
a
A
S
4G5
2
a – is IR cutoff
l3
A
2
a G5
N2
A
2
a
B
l3
G5
N 2,
( SU ( N ))
Motivation of the holographic formula
DF, hep-th/0606184
Low-energy approximation
Partition function for the bulk gravity (for the “replicated” boundary CFT)
Z CFT (  , T )  Z AdS (  , T )
Z AdS (  , T )   [ Dg  ][ D ] e  I [ g , ] ,
ln Z AdS (  , T )
I [ , g ]  
  2 n
I [  , g ,  ]  I [  , g ]  I matter [ g ,  ],
1
d 1
R
gd
x  b.t. ,
(n)

M
d 1
16 Gd 1
F (  , T )   ln Z AdS (  , T )
I [ g ,  ],
Boundary conditions
M (n) d 1  M ( n) d
The boundary manifold has conical singularities at the separating
surface.
Hence, the bulk path integral should involve manifolds with conical
singularities, position of the singular surfaces in the bulk is
specified by boundary conditions
Semiclassical approximation

M
(n)
R gd d 1 x   R regular gd d 1 x  2(2   ) A( B ),
d 1
I (  , g ,  )  I regular (  , g ,  )  (2   )
A( B )
,
8 Gd 1

S   lim n1 Tr1 1n   lim  2     1 ln Z AdS (  , T ),
n
S 
A( B )
4Gd 1
- holographic entanglement entropy
conditions for the singular surface in the bulk
Z AdS (  , T )
e
 I (  , g , )
e
I
regular
B
e
A( B )
 (   2 )
8 Gd 1
(  , g , )
e
 (   2 )
A( B )
8 Gd 1
,
B
 (   2 )
e
A( B )
8 Gd 1
,
  2
B
 A( B )  0,  2 A( B )  0
the separating surface is a minimal
least area co-dimension 2
hypersurface
Part II
entanglement entropy
in quantum gravity
entanglement has to do with quantum
gravity:
● entanglement entropy allows a holographic interpretation for CFT’s
with AdS duals
● 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
quantum gravity theory
Can one define an entanglement entropy, S(B), of fundamental degrees
of freedom spatially separated by a surface B?
How can the fluctuations of the geometry be taken into account?
the hypothesis
● S(B) is a macroscopical quantity (like thermodynamical entropy);
● S(B) can be computed without knowledge of a microscopical
content of the theory (for an ordinary quantum system it can’t)
● the definition of the entropy is possible for surfaces B of a certain
type
Suggestion (DF, 06,07): EE in quantum gravity
between degrees of freedom separated by
a surface B is
A( B)
S ( B) 
4G
1
2
B is a least area minimal hypersurface
in a constant-time slice
conditions:
● static space-times
● slices have trivial topology
the system is determind by
a set of boundary conditions;
subsets, “1” and “2” , in the bulk
are specified by the division of the
boundary
● the boundary of the slice is simply connected
a Killing symmetry + orthogonality of the Killing field to constant-time
slices:
a hypersurface minimal in a constant time slice is minimal in
the entire space-time
a “proof” of the entropy formula is the same as the motivation
of the “holographic formula”
Higher-dimensional (AdS) bulk -> physical space-time
AdS boundary
->
boundary of the physical space
Slices with wormhole topology (black holes,
wormholes)
on topological grounds, on a space-time slice which locally is R1  S n
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
slices with wormhole topology
we follow the principle of the least total area
EE in quantum gravity is:
A( B1 )
S
,
4G
A( B1 )  A( B2 )  A( B0 )
A( B2 ) A( B0 )
S

,
4G
4G
A( B1 )  A( B2 )  A( B0 )
B1, B2
are least area minimal hypersurfaces homologous,
respectively, to D , D
1
2
consequences:
if
D1   the EE is
A( B0 )
S
4G
• for black holes one reproduces the Bekenstein-Hawking formula
• wormholes may be characterized by an intrinsic entropy associated to
the area of he mouth
Entropy of a wormhole: analogous conclusion (S. Hayward,
P. Martin-Moruno and P. Gonzalez-Diaz) is based on variational formulae
tests
inequalities for the von Neumann entropy
strong subadditivity property
S1  S2  S1 2  S1
2
Araki-Lieb inequality
1 2
| S1  S2 |  S
equalities are applied to the von Neumann entropy
and are based on the concavity property
S1  S2  S1 2  S1
strong subadditivity:
c
d
1
a
c
2
f
d
f
b
a
b
S1  Aad , S 2  Abc , (4G  1)
S1  S 2  Aad  Abc  Aaf  A fd  Abf  A fc 
( Aaf  Abf )  ( Afd  Afc )  Aab  Adc  S12  S12
generalization in the presence of closed least area
surfaces is straightforward
2
Araki-Lieb inequality, case of
slices with a wormhole topology
S1  S2  S
entire system is in a mixed state
because the states on the other
part of the throat are unobervable
A( Bk )  Ak , k  0,1, 2, assume that
A2  A0
1) A1  A2  A0
then S1  A1 , S 2  A1  A0 and
2) A1  A2  A0
then S1  A2  A0 , S 2  A2 and
3) A2  A0  A1  A2  A0
then S1  A1 , S 2  A2
S2  S1  A2  A1  A0  S0 , if
S 2  S1
S1  S 2  A1  A2  A0  S0 , if
S1  S 2
S 2  S1  S 0
S1  S 2  S0
and
variational formulae
• for realistic condensed matter systems the entanglement entropy is a
non-trivial function of both macroscopical and microscopical
parameters;
• 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
simple variational formulae
 S   M z
M  mass of a particle
 z  shift
 S 1037 if M  1 g ,  z  1cm
 S  O (1) if  z is a Compton wavelength
 S  l z
  string tension
l  lenght of the segment
variational formula for a wormhole
dr 2
ds  e dt 
 r 2 d  2  2e 2 dx  dx   r 2 d  2
2 E (r )
1
r
r  rH : g rr (rH )  0
- position of the w.h. mouth (a marginal sphere)
2
E 
2
2

 S  w V
2
E  E (rH ) 
1
rH
2
A(rH )
4G
4 rH 3
V
3
w  e 2 T
- a Misner-Sharp energy (in static case)
S
  e 2     r 
stress-energy tensor of the matter on the mouth
E
 4 rH w
2
rH
- a surface gravity
For extension to non-static spherically symmetric wormholes
and ideas of wormhole thermodynamics
see S. Hayward 0903.5438 [gr-qc];
P. Martin-Moruno and P. Gonzalez-Diaz 0904.0099 [gr-qc]
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)
• entanglement entropy is given by the “Bekenstein-Hawking” formula in
terms of the area of a co-dimensiin 2 hypersurface B ; black hole
entropy is a particular case;
• entropy formula passes tests based on inequalities;
• wormholes may possess an intrinsic entropy; variational formulae for a
wormhole might imply thermodynamical interpretation
(microscopical derivation?, Cardy formula?....)
Extension of the formula for entanglement entropy
to non-static space times?
minimal surfaces on
constant time sections