Is Communication Complexity Physical?

Samuel MarcovitchBenni Reznik

Tel-Aviv University

arxiv0709.1602

2

Non-locality (NL)

• Bell: no local-hidden-variable (LHV) theory can simulate quantum-mechanical behavior

– All possible outcomes are determined in advance– Hidden variables do not propagate superluminally

• Bell’s inequality– Measure of non-locality

CausalityRespecting

NL

LHV Quantum Maximal Non-locality

3

Main Result

Nature is as non-local as QM as otherwise CC would be trivial

Trivial Communication

Complexity

LHV QM Maximal Non-locality

Conjecture

•Bipartite - Brassard et. al. PRL 96, 250401 (2006)

•Multipartite

Trivial Communication

Complexity

LHV QM Maximal Non-locality

NL

4

Space of all Physical Theories

Causality

Conjecture

Causality Non-Trivial CC QM

Non-Trivial CC

LHVQM

?

5

More Non-local than the Quantum

• Non-local boxes (NLB) Popescu & Rohrlich Found. Phys. 24, 379 (1994)

– Hypothetical devices– Respect causality– Computational power

van-Dam quant-ph/0501159

• Brassard’s et. al. – Given Nature is

sufficiently more non-local, CC is trivial

LHV QM Non-local Boxes

Trivial CC

Causality

LHVQM NLB

NL

Non-Trivial CC

6

More Non-local than the Quantum

Physics is multipartite

Brassard’s et. al. Conjecture should be tested in

the multipartite case

Generalized Conjecture

Quantum Theory is as non-local as it is in the multipartite case, since otherwise

communication complexity would be trivial

7

Communication Complexity (CC)

• What is the minimal number of bit Alice should send to Bob?– Example: Boolean Inner-Product

– Worse case O(k): Alice sends all her bits to Bob.

– Proven: Shared randomness or shared entanglement do not help

• Trivial CC: only 1 bit of information for any function and input size

2,mod),(1

k

i

ii yxyxf

8

Non-local Boxes (NLB)

10

11

11

11

01

00

00

00

1

0

0

0

11

10

01

00ababbaxyyxba

A Bx y

a b

Computing Inner-Product with trivial CC

ba

babayxk

i

ik

i

ik

i

iik

i

ii

1111

9

Brassards’ et. al. Result

• Generalize CC to the probabilistic case– compute with probability >½, independent of input

size

• Trivial CC can be achieved with non-perfect NLB

f

Trivial CC

%8.906

63

%4.85

4

2

2

1

NLB probability

LHV

75%

Quantum Theory Non-perfect NLB 100%

0

10

NLB Probability = Non-locality

Bell’s inequality:

)(4

)(22

)(2

NLB

QM

LHVyyCxyCyxCxxC BABABABA

yy

yx

xy

xxCsignInputbaInput )(

1

0

0

0

11

10

01

00

8

4

%1004

%4.8522

%752

AA

11

Physics is Multipartite

• Multipartite CC– Example: Tripartite Inner-Product

• Trivial CC: N-1 bits of communication for N parties

• Brassard’s et al Conjecture generalized to multipartite

• Objection– Any multipartite function can be computed with trivial CC using

bipartite NLB’s.

2,mod),,(1

k

i

iii zyxzyxf

iBi

Aiii

Bii

Ai

iiiiiiiiii

ccbacbca

zbzazbazyx

12

Physics is Multipartite

• Probabilistic CC– required operation probability of the non-local box

• Bipartite ~90.8%• Tripartite ~96.7%

•

• Is Brassard’s el. al. Conjecture refuted?

1)(N

NLBp

13

Multipartite Non-local Boxes

• Examples of tripartite NLB

• Found a specific class of multipartite NLB that reduces CC to triviality effectively – Requires constant probability for all N>2.

zyzxyxcba

zyxcba

Trivial CC

%4.854

2

2

1

Multipartite NLB probability

LHV Quantum Theory Non-perfect NLB 100%

%7.9312

1

2

13

12/2

1

2

1 N

2

1

2

1)(NGHZ

14

Generalized Bell’s Inequalities

• More than two parties• (More than two observables {x,y} per site)

• Suggested box corresponds to the generalized Bell’s inequality with maximal QM violation.

