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) |

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

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 = 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)

Replacing 1 bit of communication by a PR box x [X] y [Y ] ∈ ∈

α(x) β(y, 0)

Replacing 1 bit of communication by a PR box x [X] y [Y ] ∈ ∈

α(x) β(y, 0)

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

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

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

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

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

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. ⇒ f n Q = qλ = 2 | | Xλ

Theorem By duality D(f ) log Q, where D(f ) log B, where ≥ ≥

Q = minq qλ , B = max b p , λ λ | | b f λ λ· subject to Pqλd = pf subject to b d 1 λ λ | · |≤ ∀ P LHV with quasi-probabilities Bell inequality

Degorre, Kaplan, Laplante, Roland () QIP2008 12/14 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. ⇒ f n Q = qλ = 2 | | Xλ

Theorem By duality D(f ) log Q, where D(f ) log B, where ≥ ≥ Q = minq qλ , B = max b p , λ λ | | b f λ λ· subject to Pqλd = pf subject to b d 1 λ λ | · |≤ ∀ P LHV with quasi-probabilities Bell inequality

Degorre, Kaplan, Laplante, Roland () QIP2008 12/14 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. ⇒ f n Q = qλ = 2 | | Xλ

Degorre, Kaplan, Laplante, Roland () QIP2008 12/14 Characterization of PR-box complexity

Degorre, Kaplan, Laplante, Roland () QIP2008 13/14 Characterization of PR-box complexity

n bits of communication 2n 1 PR-boxes → − Is this tight? n PR-boxes n bits of communication → Theorem f may be computed with m PR-boxes in parallel (+1 bit) f may be written as ⇔ f (x, y)= m α (x)β (y) α (x) β (y) i=1 i i ⊕ 0 ⊕ 0 L Note: Minimum m Rank over Z2 −→

α1(x) β1(y) α2(x) β2(y) αm(x) βm(y)

···

a1 b1 a2 b2 am bm

Degorre, Kaplan, Laplante, Roland () QIP2008 13/14 Characterization of PR-box complexity

n bits of communication 2n 1 PR-boxes → − Is this tight? n PR-boxes n bits of communication → Theorem f may be computed with m PR-boxes in parallel (+1 bit) f may be written as ⇔ m f (x, y)= α (x)β (y) α0(x) β0(y) i=1 i i ⊕ ⊕ L Note: Minimum m Rank over Z −→ 2 Examples: IP(x, y)= x y requires n PR boxes in parallel ⊕i i i DISJ(x, y)= (x y 1) requires 2n 1 PR boxes in parallel i i i ⊕ − Both require n bitsQof communication

Degorre, Kaplan, Laplante, Roland () QIP2008 13/14 Conclusion

Summary non-signaling distributions LHV with quasi-probabilities ←→ New intuition for studying non-locality Relation: communication complexity Bell inequalities ←→ Lower bound for deterministic communication complexity Characterization of PR box complexity (quantiﬁes [van Dam 05])

Further work and open questions Communication complexity Extension to randomized/quantum communication complexity Concrete lower bounds PR-boxes Parallel vs serial PR-boxes Related to the (non-)distillability of noisy PR-boxes →

Degorre, Kaplan, Laplante, Roland () QIP2008 14/14 Conclusion

Degorre, Kaplan, Laplante, Roland () QIP2008 14/14