1. A limit on nonlocality in any world in which communication complexity is not trivial
Information primitives
and laws of nature
Mai 2008
Alain Tapp
Université de Montréal

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

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
Alice
Bob
NLB

5. NLB, classical deterministic strategies
yes no

6. 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.

7. NLB quantum strategy Alice and Bob have the same strategy.
If input=0 applies otherwise
Measure and output the result.
This strategy is optimal and works on all inputs with probability:

8. Classical Communication Complexity
Alice
Bob

9. Quantum Communication Complexity
Alice
Bob

10. 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.

11. Most functions are difficult
For most functions f

12. Inner product (IP)

13. Inner product (IP)

14. 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?

15. Equality
Bob
Alice

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

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

18. Scheduling

19. Raz separation
There exists a problem such that:

20. IP using NLB

21. 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

22. AND

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

24. 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

25. 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.

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

27. Idea
Maj
Maj
Maj
Maj
Maj
Maj
Maj
Maj
Maj
Maj
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Maj
Maj

28. Non local majority

29. 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.

30. NLM > 5/6

31. Non local equality

32. NLE implies NLM

33. 2 NLB implies NLE

34. 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.

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