Tsirelson’s problem and linear system games William Slofstra IQC, University of Waterloo October 18th, 2016 (with some corrections) includes joint work with Richard Cleve and Li Liu Tsirelson’s problem and linear system games William Slofstra
Non-local games Win/lose based on outputs a , b Referee y x and inputs x , y Alice and Bob must cooperate Alice Bob to win a Winning conditions known in b Referee advance Complication: players cannot Win Lose communicate while the game is in progress Tsirelson’s problem and linear system games William Slofstra
Strategies for non-local games Suppose game is played many Referee y x times, with inputs drawn from some public distribution π Alice Bob To outside observer, Alice and Bob’s strategy is described by: a b P ( a , b | x , y ) = the probability of Referee output ( a , b ) on input ( x , y ) Correlation matrix: collection of Win Lose numbers { P ( a , b | x , y ) } Tsirelson’s problem and linear system games William Slofstra
Classical and quantum strategies P ( a , b | x , y ) = the probability of output ( a , b ) on Referee y x input ( x , y ) Alice Bob Value of game ω = winning probability using strategy { P ( a , b | x , y ) } a b Referee What type of strategies might Alice and Bob use? Win Lose Classical: can use randomness, flip coin to determine output. Correlation matrix will be P ( a , b | x , y ) = A ( a | x ) B ( b | y ). Quantum: Alice and Bob can share entangled quantum state Bell’s theorem: Alice and Bob can do better with an entangled quantum state than they can do classically Tsirelson’s problem and linear system games William Slofstra
Quantum strategies How do we describe a quantum strategy? Use axioms of quantum mechanics: • Physical system described by (finite-dimensional) Hilbert space • No communication ⇒ Alice and Bob each have their own (finite dimensional) Hilbert spaces H A and H B • Hilbert space for composite system is H = H A ⊗ H B • Shared quantum state is a unit vector | ψ � ∈ H • Alice’s output on input x is modelled by measurement operators { M x a } a on H A • Similarly Bob has measurement operators { N y b } b on H B a ⊗ N y Quantum correlation: P ( a , b | x , y ) = � ψ | M x b | ψ � Tsirelson’s problem and linear system games William Slofstra
Quantum correlations Set of quantum correlations: � a ⊗ N y { P ( a , b | x , y ) } : P ( a , b | x , y ) = � ψ | M x C q = b | ψ � where | ψ � ∈ H A ⊗ H B , where H A , H B fin dim’l a and N y M x b are projections on H A and H B � � M x � N y a = I and b = I for all x , y a b Two variants: 1 C qs : Allow H A and H B to be infinite-dimensional 2 C qa = C q : limits of finite-dimensional strategies Relations: C q ⊆ C qs ⊆ C qa Tsirelson’s problem and linear system games William Slofstra
Commuting-operator model Another model for composite systems: commuting-operator model In this model: • Alice and Bob each have an algebra of observables A and B • A and B act on the joint Hilbert space H • A and B commute: if a ∈ A , b ∈ B , then ab = ba . This model is used in quantum field theory Correlation set: � a N y { P ( a , b | x , y ) } : P ( a , b | x , y ) = � ψ | M x C qc := b | ψ � , � a N y b = N y M x b M x a Hierarchy: C q ⊆ C qs ⊆ C qa ⊆ C qc Tsirelson’s problem and linear system games William Slofstra
Tsirelson’s problem strong C q ⊆ C qs ⊆ C qa ⊆ C qc weak Two models of QM: tensor product and commuting-operator Tsirelson problems: is C t , t ∈ { q , qs , qa } equal to C qc Fundamental questions: 1 What is the power of these models? Strong Tsirelson: is C q = C qc ? 2 Are there observable differences between these two models, accounting for noise and experimental error? Weak Tsirelson: is C qa = C qc ? Tsirelson’s problem and linear system games William Slofstra
What do we know? strong C q C qs C qa C qc ⊆ ⊆ ⊆ weak Theorem (Ozawa, JNPPSW, Fr) C qa = C qc if and only if Connes’ embedding problem is true Theorem (S) C qs � = C qc Tsirelson’s problem and linear system games William Slofstra
Other formulations (NCPV) Formulation due to Navascu´ es, Cooney, P´ erez-Garc´ ıa, Villanueva Given { P ( a , b | x , y ) } Local measurement statistics: P ( a | x ) = � b , y P ( a , b | x , y ), P ( b | y ) = similar Rather than modeling joint system, model Bob’s system: 1 For local measurement statistics, find measurements { N y b } and density matrix ρ such that N y � � P ( b | y ) = tr b ρ 2 For joint statistics, find measurements { N y b } and density matrices ρ xa such that N y b ρ xa � � P ( a , b | x , y ) = P ( a | x ) tr . Tsirelson’s problem and linear system games William Slofstra
