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The Complexity of Simulating Non-Signaling Distributions

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The Complexity of Simulating Non-Signaling Distributions The complexity of simulating non-signaling distributions Julien Degorre1 Marc Kaplan2 Sophie Laplante2 Jer´ emie´ Roland3 1Laboratoire d’Informatique de Grenoble 2Universite´ Paris-Sud - LRI 3UC Berkeley QIP 2008 Degorre, Kaplan, Laplante, Roland () QIP2008 1/14 Motivation Study of (quantum) non-locality Simulation of quantum correlations Generalized non-local theories Applications to (classical) communication complexity Lower bound on communication complexity Characterization of PR-box complexity Degorre, Kaplan, Laplante, Roland () QIP2008 2/14 Non-signaling distributions EPR experiment x [X] y [Y ] ∈ ∈ ψ (entangled| i state) a [A] b [B] ∈ ∈ p is non-signaling: p[a, b, x, y]= p(a, b x, y) | “y a”: p(a x, y)= p(a x, y ) → | | ′ “x b”: p(b x, y)= p(b x , y) → | | ′ Let : set of non-signaling distributions C Degorre, Kaplan, Laplante, Roland () QIP2008 3/14 Non-signaling distributions EPR experiment x [X] y [Y ] ∈ ∈ ψ (entangled| i state) p(a, b x, y) | a [A] b [B] ∈ ∈ p is non-signaling: p[a, b, x, y]= p(a, b x, y) | “y a”: p(a x, y)= p(a x, y ) → | | ′ “x b”: p(b x, y)= p(b x , y) → | | ′ Let : set of non-signaling distributions C Degorre, Kaplan, Laplante, Roland () QIP2008 3/14 LHV distributions (shared randomness) LHV distribution x [X] y [Y ] ∈ ∈ λ (shared randomness) α(x, λ) β(y, λ) a [A] b [B] ∈ ∈ Fixed λ Deterministic local protocol dλ = dλ dλ −→ A ⊗ B p(a, b x, y) is a convex combination of dλ’s | λ p = pλ d Xλ Let : set of LHV (local) distributions B Degorre, Kaplan, Laplante, Roland () QIP2008 4/14 LHV with quasi-probabilities Set of LHV distributions : B Convex combinations of deterministic protocols dλ Facets: Bell inequalities B p B0 · ≤ @ @ Bell inequality C @ dλ3 dλ4 @ @ B ⊂ C @ @B @ @dλ1 dλ2 @ @ @ Set of LHV distributions with quasi-probabilities ˜: B Affine combinations of deterministic protocols dλ Possibly negative weights (quasi-probabilities) −→ Degorre, Kaplan, Laplante, Roland () QIP2008 5/14 LHV with quasi-probabilities Set of LHV distributions : B Convex combinations of deterministic protocols dλ Facets: Bell inequalities B p B0 · ≤ @ @ Bell inequality C @ dλ3 dλ4 @ ˜ @ B ⊆ B ⊆ C @ @B @ @dλ1 dλ2 @ @ @ Set of LHV distributions with quasi-probabilities ˜: B Affine combinations of deterministic protocols dλ Possibly negative weights (quasi-probabilities) −→ Degorre, Kaplan, Laplante, Roland () QIP2008 5/14 Non-signaling distributions and quasi-probabilities Theorem The set of non-signaling distributions coincides with the set of LHV with quasi-probabilities ˜ = B C 2-slit experiment, quantum diffusion (walks), ... [Feynman 85] Quasi-probability LHV for quantum measurement [Groenewold 85] Characterize non-signaling polytope [Klay¨ Randall Foulis 87, Wilce 92, Barrett 07] Applications to no-cloning and no-broadcasting [Barnum Barrett Leifer Wilce 07] Degorre, Kaplan, Laplante, Roland () QIP2008 6/14 Non-signaling distributions and quasi-probabilities Theorem The set of non-signaling distributions coincides with the set of LHV with quasi-probabilities ˜ = B C 2-slit experiment, quantum diffusion (walks), ... [Feynman 85] Quasi-probability LHV for quantum measurement [Groenewold 85] Characterize non-signaling polytope [Klay¨ Randall Foulis 