Communication equivalent of non-locality

Mario Szegedy, Rutgers UniversityGrant: NSF 0523866

Emerging Technotogies(joint work with Jérémie Roland)

EPR paradox (Einstein, Podolsky, and Rosen)

e

EPR paradox (Einstein, Podolsky, and Rosen)

e e

|00 + |11 √2

EPR paradox (Einstein, Podolsky, and Rosen)

e e

|00 + |11 √2

EPR paradox (Einstein, Podolsky, and Rosen)

e

0 1

Measurement in the standard basis

EPR paradox (Einstein, Podolsky, and Rosen)

e

|0

EPR paradox (Einstein, Podolsky, and Rosen)

e e 0

1

Measurement in a rotated basis

EPR paradox (Einstein, Podolsky, and Rosen)

e

|0 + |1 √2

General EPR experiment

ψb a

EPR experiment

ψba 0

1

0 1

EPR experiment

B A

0 1

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

λb aRandom 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

λb a

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

λb a

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 1.17 unbounded 1992

G. Brassard, R. Cleve, and A. Tapp

8 8 1999

Steiner 1.48 bounded

Cerf, Gisin and Massar 1.19 bounded

Toner and Bacon 1 1 2003

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 = 0.4142 by Pironio

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) =

∫∫S δθ(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 δθ(a,b) A(a) A(b) da db ≤ 1- θ/π

Note (for what function is the extreme value taken?):

1- θ/π = ∫∫S δθ(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’)