Communication equivalent of non-locality

1. Communication equivalent of non-locality Mario Szegedy, Rutgers University Grant: NSF 0523866 Emerging Technotogies (joint work with Jérémie Roland)

2. EPR paradox (Einstein, Podolsky, and Rosen) e

3. EPR paradox (Einstein, Podolsky, and Rosen) e e |00 + |11 √2

4. EPR paradox (Einstein, Podolsky, and Rosen) e e |00 + |11 √2

5. 0 1 EPR paradox (Einstein, Podolsky, and Rosen) Measurement in the standard basis e

6. EPR paradox (Einstein, Podolsky, and Rosen) e |0

7. 0 1 EPR paradox (Einstein, Podolsky, and Rosen) Measurement in a rotated basis e e

8. EPR paradox (Einstein, Podolsky, and Rosen) e |0 + |1 √2

9. General EPR experiment b a ψ

10. 0 1 0 1 EPR experiment a ψ b

11. 0 1 0 1 EPR experiment A B

12. EPR experiment b a A B Joint distribution of A and B: P(A,B|a,b) = (1 – A∙B a.b) / 4

13. Distributed Sampling Problem Random string b a λ

14. Distributed Sampling Problem Random string b a λ Computational task B(b, λ) A(a, λ) Given distribution D(A,B|a,b), design λ, A, B s.t. P(A(a, λ), B(b, λ) | a,b) = D(A,B|a,b)

15. Distributed Sampling Problem Random string b a λ EPR paradox B(b, λ) A(a, λ) There is no distribution λ, and functions A and B for which the DSP would give the joint distribution (1 – A∙B a.b) / 4

16. Additional resources are needed such as: • Classical communication (Maudlin) or • Post selection (Gisin and Gisin) or • Non-local box (N. J. Cerf, N. Gisin, S. Massar, and S. Popescu)

17. Classical communication avg max year

18. Our result One bit of communication on average is not only sufficient, but also necessary Previous best lower bound of √2-1 = 0.4142 by Pironio

19. New Bell inequality ∫∫S( δθ(a,b)+2δ0(a,b)- 2δπ(a,b) ) E(A,B|a,b) da db ≤ 5- θ/π δθ(a,b) = ∫∫Sδθ(a,b) da db= 1. ∞, if angle(a,b)= θ 0, if angle(a,b)= θ

20. Isoperimetric inequality For every odd 1,-1 valued function on the sphere ∫∫Sδθ(a,b)A(a) A(b) da db ≤ 1- θ/π Note (for what function is the extreme value taken?): 1- θ/π = ∫∫Sδθ(a,b)H(a) H(b) da db. Here H is the function that takes 1 on the Northern Hemisphere and -1 on the Southern Hemisphere.

22. Product Theorems for Semidefinite Programs By Rajat Mittal and Mario Szegedy, Rutgers University Presented by Mario Szegedy

23. Product of general semidefinite programs Π = (J,A,b); Π’= (J’,A’,b’). ΠΠ’= (J J’, AA’, b b’),

24. Main Problem Under what condition on Π and Π’ does it hold that ω(ΠΠ’)= ω(Π) x ω(Π’)?

25. Positivity of the objective matrices Theorem: J, J’ ≥ 0→ω(ΠΠ’)= ω(Π) x ω(Π’)