87, Wilce 92, Barrett 07] Applications to no-cloning and no-broadcasting [Barnum Barrett Leifer Wilce 07] Degorre, Kaplan, Laplante, Roland () QIP2008 6/14 Communication complexity and non-signaling distributions Function f : [X] [Y ] 0, 1 × 7→ { } x [X] y [Y ] ∈ ∈ T β(y, T ) b b = f (x, y) Degorre, Kaplan, Laplante, Roland () QIP2008 7/14 Communication complexity and non-signaling distributions Function f : [X] [Y ] 0, 1 × 7→ { } x [X] y [Y ] ∈ ∈ T α(x, T ) β(y, T ) a b a b = f (x, y) ⊕ Degorre, Kaplan, Laplante, Roland () QIP2008 7/14 Communication complexity and non-signaling distributions Function f : [X] [Y ] 0, 1 × 7→ { } x [X] y [Y ] ∈ λ ∈ T α(x, T , λ) β(y, T , λ) a b 1 2 if a b = f (x, y) pf (a, b x, y)= ⊕ | 0 otherwise Degorre, Kaplan, Laplante, Roland () QIP2008 7/14 Communication complexity and non-signaling distributions Function f : [X] [Y ] 0, 1 × 7→ { } x [X] y [Y ] ∈ λ ∈ T α(x, T , λ) β(y, T , λ) a b Zero-error randomized communication complexity R0(p ) R0(f ) R0(p )+ 1 f ≤ ≤ f = D(f ) (deterministic) Degorre, Kaplan, Laplante, Roland () QIP2008 7/14 Popescu-Rohrlich box [Popescu,Rohrlich 94] PR box = black-box simulating p for f (x, y)= xy AND function f −→ x 0, 1 y 0, 1 ∈ { } ∈ { } a b = xy ⊕ a b 1 2 if a b = xy pf (a, b x, y)= ⊕ | 0 otherwise Degorre, Kaplan, Laplante, Roland () QIP2008 8/14 Replacing communication by PR boxes Replacing 1 bit of communication by a PR box x [X] y [Y ] ∈ ∈ c(x) α(x) β(y, c) Alice does not communicate c(x) to Bob, who assumes that c(x)= 0 c(x)= 0 √ x˜ = c(x) ⇒ β(y, 0)= β(y, 1) √ y˜ = β(y, 0) β(y, 1) ⇒ ⊕ c(x) = 0 AND β(y, 0) = β(y, 1) Error 6 6 ⇒ Must compute AND May be corrected by a PR-box ⇒ Degorre, Kaplan, Laplante, Roland () QIP2008 9/14 Replacing communication by PR boxes Replacing 1 bit of communication by a PR box x [X] y [Y ] ∈ ∈ α(x) β(y, 0) Alice does not communicate c(x) to Bob, who assumes that c(x)= 0 c(x)= 0 √ x˜ = c(x) ⇒ β(y, 0)= β(y, 1) √ y˜ = β(y, 0) β(y, 1) ⇒ ⊕ c(x) = 0 AND β(y, 0) = β(y, 1) Error 6 6 ⇒ Must compute AND May be corrected by a PR-box ⇒ Degorre, Kaplan, Laplante, Roland () QIP2008 9/14 Replacing communication by PR boxes Replacing 1 bit of communication by a PR box x [X] y [Y ] ∈ ∈ α(x) β(y, 0) Alice does not communicate c(x) to Bob, who assumes that c(x)= 0 c(x)= 0 √ x˜ = c(x) ⇒ β(y, 0)= β(y, 1) √ y˜ = β(y, 0) β(y, 1) ⇒ ⊕ c(x) = 0 AND β(y, 0) = β(y, 1) Error 6 6 ⇒ Must compute AND May be corrected by a PR-box ⇒ Degorre, Kaplan, Laplante, Roland () QIP2008 9/14 Replacing communication by PR boxes Replacing 1 bit of communication by a PR box x [X] y [Y ] ∈ ∈ α(x) β(y, 0) Alice does not communicate c(x) to Bob, who assumes that c(x)= 0 c(x)= 0 √ x˜ = c(x) ⇒ β(y, 0)= β(y, 1) √ y˜ = β(y, 0) β(y, 1) ⇒ ⊕ c(x) = 0 AND β(y, 0) = β(y, 1) Error 6 6 ⇒ Must compute AND May be corrected by a PR-box ⇒ Degorre, Kaplan, Laplante, Roland () QIP2008 9/14 Replacing communication by PR boxes Replacing 1 bit of communication by a PR box x [X] y [Y ] ∈ ∈ x˜ y˜ a˜ b˜ = x˜y˜ ⊕ a˜ b˜ α(x) a˜ β(y, 0) b˜ ⊕ ⊕ Alice does not communicate c(x) to Bob, who assumes that c(x)= 0 c(x)= 0 √ x˜ = c(x) ⇒ β(y, 0)= β(y, 1) √ y˜ = β(y, 0) β(y, 1) ⇒ ⊕ c(x) = 0 AND β(y, 0) = β(y, 1) Error 6 6 ⇒ Must compute AND May be corrected by a PR-box ⇒ Degorre, Kaplan, Laplante, Roland () QIP2008 9/14 PR box and quasi-probabilities x y λ (α, β) a = λ αx b = λ βy ⊕ ⊕ q(α,β) α = 0 α = 1 1 pλ = 2 λ 0,1 1 1 ∀ ∈ { } β = 0 2 2 ( 1)αβ q α,β = − 1 1 ( ) 2 β = 1 