1 Combinatorics and Quantum Information Simone Severini Department of Combinatorics & Optimization,...

1

Combinatorics and

Quantum Information Simone Severini

Department of Combinatorics & Optimization, and

Institute for Quantum Computing,

University of Waterloo, Canada

www.iqc.ca/~sseverin

Isaac Newton Institute, Cambridge, March 2008

2

Quantum Information

Quantum information may be defined as the study of the achievable limits to information processing possible within quantum mechanics.

Goals of quantum information:

• Determine limits on the class of information processing tasks which are possible in quantum mechanics (e.g., limitations on the class of measurements that may be performed on a quantum system).

• Provide constructive means for achieving information processing tasks (e.g., development of unbreakable schemes for doing cryptography based upon the principles of quantum mechanics, study of the power of quantum computational devices).

(M. Nielsen)

3

• Quantum teleportation (applications: quantum memories, quantum repeaters). • Superdense coding (primitive in quantum protocols).• Secure key distribution in cryptography.• Quantum computing (e.g., discrete Log, search in databases).• Simulation of quantum mechanical evolutions (e.g., many-body systems, applications in condensed matter physics).• Secret sharing (e.g., secure distributed computation).• Quantum Games (e.g., lower bounds in algorithms, study of phase transitions).• Quantum Lithography and Imaging (e.g., increased signal intensities, extensions of optical microscopy).• Quantum Sensors (improve the sensitivity of measurements across the electromagnetic spectrum, measuring microwaves).

Phenomena and Applications

4

Entanglement Theory

What is entanglement?

• Entanglement is a property of certain states of quantum systems.• Entanglement is a physical resource (like energy or time).• It is one of the key ingredients for a quantum computer, allowing to solve certain tasks more quickly. • It is an NP hard problem to decide whether a given state is entangled or not.

Goals of entanglement theory:

• Find easily implementable criteria for detecting and classifying entanglement.

• Isolate the properties that characterize entanglement.

• Determine how useful entanglement is.

I would not call entanglement one but rather the characteristic trait of quantum mechanics

(E. Schrödinger)

5

• Reconstruction of quantum states: Young diagrams, Marginal inequalities (Christandl et al. 2005, Klyachko, 2004)• Quantum state transformations: Schubert calculus and varieties (Hayden et al. 2004, Knutson 2004)• Classification of entangled states: Hyperdeterminants (Miyake et al., 2003), Combinatorics of permutations (Clarisse et al., 2005), Knots invariants (Kauffman et al., 2004)• Toy states and entanglement detection: graph laplacians (Braunstein et al., 2005)• One-way quantum computation: local operations on graphs (Briegel et al., 2002)• Quantum-error correction: codes over Z₄ (Calderbank et al., 2003)• Bell's inequalities: cut polytopes (Avis et al., 2005)• Non-locality, CHSH games: extremal set theory (Cleve et al., 2005)• Optimal measurements: finite geometry, projective designs (Wootters et al., 2005)• Randomization of quantum states: quantum expanders (Ta-Shma et al., 2006)• Information transfer and spin network control: spectral graphtheory (Cambridge people, Bose >2000), power domination in graphs (Aazam et al., 2008)• Topological quantum computation: Hopf algebras (Friedman, Kitaev, 1998)• Complexity parameters and simulation: (Valiant 2000, Shi, Josza 2006)

Combinatorics

6

• Classification of entangled states: combinatorics of permutations

• Toy states and entanglement detection: graph laplacians

• Quantum-inspired algorithm for graph matching

• Information transfer and spin network control: spectral graphtheory, power domination in graphs

• Non-locality, CHSH games: extremal combinatorics

Plan of this talks

7

Clarisse, Ghosh, Sudbery, S, Phys. Rev. A 72, 012314 (2005)

quant-ph/0502040

Clarisse, Ghosh, Sudbery, S, Phys. Lett. A, 365 (2007)

quant-ph/0611075

Classification of entangled states and combinatorics of

permutations

8

What is a quantum state?

†

:

Def. (

0

T r

S

(

ta t

1

e)

)

d

2 †

:

o r s.t

Def.

. 1 and

is ide

(Pure st

n tifi

a

ed w ith

te) d

d

9

What is entangled?

0

:

there are no : ,

Def. (Entang led sta te)

: and

s.t.

p qA B

A p B qi A i B i

A Bi i ii

r

r

there are no

Def. (Ent

a

an

nd

gled pure s

s.t.

ta te) p qA B

p qA B

10

How to measure entanglement?

