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

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

2. In collaboration with… • Gilles Brassard • Harry Buhrman • Naoh Linden • André Allan Methot • Falk Unger • Quant-ph/0508042

3. 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?

4. Perfect Non Local Boxes Bob Alice NLB

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

6. 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.

7. NLB, classical deterministic strategies

8. 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

9. NLB quantum strategy 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:

10. NLB quantum strategy Tsirelson proved in 1980 that this is optimal whatever the entanglement shared by the players.

11. 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.

12. Classical Communication Complexity Alice Bob

13. Quantum Communication Complexity Alice Bob

14. Communication Complexity 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.

15. Inner product (IP)

16. Inner product (IP)

17. Most functions are difficult For most functions f

18. 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?

19. Equality Bob Alice

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

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

22. Scheduling

23. Raz separation There exists a problem such that:

24. IP using NLB

25. Perfect NLB implies trivial CC Any function can be computed with a serie of AND gates and negations. Distributed bit Input bit Negation: Two NLBs AND Bob sends to Alice Outcome

26. AND

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

28. 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

29. 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.

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

31. Idea Maj Maj Maj Maj Maj Maj Maj Maj Maj Maj Maj Maj Maj

32. Non local majority

33. 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.

34. NLM > 5/6

35. Non local equality

36. NLE implies NLM

37. 2 NLB implies NLE

38. 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.

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