N22

Multipartite NLB

Generalized Bell’s inequality

N2

j

N

i

N

iji

N

ii zza

1

1 11

Ardehali’s Inequality

15

Ardehali’s inequality

• N even: Ardehali’s inequality coincides with the maximal violating inequality (Klyshko’s inequality)

• N odd: Maximal violation among all CHSH inequalities that have corresponding NLB

k is the number of y’s in

2'2

1

2

111 yxAyxAA m

NKN

mN

)(' yxAA

yyyyyxyxyyxxxyyxyxxxyxxxA

yyxxAyxxyAA

A

mN

N

mN

kAN

2

1

2'2

1

2

1

3

2

1

2

mNA

16

Summary of the results

NLB probability

%4.854

2

2

1

Multipartite NLB probability

LHV Quantum Theory Non-perfect multipartite NLB

100%

%7.9312

1

2

13

12/2

1

2

1 N

Non-perfect bipartite NLBArdehali’s Inequality

Maximal quantum non-locality

Bipartite

Multipartite

Trivial CC

%8.906

63

%4.85

4

2

2

1

LHV

75%

Quantum Theory Non-perfect NLB 100%

Trivial CC

17

Space of all Physical Theories

Causality

Conjecture

Causality Non-Trivial CC QM

Non-Trivial CC

LHVQM

?

18

Obtaining the BoundBipartite case (Brassard et. al)

• Non-local computation = trivial CC• With shared randomness can be

computed non-locally with .

• Non-local Majority:

• If Alice and Bob can compute non-local Majority with , CC is trivial.

),,( 33221121 yxyxyxMajaa

21p

f

65q

32

23

)1()1(3)1(

)1(3)(

pppp

pppqph

312

3

2

1,6

5

)(21

sq

sphpsp

19

Obtaining the BoundBipartite case (Brassard et. al)

• Non-local equality:

• Non-local equality can be computed with only two NLB’s

• Non-local Majority can be computed with only two NLB’s

• Required probability

otherwise

yxyxyxifaa

332211

21 0

1

32213221

332211

'',','','

''''''

yyyyyyxxxxxx

yxyxyxyxyx

yxyx

)()(),,( 3212

3211

332211 yyyaxxxayxyxyxMaj

65)(1)( 22 NLBpNLBp

20

Obtaining the BoundMultipartite case

• Non-local equality:

• N(N-1) bipartite NLB are required • 3 optimal multipartite NLB are required

otherwise

xxxifa

N

ii

N

ii

N

ii

N

ii 1

3

1

2

1

1

1 0

1

2,'',','','

'''''''''

31

21

21

11

31

211

21

111

1,1111

3

1

2

1

1

ixxxxxxxxxxxx

xxxxxxxxx

ii

j

N

jiji

ii

N

ii

N

ii

N

ii

N

ii

N

ii

N

ii

1)(N

NLBp

21

Obtaining the BoundMultipartite case

• Box 1:

• Box 2:

• Box 3:

1

1 11

1 ''N

ij

N

iji

N

ii xxa

1

1 11

2 ''''N

ij

N

iji

N

ii xxa

j

N

jiji

i

N

ij

N

iji

N

ij

N

iji

N

ijj

N

ijii

N

ii

xxxxxx

xxxxa

'''''''''

''''''

1,

1

1 1

1

1 1

1

1 11

3

22

Obtaining the BoundMultipartite case

• Non-local Majority is obtained in the same way:

• Required probability

65)(1)(3)( 23 NLBpNLBpNLBp

N

iiiii

N

ii

N

ii

N

ii xxxaxxxaxxxMaj

2

32131

21

111

1

3

1

2

1

1 )()(),,(

23

Correspondence to Ardehali’s Inequality

• f = 1 for q 1’s such that q(q-1)/2 is odd

• Correspondence proved by induction

• Classic bound decreases as N increases– Classically, one can simulate f = 1 for odd q

(or f = 1 for q equals 0/N)

• QM bound is constant and satisfied by the generalized GHZ state

0

0

1

1

1

1

1

1

0

1

1

1

0

1

0

0

0

10000

01111

01110

01101

01100

01011

01010

01001

01000

00111

00110

00101

00100

00011

00010

00001

00000

1

1 1321

jN

i

N

ij

iN xxfzzzz

2

1

2

1)(NGHZ

24

Summary of the resultsTrivial

communication complexity

%8.906

63

%4.85

2

21

NLB Operationprobability

Classic Theory

75%

Quantum Theory Non-perfect NLB 100%

Trivial communication

complexity

%4.852

21

Multipartite

NLB Operationprobability

Classic Theory Quantum Theory Non-perfect NLB 100%

%7.9312

1

2

13

12/2

1

2

1 N

Non-perfect bipartite NLBArdehali’s Inequality

Maximal quantum non-locality

Bipartite

Multipartite