2

:

S : (1 T r( ))1

is the reduced density m atrix w .r

Def. (L

.

inear entrop

t

)

y

.

d dA B

A

dA B

d

d

11

How to entangle?

†

,

( ) . .

( )

if S 0 then entang ls edi

p qA B

U U pq i e UU I

U

12

What is required to be a good entangler ?

2

, 1 , 1

(

2

)

,

( ) :

, norm alize

Def

d p

. (Entang ling po

rob. m easures on un it spheres

w er o f ( )

m x ( )1

)

a

d dA B

U d

e U S U d d

d d

de U

U U d

d

13

What are the best entanglers among all the elements of the unitary group?

14

Latin square

1 2 3

2 3 1

3 1 2

15

Mutually Orthogonal Latin Squares (MOLS)

1 2 3

2 3 13 1 2

1 3 22 1 33 2 1

1,1 2,3 3,22,2 3,1 1,33,3 1,2 2,1

16

MOLS permutation

1,1 2,3 3,22,2 3,1 1,33,3 1,2 2,1

1 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

0 1 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1 0 0

0 0 1 0 0 0 0 0 0

17

What are the best entanglers ?

2

2

Let 6.

Then ( )1

if is a MOLS permutation of size .

This is max. in (

Theorem.

).

d

de U

d

U d

U d

18

Open problemsCombinatorial problems: • What about dimension 6?• Classify permutations according to the their entangling power.• What about multipartite systems (i.e., many subsystems)?

Notice: The max. # of MOLS of side l is • For a larger # of subsystems can we do it with other unitaries? • If we can not? 1. e(U) is not plausible; 2. entanglement for a few subsystems is special(or MOLS are objects from a larger class?).

1l

19(Pictures - Courtesy of filmforum.org)

1/4

20

Braunstein, Ghosh, S, Ann. Comb. 10 (2006), 291-327 quant-ph/0406165

Braunstein, Ghosh, Mansour, S, Wilson, Phys. Rev. A 73, 012320 (2006) quant-ph/0508020

Hildebrand, Mancini, S, Math. Structures Comput. Sci. 18:1 (2008), 205-219cs.CC/0607036

Toy states and entanglement detection

21

Associating graphs to states

( , ), undirected graph, with no self-loops

, adjacency matrix

, degree matrix

, combinatorial la

Def. (Laplacian

placian

0,

1 ( ) is a quantum state

Tr( )

state)

T

G V E

M

L M

L L L

G L

(Spin networks, bosonic graphs, graph states, etc.)

22

3/

How hard is to detect entanglement?

Give

Decision p

n: :

Question: is entangled?

This is NP-comple

ro .

te

blemp qA B

uuuuuuuuuuuuuuuuuuux

£ £

23

11 1

1

11 1

1

:

is

Def. (Partia l transpose)

B

p qA B

p

p pp

ii

T Tp

T Tp pp

M M

M M

M q q

M M

M M

Partial transpose

24

Positivity of Partial Transpose (PPT) Criterion

2 2 2 3

:

If is separab le then 0

The converse is not necessarily true

: or ,

Theore

i

m .

Theor

s separab le iff 0

em .

B

B

p qA B

A B A B

25

Associating a lattice to the graph

( ) :

| ( ) | .

Le t { } and { } standard bases for and .

Labe l each vertex by a ordered pa ir ( , ).

D raw on the { } X { } la ttice ,

P r

accord ing to the chosen la

oce

be l

dure .

ing .

p qA B

p qA B

G

V G pq

i j

i j

G i j

26

Partial transpose on the latticeA B

C D

BG

G H

BH

27

Degree Criterion

( ) :

If ( ) is separab le in

T he

then ( ) (

orem .

).B

p qA B

p qA B

G

G

G G

28

Degree Criterion and PPT

( ) :

T hen

T heor

( ) ( ) iff

em

.

.

( ) 0B B

p qA BG

G G G

29

Degree Criterion and entanglement

2 Let ( ) :

Then ( ) ( ) iff ( ) is separable.

Then ( ) 0 iff

Theore

( ) is separable

m.

.

B

B

qA BG

G G G

G G

£ £

30

• The concurrence of entangled laplacian states in dimension four is exactly fractional: 1/(# edges).• Characterize graphs whose laplacian state is entangled independently of the labelling.• Consider multipartite systems: Wang, Wang, Elec. J. Comb. 14 (2007) Hassan, Joag, J. Phys. A: Math. Theor. 40 (2007); J. Math. Phys. 49 (2008)

Open problems

Let be a connected graph.