Other formulations (NCPV continued) 1 For local measurement statistics, find measurements { N y b } and density matrix ρ such that N y � � P ( b | y ) = tr b ρ 2 For joint statistics, find measurements { N y b } and density matrices ρ xa such that N y � b ρ xa � P ( a , b | x , y ) = P ( a | x ) tr . Question: Can Bob build a model of his local statistics which is consistent with Alice’s observed inputs/outputs? Answer: If and only if there are ρ xa as above with a P ( a | x ) ρ xa = ρ (independent of x ) � Tsirelson’s problem and linear system games William Slofstra
Other formulations (NCPV continued) Question: Can Bob build a model of his local statistics which is consistent with Alice’s observed inputs/outputs? Answer: If and only if there are ρ xa as above with a P ( a | x ) ρ xa = ρ (independent of x ) � Fact: This happens if and only if { P ( a , b | x , y ) } belongs to C qs General state: a linear functional f : B → C such that f ( I ) = 1 and f ( A ) ≥ 0 if A is positive If ρ density matrix, then f ( A ) = tr( A ρ ) is general state Not every general state comes from a density matrix What if Bob uses general states instead of density matrices? Tsirelson’s problem and linear system games William Slofstra
Other formulations (NCPV continued) Condition (*): Bob can build a model of his local statistics which is consistent with Alice’s observed inputs/outputs If Bob uses density matrices, then (*) holds if and only if { P ( a , b | x , y ) } belongs to C qs If Bob uses general states, then (*) holds if and only if { P ( a , b | x , y ) } belongs to C qc Conclusion: Since C qs � = C qc , modeling power of general states is greater than modeling power of density matrices, even for Bell scenarios Tsirelson’s problem and linear system games William Slofstra
Other formulations (Ozawa) Correlations with limited interactions: � a ◦ N y { P ( a , b | x , y ) } : P ( a , b | x , y ) = � ψ | M x C qc ( ǫ ) = b | ψ � a N y b − N y � ≤ ǫ � � M x b M x � a � | ψ � ∈ finite-diml H These correlations are non-signalling Theorem (Ozawa,Coudron-Vidick) C qc = � ǫ> 0 C qc ( ǫ ) If { P ( a , b | x , y ) } has finite-dimensional limited interaction models for every ǫ > 0, does it belong to C q or C qa ? (Answer: no) Tsirelson’s problem and linear system games William Slofstra
Other fundamental questions 1 Given a non-local game, can we compute the optimal value ω t over strategies in C t , t ∈ { qa , qc } ? 2 Is C q = C qa ? (In other words, does every non-local game have an optimal finite-dimensional strategy?) 3 Given P ∈ C q , is there a computable upper bound on the dimension needed to realize P ? Tsirelson’s problem and linear system games William Slofstra
What do we know? Theorem (Navascu´ es, Pironio, Ac´ ın) Given a non-local game, there is a hierarchy of SDPs which converge in value to ω qc Problem: no way to tell how close we are to the correct answer Theorem (S) It is undecidable to tell if ω qc < 1 General cases of other questions completely open! Tsirelson’s problem and linear system games William Slofstra
Two theorems Theorem (S) C qs � = C qc Theorem (S) It is undecidable to tell if ω qc < 1 Proofs: make connection to group theory via linear system games Tsirelson’s problem and linear system games William Slofstra
Linear system games Start with m × n linear system Ax = b over Z 2 Inputs: • Alice receives 1 ≤ i ≤ m (an equation) • Bob receives 1 ≤ j ≤ n (a variable) Outputs: • Alice outputs an assignment a k for all variables x k with A ik � = 0 • Bob outputs an assignment b j for x j They win if: • A ij = 0 (assignment irrelevant) or • A ij � = 0 and a j = b j (assignment consistent) Tsirelson’s problem and linear system games William Slofstra
Quantum solutions of Ax = b Observables X j such that 1 X 2 j = I for all j j =1 X A ij = ( − I ) b i for all i 2 � n j 3 If A ij , A ik � = 0, then X j X k = X k X j (We’ve written linear equations multiplicatively) Theorem (Cleve-Mittal,Cleve-Liu-S) Let G be the game for linear system Ax = b. Then: • G has a perfect strategy in C qs if and only if Ax = b has a finite-dimensional quantum solution • G has a perfect strategy in C qc if and only if Ax = b has a quantum solution Tsirelson’s problem and linear system games William Slofstra
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