A limit on nonlocality in any world in which communication complexity is not trivial IFT6195 Alain Tapp

In collaboration with…  Gilles Brassard  Harry Buhrman  Naoh Linden  André Allan Methot  Falk Unger  Quant-ph/

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

NLB and communication One bit of communication is enough to implement a NLB. Alice sends a to Bob and output x=0 Bob outputs

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 no yes noyes noyes noyes

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 otherwise Measure and output the result. This strategy works on all inputs with probability: NLB quantum strategy

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.

Classical Communication Complexity AliceBob

Quantum Communication Complexity AliceBob

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

Inner product (IP)

Most functions are difficult For most functions f

Equality 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 Alice Bob

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

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

Scheduling

Raz separation There exists a problem such that:

IP using NLB

Perfect NLB implies trivial CC 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 Alice

AND

Main result 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

CC with a bias Every function can be computed with a bias. Alice’s input: x Bob’s input: y Alice and Bob share z Alice outputs a=f(x,z) Bob outputs b=0 if y=z and a random bit otherwise.

Idea  We want a bounded bias.  Let’s amplify the bias.  Repetition and majority?

Idea Maj

Non local majority

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

Non local equality

NLE implies NLM

2 NLB implies NLE

To conclude the proof 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.