Contextuality in Multipartite Pseudo-Telepathic Graph Games Simon Perdrix CNRS, Inria Project team CARTE, LORIA simon.perdrix@loria.fr joint work with Anurag Anshu, Peter Høyer and Mehdi Mhalla FCT’17 – Bordeaux
Introduction Contextuality / non locality: very foundation of quantum mechanics. • Quantum mechanics is contextual [Kochen–Specker67] • Active area of research: understand the mathematical structures of contextuality [Abramsky et al.; Ac´ ın et al.] 2/12
Introduction Contextuality / non locality: very foundation of quantum mechanics. • Quantum mechanics is contextual [Kochen–Specker67] • Active area of research: understand the mathematical structures of contextuality [Abramsky et al.; Ac´ ın et al.] Pseudo-telepathic games • Collaborative games: all the players win or all the players lose. • No communication between the players. • Shared resources (random bits, entangled states...) • Graph games. • Interesting cases: the games that can be won quantumly but not classically (pseudo-telepathic game). 2/12
Multipartite Games Definition. Given a set V of players, a game is characterized by a set of losing pairs L ⊆ { 0 , 1 } V × { 0 , 1 } V : • Question: x ∈ { 0 , 1 } V : each player u ∈ V receives a single bit x u ∈ { 0 , 1 } of the question. • Answer: a ∈ { 0 , 1 } V : each player u ∈ V produces a single bit a u ∈ { 0 , 1 } of the answer, without communication. • The players win if ( a | x ) / ∈ L . 3/12
Multipartite Games Definition. Given a set V of players, a game is characterized by a set of losing pairs L ⊆ { 0 , 1 } V × { 0 , 1 } V : • Question: x ∈ { 0 , 1 } V : each player u ∈ V receives a single bit x u ∈ { 0 , 1 } of the question. • Answer: a ∈ { 0 , 1 } V : each player u ∈ V produces a single bit a u ∈ { 0 , 1 } of the answer, without communication. • The players win if ( a | x ) / ∈ L . Example. Mermin Game : L Mermin = { ( a | x ) : 2 | a | = | x | + 1 mod 4 } , where | x | is the Hamming weight of x . 3/12
Multipartite Games Definition. Given a set V of players, a game is characterized by a set of losing pairs L ⊆ { 0 , 1 } V × { 0 , 1 } V : • Question: x ∈ { 0 , 1 } V : each player u ∈ V receives a single bit x u ∈ { 0 , 1 } of the question. • Answer: a ∈ { 0 , 1 } V : each player u ∈ V produces a single bit a u ∈ { 0 , 1 } of the answer, without communication. • The players win if ( a | x ) / ∈ L . Example. Mermin Game : L Mermin = { ( a | x ) : 2 | a | = | x | + 1 mod 4 } , where | x | is the Hamming weight of x . E.g., when V = { 1 , 2 , 3 } , a ∈ { 0 , 1 } V should satisfy question x ∈ { 0 , 1 } V 100 2 | a | � = 2 mod 4 ⇔ | a | � = 1 mod 2 ⇔ | a | = 0 mod 2 010 | a | = 0 mod 2 001 | a | = 0 mod 2 111 | a | = 1 mod 2 3/12
No Classical Winning Strategy for the Mermin Game Property. There is no winning strategy for the Mermin game when | V | ≥ 3 . Proof. For any u ∈ V , let f u : { 0 , 1 } → { 0 , 1 } be the deterministic strategy of player u : a u = f u ( x u ) . x = 100 f 1 (1) + f 2 (0) + f 3 (0) = 0 mod 2 x = 010 f 1 (0) + f 2 (1) + f 3 (0) = 0 mod 2 x = 001 f 1 (0) + f 2 (0) + f 3 (1) = 0 mod 2 x = 111 f 1 (1) + f 2 (1) + f 3 (1) = 1 mod 2 4/12
No Classical Winning Strategy for the Mermin Game Property. There is no winning strategy for the Mermin game when | V | ≥ 3 . Proof. For any u ∈ V , let f u : { 0 , 1 } → { 0 , 1 } be the deterministic strategy of player u : a u = f u ( x u ) . x = 100 f 1 (1) + f 2 (0) + f 3 (0) = 0 mod 2 x = 010 f 1 (0) + f 2 (1) + f 3 (0) = 0 mod 2 x = 001 f 1 (0) + f 2 (0) + f 3 (1) = 0 mod 2 x = 111 f 1 (1) + f 2 (1) + f 3 (1) = 1 mod 2 0 = 1 mod 2 4/12
A Quantum Strategy Assume the players share a graphe state | G △ � . 2 1 3 Quantum strategy: • If x u = 0 , a u is the outcome of the measure according to Z of qubit u ; • If x u = 1 , a u is the outcome of the measure according to X of qubit u ; 5/12
A Quantum Strategy Assume the players share a graphe state | G △ � . 2 X Z Z 1 3 Quantum strategy: • If x u = 0 , a u is the outcome of the measure according to Z of qubit u ; • If x u = 1 , a u is the outcome of the measure according to X of qubit u ; • When x = 100 : Fundamental property of Graph state: X 1 Z 2 Z 3 | G △ � = | G △ � which implies a 1 + a 2 + a 3 = 0 mod 2 5/12
