Quantum info tools & toys

for quantum gravity

LOOPS `05LOOPS `05

Daniel TernoPerimeter Institute

MEASUREMENTS

OutlineOutline

POVMInformation gain

DYNAMICSCompletely positive mapsNon completely positive maps

ENTANGLEMENTEntang’t 101BH applications

0iE iiE 1 ( ; ) tr iP i E

( )X

dE x 1 ( ; ) tr ( )p x E x

tr 0 i j i iE E E P

POVMdiscrete

continuous

Projections/von Neumann

Realization ancillary system+ unitary evolution+ PVM

1

22

( )

( )X

X

M x dE x

M x dE x

22 1M MMoments

MEASUREMENTSMEASUREMENTS

Construction: covariance considerations and /or optimization

Use: decision/identification unsharp properties non-commuting variables/ phase space observables

Coexistence & uncertainty

( )QQ xdE x x x x dx ( )PP pdE p

,

( , ) tr ( , )X Z

Prob q X p Z dqdp E q p ( ) ( , )s

QE q E q p dp ( ) ( , )sPE p E q p dq

( ) ( ) ( )sQ QE q E q s q

, s sQ Q P P

2 2 2sQ Q s

12Q P

s sQ P

1e

2e

3e

12

2313

1n

2n 3n

Classical geometry

• 6 edges1 2 3 4 5 6, , , , ,e e e e e e

1 2 3 12 13 23, , , , , e e e

1 2 3 1 2 3 2 1 3, , , , , e e e e e e e e e

• 3 edges, 3 angles

• 3 areas, 3 dihedral angles1 2 3 12 13 23, , , , , n n n

• 4 areas, 2 dihedral angles

• 3 edges, 3 products

Volume1 2 3

21 2 3

2

( )

36V

n n n

e e e

1 2 3 4 0 n n n n

TETRAHEDRONTETRAHEDRON

1 2 3 4 1

4

1j j j j jiH H

0H 1 2 3 4ˆ ˆ ˆ ˆ 0 J J J J

ˆ ˆ ˆij i jJ J J 12 23 1 2 3

ˆ ˆ ˆ ˆ ˆ ˆ[ , ]J J i iU J J J

5 commuting observables

1212 23

ˆ ˆ ˆJ J U Standard uncertainty relation

24-vertex

ˆ ˆU V1j

3j

4j

2j

0 1 2 3 4 1 2 3 4dim min( , ) max( , ) 1j j j j j j j j H2 1j

Basis: eigenvectors of

12j

21 2

ˆ ˆ( )J J

Quantum mechanics

QuestionQuestionHow uncertain is the shape and how this uncertainty decreases in the classical limit?

1 , i ij j j a j j

Observation 1 [numeric]2 3

max 1 2 3 4( ) max ( , , , )u j V j j j j j

212 1 2 12(cos )J j a a

12 231(cos ) (cos )j

Naïve bound

Observation 2

223 2 3 23(cos )J j a a

More precise formulation: quantum communication problem

1. Fix the areas

12 23 12 23 0( , ) ( , ) H2. Encode the angles

3. Decode 12 23 12 23( , ) ( ' , ' )

4. Calculate the figure of merit ( , ')D

5. Average over all angles ( , ')D

6. Take the limit j

PriorsPriors

1e

2e

3e

12

2313

1n

2n 3n

At least two natural probabilitydistributions

12 13 23 12 13 23( , , ) cos cos cosdP d d d

12

23

12

23

(1:1:1:1)(1: 2 : 2 : 2)

12 13 23 12 13 23( , , ) cos cos cosdP d d d or

Fixing 4 areas

Encoding & distanceEncoding & distanceˆij ijJ J

Condition

Figure of merit 12 23J J

POVMPOVM(2 1)( , ) , ,

4jE Spin POVM

( 1) ( , ) ( , )j E d J n

12 1 2 1 1 2 2 1 1 2 2 1 2( 1)( 1) ( , ) ( , ) ( , ) ( , )J j j E E d d n n

0

0.25

0.5

0.75

1 0

0.5

1

1.5

0.65

0.7

0.75

0.8

0.85

0

0.25

0.5

0.75

1

(1,1,1,1) tetrahedron

12 2323

J J max 3u

12 2320 7 3ˆ ˆ 0.6533

81 2J J

32 2 21 2 1 2( , , 1 )iia a e a a e

12 232ˆ ˆ3

J J

12 2389

J J

ILLUSTRATION

Optimization:

Constraint:

Independent variables: phases

2 3 0

ˆ 0U

12 23ˆ ˆJ J

2

3

Unitary

Completely positive

†U U

†a a

aA A

†( ) tr ( )B BU U

†' Ap U p U

A p U

Def: unital map ( ) 1 1

Definition: ( ) 0A B AB 1

Physics:

DYNAMICSDYNAMICS

1 A B A BAB A B i i B j A j ij i j

A Bd d \1 1 1 1

† †

,( ) A B

ij i jM M U U

ij ij i j A Bd d

Unitary evolution & partial trace

Non completely positive † †a a b b

a bA A B B

†( )A a A a i iaM M c Physically acessible

Causal setsCausal sets

( )H

( )HU

CNOT gate 0 0 0 0

0 1 0 1

1 0 1 0

1 1 01

( ) ( ) ( )x y H H H( ) ( ) ( )w z H H H

x y

w z

Hawkins, Markopoulou, SahlmannCQG 20, 3839 (2003)

Causal setsCausal setsPartial sets: unital CP dynamics?

Lemma: physically accessible and unital => CP

y

w

( )A

( )A

( )wA

( )yA

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

ENTANGLEMENTENTANGLEMENT

a 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 iw w w

, 0, 1i ii A B i i

i iw w w

ENTANGLEMENTENTANGLEMENT

Entanglement of formation

i i iiw

({ }) ( )i i ii

S w S tr

{ }( ) inf ({ })FE S

Minimal weighted averageentanglement of constituents

measuresENTANGLEMENTENTANGLEMENT

“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

Entropy and entanglementEntropy and entanglementon the horizonon the horizon

gr-qc/0508085gr-qc/0505068Phys. Rev. A 72 022307 (2005)

Etera Livine, Tuesday I, 16:00

in grav mat

out inU

mat grav outtr

mat out( ) ( )ES S

EvaporationEvaporation

MEASUREMENTS

SummarySummary

POVMInformation gain

DYNAMICSCompletely positive mapsNon completely positive maps

ENTANGLEMENTEntang’t 101BH applications

Thanks toHilary CarteretViqar HusainNetanel LindnerEtera LivineLee SmolinOliver WinklerKarol Życzkowski