Danny Terno

Entropy and entanglementEntropy and entanglementon the horizonon the horizon

joint work with

Etera Livinegr-qc/0508085

gr-qc/0505068Phys. Rev. A 72 022307 (2005)

Gauge invariance: SU(2) invariance at each vertexbecomes SU(2) invariance for the horizon states

Object: static black hole

0J

States: spin network that crosses the horizon

in LQGBlack hole

Comment 1: no dynamicsComment 2: closed 2-surface

Definition of a “black hole”:complete coarse-graining of the spin network inside

Microscopic states: intertwiners

Features & assumptions

12

( 1)j

j ja

j

Area spectrumThe probing scale

312 2,1, ,...j

The flow: scaling and invariance of physical quantities

We work at fixed jComment: reasons to be discussed

For starters: a qubit black hole

SummaryQubit black hole 3

2log 2 log 2 logS N n n

Spin-j black hole 32log 2 log(2 1) logS N n j n

Entanglement between halves of the horizon

32( : ) ( ) ( ) ( ) logA BI A B S S S n

12: ( ) logn nE n

Logarithmic correction = quantum mutual information

Area rescaling micro

macro

A n

A np

1/ 2 2A a n

1

1 N

k kkN

2 0k J

tr log logS N

density matrix

2 2

3

21

n nnCNn n

Standard counting story

area

constraint

2n spins

number of states

entropy 32log 2 log 2 logS N n n

Fancy counting story

entropy

CombinatoricsSchur’s duality

22

,0 0

n n nj

j n jj j

V

C H

,n j [ , ]n j n j is the irrep of the permutation group

41 2 0 1 22 3V V V V Example:

(2 ),dimn

j n jd =#standard tableaux

1

3

2

4

1

2

3

4

(2 )njN d

0 0 ,n jV H

Entanglement

a brief historyAncient times: 1935-1993

“The sole use of entanglement was to subtly humiliate the opponents of QM”

Modern age: 1993-

Resource of QITTeleportation, quantum dense coding, quantum computation….

Postmodern age: 1986 (2001)-Entanglement in physics

1/3

Entanglementa closer encounter

2

2

| | 0

0 1 | |

Pure states

, ( ) tr logA B S

S

0.2

0.2

0.4

0.4

0.6

0.6

0.8

10.8

1

Mixed states hierarchy

Direct product

Separable

Entangled

A B

, 0, 1i ii A B i i

i i

w w w

, 0, 1i ii A B i i

i i

w w w 2/3

Entanglement of formation

i i ii

w ({ }) ( )i i i

i

S w S tr { }( ) inf ({ })FE S

Minimal weighted averageentanglement of constituents

Entanglementmeasures

“Good” measures of entanglement: satisfy three axioms

Coincide on pure states with ( )E S

sep( ) 0E

Do not increase under LOCC

Zero on unentangled states

Almost never known

3/3

Entanglementcalculation

,

2 2 2 22 2 2

, ,(( ) )A B

A BA B

n k n kj j

n j n k jj jV V

C C C

0 0 ( ) , ,

0( )

A B

k

j n j n k jjV

H

Clever notation j jBA

j jABj a b

1( ) log(2 1)B AN j j

j

E d d j

2 vs 2n-2

States

0 00,0, 0,0,a b

Unentangled fraction 140f

Entanglement 34( | 2) log3E

degeneracy 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

Entanglement

n vs n 12( ) logE n

half( ) 2 ( ) 3 ( )S S E

Entropy of the whole vs. sum of its parts

Reduced density matrices

( ) ( ) log 2A BS S n

( ) ( ) ( )A BS S S

BH is not madefrom independent

qubits,but…

Logarithmic correction equals quantum mutual information :( : ) ( ) ( ) ( ) 3A B A BI A B S S S E

Why qubits (fixed j)?Answer 1: Dreyer, Markopoulou, Smolin

Comment: spin-1

Answer 2: if the spectrum is ja j

Answer 3: irreducibility2

2,

0 0

n n nj

j n jj j

V

C H

Decomposition into spin-1/2. 1-1 relation between the intertwiners.No area change

Entropy

1 2 34-1

32( ) 2 log(2 1) logj jS n j n

Explanation: a random walk with a mirror

-4 -3 -2

j

Practical calculation: RWM(0)=RW(0)-RW(1)

3log

2

Universalityand the random walks

1 1 1k k k kV V V V V

Calculations & asymptotics

( ) 1( ) ( 1 )

2k ij ik ik n

nRW j d e e e

2(2 1) njN

n

2

0

(2 1) njN

n n

12 (2 ) 2

2

1( ) ( )

2i j i i n

nRW j d e e e

Asymptotics

Entanglement:

12( ) logE n half( ) 2 ( ) 3 ( )S S E

1( ) log(2 1)B AN j j

j

E d d j n vs n

Area renormalizationGeneric surface, 2n qubits

22

,0 0

n n nj

j n jj j

V

C H

micro 1 22A na

Complete coarse-grainingThe most probable spin: maximal degeneracy

12

max 2n

j macroA n

Horizon, 2n qubits split into p patches of 2k qubits

(2 )0 0

1(2 1)

Aj

j

dkB

k j j j jj a

d a a jN

1

The most probable spin: maximal degeneracy

p :J

(2 ) (2 2 )

(2 )0

tr[ ] ( )k n kk

j jk

j

d dJ J J j

N

different options

p maxJ j k

The average spin:

p

2J J k

micro 1 22A na

macro (2 )kA pA np

Area rescaling:

p(2 )k JA a k

Open questionsOpen questions

Dynamics: evolution of entanglement dynamical evolution of evaporation "H=0" section & the number of states

Semi-classicality: requiring states to represent semi-classical BH rotating BH

Open questions

EvaporationA model for Bekenstein-Mukhanov spectroscopy (1995)

Minimal frequency <= fundamental j

Probability for the jump is proportional to the unentangled fraction

11( |1) 2 k

tP k 1

1( |1) 2 ktP k

11 0(2 |1) ( )mtP m f

140 ( )f

number of blocks

unentangled fraction(of 2-spin blocks)

1m

Entanglementcalculation

( ) ( )A j j jj

w

•Alternative decomposition: linear combinations •Its reduced density matrices: mixtures•Entropy: concavity

, A Bj jj j

j j

ja b j jAB D Dja b

c j a b

Clever notation (2):*

' 'j j j j j j

j

ja a ja b ja bb

c c j j

j

j ja aa

Clever notation (3):

1

( ( )) [ ( ) ( ( ))]

( ) ( )

A j j jj

A BNj j j j j

j j

w S w S S

w S c c S

Coup de grâce: