a limit on nonlocality in any world in which communication complexity is not trivial

A limit on nonlocality in any

world in which communication

complexity is not trivial

IFT6195Alain Tapp

In collaboration with… Gilles Brassard Harry Buhrman Naoh Linden André Allan Methot Falk Unger

Quant-ph/0508042

Motivation What would be the consequences if the non local collerations in our world were stronger than the one given by quantum mechanics?

Theoretical computer science? Foundation of physics? Philosophy?

Perfect Non Local Boxes

Alice Bob

NLB

yxba

a byx

NLB and communication

One bit of communication is enough toimplement a NLB.

Alice sends a to Bob and output x=0 Bob outputs bay

NLB and communication

NLBs does not allow for communication.

We can have a perfect box for which x and y are uniformly distributed and independent of (respectively) a and b.

NLB, classical deterministic strategies

yes yes yes no

yes yes no yes

yes no yes yes

no yes yes yes

0 0 0 a b 0 ba0 01 00 11 1

NLB classical implementation There is a probabilistic strategy with succes probability ¾ on all input.

There is no classical déterministic strategy with success proportion greater than ¾.

There is no probabilistic strategy with success probability greater than ¾. ¾

Alice and Bob have the same strategy.If input=0 applies otherwiseMeasure and output the result.This strategy works on all inputs with probability:

)cos()sin(

)sin()cos()(

R

NLB quantum strategy

)16/( R

11002

1

)16/3( R

%85)8/(cos2

NLB quantum strategy

%85)8/(cos2

Tsirelson proved in 1980 that this is optimal whatever the entanglement shared by the players.

Bell theorem

The classical upper bound and the quantum lower bound do not match.

We can derive an inequality from this that provides a Bell theorem proof.

This is known as the CHSH inequality.

4/3 )8/(cos %85 2

Classical Communication Complexity

Alice Bobk

Rz }1,0{

x y

),( yxf

Quantum Communication Complexity

Alice Bobk

11

2

100

2

1

x y

),( yxf

The classical/quantum probabilistic communication complexity of f, C(f)/Q(f) is the amount of classical communication required by the best protocol that succeeds on all input with probability at least when the players have unlimited prior classical/quantum correlation.

Communication Complexity

2/1

Inner product (IP)

)2(mod

)()()(

),(

1

3211

2121

n

iii

nn

nn

yxyx

yxyxyxyx

yyyyxxxx

yxyxIP

Inner product (IP)

)()(

)1()(

nIPQ

OnIPC

Most functions are difficultFor most functions f

)1()(

)1()(

OnfQ

OnfC

Equality

0),EQ(

1),EQ(

yxyx

yxyx

Alice and Bob each have a very large file and they want to know if it is exactly the same.

How much do they need to communicate?

Equality

nRz 1,0

Alice Bobzymb yx

zxma

bm

ba mm Output

Equality

2)EQ( C

2

1

1

yzxzPyx

yzxzPyx

By repeating the protocol twice we have success probability of at least ¾.

Scheduling

0),(

)()()(),(

1

3211

2121

n

iii

nn

nn

yxyxS

yxyxyxyxS

yyyyxxxx

Alice and Bob want to find a time where they are both available for a meeting.

Scheduling

)()(

)()(

nSQ

nSC

Raz separationThere exists a problem such that:

))(log()(

)log()(

4/1

nOSQ

n

nSC

IP using NLB

)()(

)()()(

)()()(

),(),(

2121

2211

3211

nn

nn

nn

iiii

iiii

BBBAAAyx

BABABAyx

yxyxyxyx

yxBA

yxNLBBA

Perfect NLB implies trivial CC

ba xxx

ba xxx

0,1 ba xxx

Any function can be computed with a serie of AND gates and negations.

Distributed bit

Input bit

Negation:

AND Two NLBs

Outcome Bob sends to Aliceba yyy by

AND

))(())((

)()()()(

)()(

),(),(

),(),(

2121

22

11

22

11

bbaa

bbabbaaa

baba

ba

ba

ba

ba

yxBBAAyxyx

yxyxyxyxyx

yyxxyx

xyBA

yxBA

xyNLBBA

yxNLBBA

Main result

1)(, 6

1

2

1* fCfNLB

In any world where non local boxes can be implemented with accuracy larger than 0.91 communication complexity is trivial.

CC with a bias We say that a function f can be computed with a bias if Alice and Bob can produce a distributed bit z such that

2

1]),([ zyxfP

ba zzz

CC with a biasEvery function can be computed with a bias.

Alice’s input: xBob’s input: yAlice and Bob share z

Alice outputs a=f(x,z)Bob outputs b=0 if y=z and a random bit otherwise.

2

1

2

1

2

11

2

1 ]),([

nnbayxfP

Idea We want a bounded bias.

Let’s amplify the bias.

Repetition and majority?

IdeaMaj

Maj Maj Maj

Maj Maj Maj Maj Maj Maj Maj Maj Maj

)(~

)(~

)(~

xfxfxf )(~

)(~

)(~

xfxfxf )(~

)(~

)(~

xfxfxf )(~

)(~

)(~

xfxfxf )(~

)(~

)(~

xfxfxf )(~

)(~

)(~

xfxfxf )(~

)(~

)(~

xfxfxf )(~

)(~

)(~

xfxfxf )(~

)(~

)(~

xfxfxf

)(~xf

Non local majority

),,(

2 iff 1),,(

332211

333

222

111

321321

babababa

ba

ba

ba

ba

xxxxxxMajyy

yyy

xxx

xxx

xxx

xxxxxxMajy

NLM > 5/6 If NLM can be computed with probability stricly greather than 5/6 than every fonction can be computed with a bounded bias.

Below that treshold NLM makes things worst.

NLM > 5/6

pphsp

s

pppqpppqph

q

p

)(2/1

2

1

312

3

2

1

))1()1(3)(1())1(3()(

0)( 6/5

)0( 2/1

3223

Non local equality

),,(

iff 0),,(

332211

333

222

111

321321

babababa

ba

ba

ba

ba

xxxxxxNLEyy

yyy

xxx

xxx

xxx

xxxxxxNLEy

NLE implies NLM

),,(

),,(

332211

321

321

332211

babababa

bbbbb

aaaaa

babababa

xxxxxxMajzz

xxxyz

xxxyz

xxxxxxNLEyy

2 NLB implies NLE

213221

213221

33222211

3221

321

213222

322111

333222111

))()((

))()((

) () (

)()(

),,(

) ,1(

),(

bbbbbb

aaaaaa

babababa

bbaaba

bbaaba

bababa

zzxxxx

zzxxxx

xxxxxxx

xxxx

xxxNLE

xxxxNLBzz

xxxxNLBzz

xxxxxxxxx

To conclude the proof

6

5

6

5

6

1

2

1 MajNLENLB

•Compute f several times with a bias•Use a tree of majority to improve the bias.•Bob sends his share of the outcome to Alice.

Open question

Show some unacceptable consequences of correlations epsilon-stronger than the one predicted by quantum mechanics.