Then ( ) is entangl

Conjec

ed

ture.

iff 0.B

G

G

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2/4

32

Emms, Hancock, S, Wilson, Elec. J. Comb. 13 (2006), #R34. quant-ph/0505026See this morning talk by E. Hancock, for applications in pattern recognition: Emms, Hancock, Wilson, ICPR, IEEE Computer Society (2006).

For a different approach seeAudenaert, Godsil, Royle, Rudolph, J. Comb. Theory, Ser. B 97(1): 74-90 (2007).math.CO/0507251

A quantum-inspired graph matching algorithm

33

Graph matching

Problem.

• How “similar” graphs G and H are?

(Pictures - Courtesy of M. Schultz)

34

Graph isomorphism

Problem.

• Are graphs G and H actually the “same graph”?

(Pictures - Courtesy of D. Bacon)

35

Let ( , )

Then D ( ( ), ), where ( , )

Def. (Bi-or

,( , ) ( ) iff

ientation)

{ , } ( )G G

G V E

V G A i j j i A D i j E G

( ) #{

Def. (D

:

egree)

( , ) ( )}Gd i j i j A D

,

( , ),( , )

For all ( , ),( ,

Def. (Orthogonal matrix represe

) ( )

2 / ( ) , if ; ( )

0, ot

ntation

herwise.

)

G

i li j k l

i j k l A D

d j j kU G

( ) (

Lemm

)

a.TU G U G I

Representing a graph with an orthogonal matrix

36

,

Let ( , )

1, if { , } ( );

Def. (Adjacency matr

( )0, otherwis

)

.

ix

ei j

G V E

i j E GM G

GraphsDef. (Isomor and

iff : ( ) ( )

phic graphs)T

G H

G H P PM G P M H

Graphs and

If then Sp( ( )) Sp( ( )).

The converse is not nec

Lemma.

essarily true.

G H

G H M G M H

Graphs and

Prob

?

lem. (Graph Isomo

So far the prob

rp

l

hism Pro

em is in

b

lem

NP A

)

co M

G H

G H

Graph Isomorphism Problem

37

is a srg( , , , )

| ( ) | ;

the degree of each vertex is ;

, , if { , } ( ) then , have exactly common neigh;

Def. (Strongly

, , if { , } ( ) then , have exactly co

regular graph) G n m a b

V G n

m

i j i j E G i j a

i j i j E G i j b mmon neigh.

Let and be srg( , , , )

Then Sp( ( )) Sp( ( )), even if

Lemma.

non-isomorphic.

G H n m a b

M G M H

1/3( log )

and are srg( , , , ).

Problem.

?

So

(SRG I

far the

somorphism

problem is .

Problem)

O n n

G H m n a b

G H

n

Strongly regular graphs

38

,

,

Let be a real matrix.

1, if 0;

Def. (Support of a matr

0, o

ix

therwise

)

.i j

i j

M

MM

3 3

Let and be srg( , , , ), where 64.

Then iff Sp( ( ) ) Sp( ( )

Fact. (Theorem proved by inspecti

)

on)

G H n m a b n

G H U G U H

3 3

Let and be srg( , , , ), where 64.

Then iff Sp

Conjec

( ( ) )

ture.

Sp( ( ) ).

G H n m a b n

G H U G U H

Result

39

3/4

40

Aazami, S, A covering problem with a propagation rule: formulations and algorithms, submitted to SWAT08. Important reference:Burgardt, Giovannetti, Phys. Rev. Lett. 99 (2007).

A game for controlling spin systems

Perfect state transfer Saxena, S, Shparlinski, Int. J. Quantum Inf. (2007), 417-430.quant-ph/0703236Important reference:Christandl, Datta, Ekert, Landahl, Phys. Rev. Lett. 92 (2004).

41

Nondiscriminatory propagation

42


43


44


45


46


47


48


49


50


51


52

Nondiscriminatory propagation problem

Given a graph .

Find a minimum cardinality set of initially colored vertices

required to propagate the color in the entire .

G

G

53

Example

54

Example

55

Example

56

Example

57

Result

We can solve NONDISCRIMINATORY PROPAGATION

optimally in polynomial time on graphs of bounded tree-width.

(This is done by reformulating the problem as an orientation

one, and by making u

Theo

se

rem.

of a dynamic-programming algorithm.)

1

Determine complexity. (It is NP-hard for weigh

Open

ted graphs.)

Opt(HC ) 2 ?

Good expanders have

pr

la

oblems

rge t

.

Op .

dd

58

D

A graph with quant

ef. (Spin system on a

um particles on the ve

gra

rt

ph

.

