quantum info tools & toys for quantum gravity loops `05 daniel terno perimeter institute
DESCRIPTION
POVM discrete continuous Projections/von Neumann Realization ancillary system+ unitary evolution+ PVM Moments MEASUREMENTSTRANSCRIPT
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