PRESENTS

DANNY TERNO & ETERA LIVINE

With contributions fromAsher Peres,

Viqar Hussain and Oliver Winkler

PRODUCTION

when

met

Noncovariance of reduced density matrices Noninvariance of entropy Implications to holography and thermodynamics

Outline

Entanglement and black hole entropy Entanglement and Hawking radiation

Volume 1: new properties

Volume 2: old applications

Volume 1: new properties

Noncovariance (1)

spin

momentum

classical info

DExample: Lorentz transform of a single massive particle

( ) ( ) ,d p p p

*( ) ( ) ( )d p p p 2 2/ 2( ) ,

1

p N e

m

p

,)],([,)( ppWDpUtransform: valong the z-axis

important parameter:

21 1m

vv

sin

cos

Partial trace is not Lorentz covariantSpin entropy is not scalarDistinguishability depends on motion

1

2

3

0.2

0.4

0.6

0

0.2

0.4

0.6

0.8

1

2

3

0

0.2

0.4

0.6

0.8

0

3.0

6.0

Entropy

Peres and Terno,Rev. Mod. Phys. 76, 93 (2004)

sin

cos

21 1m

vv

herethere there

here( ) 0S

here

trace out “there”

2( / )cS f A l

Bombelli et al, Phys. Rev. D34, 373 (1986)

i i ii

c Holzhey, Larsen and Wilczek, Nucl. Phys. B424, 443 (1994) Callan and Wilczek, Phys. Lett.B333, 55 (1994).

Noncovariance (2)

Geometric entropy

c Pl l

1

=?

no correlationsno Bell-type violations

1 2( ) ( )U U not irreducible

Transformations do not split into here and there spaces

i i ii

c

Decomposition of Lorentz transformations

Terno, Phys. Rev. Lett. 93, 051303 (2004)

1 2( ) ( ) ( )U U U

trivial 1D rep

irrep of 1-particle states

Noninvariance (1)

Yurtsever, Phys. Rev. Lett. 91, 041302 (2003)

24 P

AS

l

Bekenstein, Lett. Nuovo Cim. 4, 737 (1972) ….Busso, Rev. Mod. Phys. 74,825 (2002)

3 3PN L l

4max ( )E c G L

2 2PS L l

Pm

maxE

Boundary conditions & cut-offs

Model

Number of degrees of freedom " " SN e

N is frame-dependent

Lorentz boost: factors 1/γ

26A L 2' 2 (1 2 / )A L

21 v

max max' (1 )S S v

Spacelike holographic bound

both area and entropy change

Saved ?

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

v

' 'S A

Terno, Phys. Rev. Lett. 93, 051303 (2004)

Black holes: invariance

( )S S A

Hawking’s area theorem constS

Model 1+1 calculations: the same crossing point, relative boost

Two observers with a relative boost

-

Fiola, Preskill, Strominger,Trivedi, Phys. Rev. D 59, 3987 (1994)

-

constS

Noninvariance (2)Accelerated cavity

Moore, J. Math. Phys. 11, 2679 (1970)Levin, Peleg, Peres, J.Phys.A 25, 6471 (1992)

Accelerated observers & matter beyond the horizon

Terno, Phys. Rev. Lett. 93, 051303 (2004)

Volume 2: old applications

Entanglement on the horizon

Requirement: SU(2) invariance of the horizon states

Object: static black hole

0J

States: spin network that crosses the horizon

Qubit BH

1/ 2 2A a n

1

1 N

k kkN

2 0k J

tr log logS N

density matrix

2 2

3

21

n nnC

Nn n

Standard counting story

area

constraint 2n spins

number of states

entropy 32log 2 log 2 logS N n n

Fancy counting story

entropy

( 1)ja j j

Entanglement

Measure: entanglement of formation

i i ii

w ({ }) ( )i i ii

S w S tr

{ }( ) min ({ })ES S

2 vs 2n-2

States of the minimal decomposition

0 00,0, 0,0,a bdegeneracy indices

11 1 1 1 1 13

1, 1, 1,1, 1,0, 1,0, 1,1, 1, 1,a b a b a b

Alternative decomposition: linear combinations Its reduced density matrices: mixturesEntropy: concavity

unentangled fraction 140f

entanglement 34( | 2) log3ES

n vs n 12( ) logES n

Entropy of the whole vs. sum of its parts

half( ) 2 ( ) 3 ( )ES S S

reduced density matrices ( ) ( ) log 2A BS S n

( ) ( ) ( )A BS S S

BH is not madefrom independent qubits,

but…Livine and Terno, gr-qc/0412xxx

Entanglement and Hawking radiation

Hussein,Terno and Winkler, in preparation

in grav mat

out inU

mat grav outtr

mat out( ) ( )ES S

Summary

when

met

Reduced density matrices are not covariant Entropy (and the # of degrees of freedom) are observer-dependent

Entanglement is responsible for the logarithmic corrections of BH entropy

Entropy of the BH radiation = entanglement entropy between gravity and matter

Thanks to

Jacob BekensteinIvette Fuentes-SchullerFlorian GirelliNetanel LindnerRob MyersJohnathan OppenheimDavid PoulinTerry RudolphFrederic SchullerLee SmolinRafael SorkinRowan Thomson

TechniqueTechnique:

Unruh effect

'', ' ', '

', ' '

1kr

k m k mrk m k

e r rZ

( )S

Entropy usually diverges maxS

General: cut-off

: lim ( , ) ( , )l

S S l S l

''

2

krk

r

Z e

a

trB

renormalization of entropy

UnruhUnruh ++

Audretsch and Müller, Phys. Rev. D 49, 4056 (1994)

Matter outside the horizon

'', ' ', ' , ,

', ' ',

1 1 ( )!! !

k kqrk m k m n k m k m

r qk m k kk m

n qe r r e r r

Z Z n q

n particles in the mode (k,m)

''

2

krk

r

Z e

a

Splitting: usual + super

1ne Special case

renormalized quantitiesE ne

( ( 1) log( 1))S n n n n e

temperature

1 ln( 1)ndS dEdn dnT

Of what?

/ ( ln( 1)) 0S E T n n e

two subsystemsGeneral case:

temperature is undefined

Two observers: the same acceleration,

different velocities

1 2S S

/S E T