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

EPR paradox (Einstein, Podolsky, and Rosen)EinsteinPodolskyRosen e

EPR paradox (Einstein, Podolsky, and Rosen)EinsteinPodolskyRosen ee |00  + |11  √2

EPR paradox (Einstein, Podolsky, and Rosen)EinsteinPodolskyRosen ee |00  + |11  √2

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

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

EPR paradox (Einstein, Podolsky, and Rosen)EinsteinPodolskyRosen ee 0 1 Measurement in a rotated basis

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

General EPR experiment ψ ba

EPR experiment ψ b a 0 1

EPR experiment B A 0 1

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

Distributed Sampling Problem λ ba Random string

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) Distributed Sampling Problem λ ba B(b, λ)A(a, λ) Computational task Random string

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 Distributed Sampling Problem λ ba B(b, λ)A(a, λ) EPR paradox Random string

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)

Classical communication Maudlin 1.17unbounded1992 G. Brassard, R. Cleve, and A. Tapp Steiner 1.48bounded Cerf, Gisin and Massar 1.19bounded Toner and Bacon avg max year

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

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

Isoperimetric inequality For every odd 1,-1 valued function on the sphere ∫∫ S δ θ (a,b) A(a) A(b) da db ≤ 1- θ/ π Note (for what function is the extreme value taken?): 1- θ/ π = ∫∫ S δ θ (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.

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

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

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

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