A Quantum Strategy Assume the players share a graphe state | G △ � . 2 X X X 1 3 Quantum strategy: • If x u = 0 , a u is the outcome of the measure according to Z of qubit u ; • If x u = 1 , a u is the outcome of the measure according to X of qubit u ; • When x = 100 : Fundamental property of Graph state: X 1 Z 2 Z 3 | G △ � = | G △ � which implies a 1 + a 2 + a 3 = 0 mod 2 • When x = 111 : X 1 Z 2 Z 3 X 2 Z 1 Z 3 X 3 Z 1 Z 2 | G △ � = | G △ � ⇔ X 1 Z 2 X 2 Z 2 X 3 | G △ � = | G △ � since Z 2 = I ⇔ X 1 X 2 X 3 | G △ � = − | G △ � since XZ = − ZX . which implies a 1 + a 2 + a 3 = 1 mod 2 5/12
Graph Games Definition. [Anshu, Mhalla’13] Given a graph G = ( V, E ) , let L G := { ( a | x ) : ∃ D involved in x s.t. � u ∈ D ∪ Odd ( D ) a u = | G [ D ] | + 1 mod 2 } , where • Odd ( D ) = { u ∈ V | | N ( u ) ∩ D | = 1 mod 2 } is the odd neighbourhood of D ; • | G [ D ] | is the number of edges of the subgraph induced by D ; • D is involved with x if ∀ u ∈ D, x u = 1 and ∀ u ∈ Odd ( D ) , x u = 0 . 1 0 D 1 Losing condition: 1 � a u = | G [ D ] | + 1 mod 2 u ∈ D ∪ Odd ( D ) 1 = 1 mod 2 Odd ( D ) 0 0 6/12
Quantum Strategy QStrat: Given a graph G = ( V, E ) , the players share the graph state | G � , • If x u = 0 , a u is the outcome of the Z -measurement of qubit u • If x u = 1 , a u is the outcome of the X -measurement of qubit u Property. [Anshu,Mhalla’13] For any graph G , QStrat is a winning strategy. � L G := { ( a | x ) : ∃ D involved in x s.t. a u = | G [ D ] | + 1 mod 2 } u ∈ D ∪ Odd ( D ) 7/12
Quantum Strategy QStrat: Given a graph G = ( V, E ) , the players share the graph state | G � , • If x u = 0 , a u is the outcome of the Z -measurement of qubit u • If x u = 1 , a u is the outcome of the X -measurement of qubit u Property. [Anshu,Mhalla’13] For any graph G , QStrat is a winning strategy. � L G := { ( a | x ) : ∃ D involved in x s.t. a u = | G [ D ] | + 1 mod 2 } u ∈ D ∪ Odd ( D ) Proof. Given a x ∈ { 0 , 1 } V and D ⊆ V involved with x , ⇔ X D Z Odd ( D ) | G � = ( − 1) | G [ D ] | | G � � X u Z N ( u ) | G � = | G � u ∈ D Since D is involved, qubits in D are X -measured, those in Odd ( D ) are Z -measured, so � a u = | G [ D ] | mod 2 u ∈ D ∪ Odd ( D ) 7/12
Quantum Strategy QStrat: Given a graph G = ( V, E ) , the players share the graph state | G � , • If x u = 0 , a u is the outcome of the Z -measurement of qubit u • If x u = 1 , a u is the outcome of the X -measurement of qubit u Property. [Anshu,Mhalla’13] For any graph G , QStrat is a winning strategy. Property. QStrat produces the good answers uniformly: � 0 if ( a | x ) ∈ L p ( a | x ) = |{ D involved in x }| otherwise. 2 | V | 8/12
Quantum Strategy QStrat: Given a graph G = ( V, E ) , the players share the graph state | G � , • If x u = 0 , a u is the outcome of the Z -measurement of qubit u • If x u = 1 , a u is the outcome of the X -measurement of qubit u Property. [Anshu,Mhalla’13] For any graph G , QStrat is a winning strategy. Property. QStrat produces the good answers uniformly: � 0 if ( a | x ) ∈ L p ( a | x ) = |{ D involved in x }| otherwise. 2 | V | Property. When the players share a graph state | G � , they can win any game described by a graph pivot-equivalent to G . u v v u C C A B A B D D 8/12
Classical Strategy CStrat: Given a graph G = ( V, E ) , pick uniformly at random λ ∈ { 0 , 1 } V . Each player u ∈ V receives ( λ u , µ u ) , where µ u = � v ∈ N G ( u ) λ u mod 2 . Given a question x ∈ { 0 , 1 } V , each player u ∈ V locally computes and answers a u = (1 − x u ) .λ u + x u .µ u mod 2 . ( λ b , λ a ⊕ λ c ⊕ λ e ) c ( λ c , λ b ⊕ λ d ) ( λ a , λ b ⊕ λ e ) b a a d e ( λ d , λ c ⊕ λ e ) ( λ e , λ a ⊕ λ b ⊕ λ d ) Property. For any bipartite graph G , CStrat is a winning strategy. 9/12
Classical Strategy CStrat: Given a graph G = ( V, E ) , pick uniformly at random λ ∈ { 0 , 1 } V . Each player u ∈ V receives ( λ u , µ u ) , where µ u = � v ∈ N G ( u ) λ u mod 2 . Given a question x ∈ { 0 , 1 } V , each player u ∈ V locally computes and answers a u = (1 − x u ) .λ u + x u .µ u mod 2 . ( λ b , λ a ⊕ λ c ⊕ λ e ) c ( λ c , λ b ⊕ λ d ) ( λ a , λ b ⊕ λ e ) b a a d e ( λ d , λ c ⊕ λ e ) ( λ e , λ a ⊕ λ b ⊕ λ d ) Property. For any bipartite graph G , CStrat is a winning strategy. Theorem [Anshu, Mhalla’13] If G is not bipartite, there is no winning classical strategy. 9/12
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