)

icesG

r

r

¡0

( )0

Assign each ( ) to a stand

Def. (Rule of th

ard basis vector v.

v : .

e dynamics

, where .

)

iA G tt

v V G

e t

r r( )

( ,

Def. (F

,

idelity

) | | .

)T iA G t

Gf v w t w e v

r r( )

Def. (Perfect state transfer, PST

( , , ) |

)

| 1.T iA G tGf v w t w e v

For the purpose of studying state transfer (in the XY model), we can introduce the following:

Perfect state transfer

59

¤

Given :

if s.t. ( , , ) 1

then { , , , } Sp( ( ), .

The converse is not necessarily true.

However

Lemma

, s.t. , ( , ) .

.

,

1

G

i ji j k l

k l

G

G

t f v w t

M G

t v f v v t

Construct graphs:

Small degree;

Large number of vertices;

Gives PST between very distant vertices;

( . ., -cubes give PST between antipodal

Proble

vertic

m.

es).e g d

60

If a circ. gives PST then it has integral spectrum.

(We characterize those graphs; small diam

Lemma

)

.

eter.

G

Result: circulant graphs

¢ ¢

The Cayley graphs, ( , ), where .

Equivalently, the graphs with circulant adjacency matrix.

(Many uses: VLSI design

Def. (Circulan

, computer net

t graph)

works, .)

n nX S S

etc

Circ. graphs on 2 1 vertices do not

Theore

give T.

m.

PSn k

61

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Parallel repetitions of Clauser-Horne-Shimoni-Holt

(CHSH) Games

R. Cleve, W. Slofstra, F. Unger, S. Upadhyayquant-ph/0608146

63

The CHSH game

BobAlice

Referee

No communication

{0,1}x {0,1}y

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The CHSH game

BobAlice

Referee

No communication

{0,1}a {0,1}b

x y a b

65

Best classical strategy

0 1

0 0 0

1 0 1

x y a b

Input Output

66


0 1

0 0 0

1 0 1

x y a b

Input Output

0 0

0 0 0

0 0 0

67


0 1

0 0 0

1 0 1

x y 0 0

0 0 0

0 0 0

a b

Input Output

68


Pr[Alice & Bob win] 3 / 4

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Best quantum strategy

2

Pr[Alice & Bob win*]

cos ( / 8) 0.85

-* when they share |

70

Best quantum strategy

2

Pr[Alice & Bob win*]

cos ( / 8) 0.85

-* when they share |

Optimal by the Tsirelson’s Inequality

71

There is a quantum strategy which is better than any classical strategy

0.85 3 / 4

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The repeated CHSH game

BobAlice

Referee

No communication

{0,1}nx {0,1}ny

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The repeated CHSH game

BobAlice

Referee

No communication

{0,1}na {0,1}nb

bitwisex y a b

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Input for n = 2

00 01 10 11

00 00 00 00 00

01 00 01 00 01

10 00 00 10 10

11 00 01 10 11

x y

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Best quantum strategy for playing the game n times

2

Pr[Alice & Bob win]

(cos ( / 8))n

76

A classical strategy

*Pr[Alice & Bob win ]

(3 / 4)n

* concatenating strategies

on the single bits

77

Is this the best strategy?If not, what is the best strategy?

78

For n = 2 there is better

00 01 10 11

00 00 00 00 00

01 00 01 00 01

10 00 00 10 10

11 00 01 10 11

Input Output

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00 01 10 11

00 00 00 00 00

01 00 01 00 01

10 00 00 10 10

11 00 01 10 11

00 00 00 10

00 00 00 00 00

00 00 01 00 01

00 00 00 10 10

01 00 01 10 11

Input Output

80


00 01 10 11

00 00 00 00 00

01 00 01 00 01

10 00 00 10 10

11 00 01 10 11

00 00 00 10

00 00 00 00 10

00 00 00 00 10

00 00 00 00 10

01 01 01 01 11

Input Output

81

Best classical strategy for playing the game 2 times

2

Pr[Alice & Bob win]

10 /16

(3 / 4)

9 /16

82


Pr[Alice & Bob win] 31/ 64

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M. Bordewich, S, Ann. of Benians Court, Vol.1:1 (2008), p. 1.

84

Pr[Alice & Bob win] ?


85

Best classical strategy for playing the game for an arbitrary

number of times

Pr[Alice & Bob win] ?

86

Upper bound

• Semidefinite programming (Feige-Lovasz relaxation)

• Density of squares in (0,1)-matrices (Peleg)

87

Natural upper bound

2(cos ( / 8))n

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Open problem

2Pr[Alice & Bob win] (cos ( / 8))

for ?

n

n

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