2 2 (α,β) 0,1 2 − ∀ ∈ { } Proof sketch Claim a b = αx βy ⊕ ⊕ 1 xy = 0 (let x = 0 wlog) a b = 0 √ p(a b = xy x, y)= 1 ⇒ ⊕ ⊕ | 2 xy = 1 a b = 1 √ ⇒ ⊕ Degorre, Kaplan, Laplante, Roland () QIP2008 10/14 PR box and quasi-probabilities x y λ (α, β) a = λ αx b = λ βy ⊕ ⊕ q(α,β) α = 0 α = 1 1 pλ = 2 λ 0,1 1 1 ∀ ∈ { } β = 0 2 2 ( 1)αβ q α,β = − 1 1 ( ) 2 β = 1 2 2 (α,β) 0,1 2 − ∀ ∈ { } Proof sketch Claim a b = αx βy ⊕ ⊕ 1 xy = 0 (let x = 0 wlog) a b = 0 √ p(a b = xy x, y)= 1 ⇒ ⊕ ⊕ | 2 xy = 1 a b = 1 √ ⇒ ⊕ Degorre, Kaplan, Laplante, Roland () QIP2008 10/14 PR box and quasi-probabilities x y λ (α, β) a = λ αx b = λ βy ⊕ ⊕ q(α,β) α = 0 α = 1 1 pλ = 2 λ 0,1 1 1 ∀ ∈ { } β = 0 2 0 2 0 ( 1)αβ q α,β = − 1 1 ( ) 2 β = 1 2 y 2 y (α,β) 0,1 2 − ∀ ∈ { } Proof sketch Claim a b = αx βy ⊕ ⊕ 1 xy = 0 (let x = 0 wlog) a b = 0 √ p(a b = xy x, y)= 1 ⇒ ⊕ ⊕ | 2 xy = 1 a b = 1 √ ⇒ ⊕ Degorre, Kaplan, Laplante, Roland () QIP2008 10/14 PR box and quasi-probabilities x y λ (α, β) a = λ αx b = λ βy ⊕ ⊕ q(α,β) α = 0 α = 1 1 pλ = 2 λ 0,1 1 1 ∀ ∈ { } β = 0 2 0 2 1 ( 1)αβ q α,β = − 1 1 ( ) 2 β = 1 2 1 2 0 (α,β) 0,1 2 − ∀ ∈ { } Proof sketch Claim a b = αx βy ⊕ ⊕ 1 xy = 0 (let x = 0 wlog) a b = 0 √ p(a b = xy x, y)= 1 ⇒ ⊕ ⊕ | 2 xy = 1 a b = 1 √ ⇒ ⊕ Degorre, Kaplan, Laplante, Roland () QIP2008 10/14 Getting rid of communication Using PR boxes Using quasi-probabilities 1 bit of communication ↓ 1 PR-box 1 bit of communication 4 local protocols↓ with 1 qλ = P qλ = 2 ± 2 λ | | n bits of communication 2n 1 PR-boxes↓ − n bits of communication 4n local protocols↓ with Example (next talk) 1 n qλ = 2n Pλ qλ = 2 2 bit protocol for quantum ± | | correlations [Regev,Toner07] Application (next slide) 3 PR-box↓ protocol Communication complexity Degorre, Kaplan, Laplante, Roland () QIP2008 11/14 Getting rid of communication Using PR boxes Using quasi-probabilities 1 bit of communication ↓ 1 PR-box 1 bit of communication 4 local protocols↓ with 1 qλ = P qλ = 2 ± 2 λ | | n bits of communication 2n 1 PR-boxes↓ − n bits of communication 4n local protocols↓ with Example (next talk) 1 n qλ = 2n Pλ qλ = 2 2 bit protocol for quantum ± | | correlations [Regev,Toner07] Application (next slide) 3 PR-box↓ protocol Communication complexity Degorre, Kaplan, Laplante, Roland () QIP2008 11/14 Getting rid of communication Using PR boxes Using quasi-probabilities 1 bit of communication ↓ 1 PR-box 1 bit of communication 4 local protocols↓ with 1 qλ = P qλ = 2 ± 2 λ | | n bits of communication 2n 1 PR-boxes↓ − n bits of communication 4n local protocols↓ with Example (next talk) 1 n qλ = 2n Pλ qλ = 2 2 bit protocol for quantum ± | | correlations [Regev,Toner07] Application (next slide) 3 PR-box↓ protocol Communication complexity Degorre, Kaplan, Laplante, Roland () QIP2008 11/14 Getting rid of communication Using PR boxes Using quasi-probabilities 1 bit of communication ↓ 1 PR-box 1 bit of communication 4 local protocols↓ with 1 qλ = P qλ = 2 ± 2 λ | | n bits of communication 2n 1 PR-boxes↓ − n bits of communication 4n local protocols↓ with Example (next talk) 1 n qλ = 2n Pλ qλ = 2 2 bit protocol for quantum ± | | correlations [Regev,Toner07] Application (next slide) 3 PR-box↓ protocol Communication complexity Degorre, Kaplan, Laplante, Roland () QIP2008 11/14 Lower bound on communication complexity Theorem f may be computed with n bits of communication p may be written as a LHV with quasi-probabilies s.t.
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