Entanglement requirements in non-local games William Slofstra IQC, University of Waterloo August 31, 2017 Entanglement requirements in non-local games William Slofstra
Non-local games (aka Bell scenarios) 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 Entanglement requirements in non-local games William Slofstra
Example: the CHSH game Compare with: Referee x ∈ { 0 , 1 } y ∈ { 0 , 1 } A 0 B 0 + A 0 B 1 + A 1 B 0 − A 1 B 1 Alice Bob b ∈ { 0 , 1 } a ∈ { 0 , 1 } Referee a ⊕ b = x ∧ y otherwise Win Lose Entanglement requirements in non-local games William Slofstra
Non-local games more formally A non-local game consists of: Referee y x Finite input sets: I X , I Y Alice Bob Finite output sets: O X , O Y a b Referee A prob. distribution π on I X × I Y Win Lose A function V : O X × O Y × I X × I Y → { 0 , 1 } Entanglement requirements in non-local games William Slofstra
Non-local games more formally A non-local game consists of: Referee y x Finite input sets: I X , I Y Alice Bob Finite output sets: O X , O Y a b Referee A prob. distribution π on I X × I Y Win Lose A function V : O X × O Y × I X × I Y → { 0 , 1 } Interpretation: If Alice and Bob win on inputs ( x , y ) and outputs ( a , b ) then V ( a , b | x , y ) = 1. Otherwise V ( a , b | x , y ) = 0. Entanglement requirements in non-local games William Slofstra
Strategies: what can Alice and Bob do? Deterministic local strategies: Referee y x Choose a x ’s and b y ’s ahead of time Alice Bob Alice outputs a x on input x a b Referee Bob outputs b y on input y Win Lose Entanglement requirements in non-local games William Slofstra
Strategies: what can Alice and Bob do? Deterministic local strategies: Referee y x Choose a x ’s and b y ’s ahead of time Alice Bob Alice outputs a x on input x a b Referee Bob outputs b y on input y Win Lose The winning probability for this strategy S is � ω ( S ) = π ( x , y ) V ( a x , b y | x , y ) . x I A , y ∈I B The classical value of the game G is ω c ( G ) = max { ω ( S ) : deterministic strategies S } . Entanglement requirements in non-local games William Slofstra
What can the players do? Quantum strategy: Referee y x Alice and Bob share quantum state Alice Bob | ψ � ∈ H A ⊗ H B a b Referee Choose outputs according to PVMs a } , { Q y { P x b } Win Lose Entanglement requirements in non-local games William Slofstra
What can the players do? Quantum strategy: Referee y x Alice and Bob share quantum state Alice Bob | ψ � ∈ H A ⊗ H B a b Referee Choose outputs according to PVMs a } , { Q y { P x b } Win Lose The winning probability for this strategy S is � a ⊗ Q y π ( x , y ) V ( a x , b y | x , y ) � ψ | P x ω ( S ) = b | ψ � . x I A , y ∈I B The quantum value of the game G is ω q ( G ) = sup { ω ( S ) : quantum strategies S } . Note: no bound on dim H A , H B assumed Entanglement requirements in non-local games William Slofstra
Entanglement requirements If ω c ( G ) < ω q ( G ), then G is a distributed computational task with quantum advantage Entanglement requirements in non-local games William Slofstra
Entanglement requirements If ω c ( G ) < ω q ( G ), then G is a distributed computational task with quantum advantage We’d like a resource theory for non-local games How much “entanglement” is required to achieve ω q ( G )? (the quantum value) ω q ( G ) − ǫ ? (near the quantum value) Entanglement requirements in non-local games William Slofstra
Entanglement requirements If ω c ( G ) < ω q ( G ), then G is a distributed computational task with quantum advantage We’d like a resource theory for non-local games How much “entanglement” is required to achieve ω q ( G )? (the quantum value) ω q ( G ) − ǫ ? (near the quantum value) Possible resources: local Hilbert space dimension, von Neumann entropy, “non-locality” Entanglement requirements in non-local games William Slofstra
Why do we care about entanglement requirements? • Test cases for power of entanglement • Certify presence of entanglement • Self-test quantum states: For some games G , achieving ω q ( G ) or ω q ( G ) − ǫ can require states or strategies of a certain form. • Device independent protocols in cryptography Entanglement requirements in non-local games William Slofstra
Why do we care about entanglement requirements? • Test cases for power of entanglement • Certify presence of entanglement • Self-test quantum states: For some games G , achieving ω q ( G ) or ω q ( G ) − ǫ can require states or strategies of a certain form. • Device independent protocols in cryptography There are other important questions, like: Can we compute value of ω q ( G )? (Note: ω c ( G ) is relatively easy to compute) Entanglement requirements in non-local games William Slofstra
What do we know about entanglement requirements? Bounded entanglement is not enough: there are games with O ( n ) questions requiring dimension 2 Ω( n ) to play optimally Entanglement requirements in non-local games William Slofstra
What do we know about entanglement requirements? Bounded entanglement is not enough: there are games with O ( n ) questions requiring dimension 2 Ω( n ) to play optimally Is a finite amount of entanglement required for every fixed G ? Entanglement requirements in non-local games William Slofstra
What do we know about entanglement requirements? Bounded entanglement is not enough: there are games with O ( n ) questions requiring dimension 2 Ω( n ) to play optimally Is a finite amount of entanglement required for every fixed G ? Conjecture [PV10]: there is a game with three questions and two answers per player for which finite local dimensions are not enough Finite dimensions are not sufficient for variants of non-local games: [LTW13], [MV13], [RV15] Entanglement requirements in non-local games William Slofstra
What do we know about entanglement requirements? Bounded entanglement is not enough: there are games with O ( n ) questions requiring dimension 2 Ω( n ) to play optimally Is a finite amount of entanglement required for every fixed G ? Conjecture [PV10]: there is a game with three questions and two answers per player for which finite local dimensions are not enough Finite dimensions are not sufficient for variants of non-local games: [LTW13], [MV13], [RV15] Theorem (S): there is a non-local game (with several hundred questions per player) for which finite local dimensions does not suffice to achieve ω q ( G ) Entanglement requirements in non-local games William Slofstra
New tool: connection to group theory Linear system game: Start with m × n linear system Ax = b over Z 2 Inputs: Alice receives 1 ≤ i ≤ m (equation) Bob receives 1 ≤ j ≤ n (variable) Outputs: Alice: assignment to variables x k with A ik � = 0 Bob: assignment to variable x j Win if Alice’s assignment satisfies equation i , and either A ij = 0 or Alice’s assignment agrees with Bob’s Entanglement requirements in non-local games William Slofstra
New tool: connection to group theory Linear system game: Start with m × n linear system Ax = b over Z 2 Inputs: Alice receives 1 ≤ i ≤ m (equation) Bob receives 1 ≤ j ≤ n (variable) Outputs: Alice: assignment to variables x k with A ik � = 0 Bob: assignment to variable x j Win if Alice’s assignment satisfies equation i , and either A ij = 0 or Alice’s assignment agrees with Bob’s Classically: can play perfectly iff Ax = b has a solution (Play perfectly = win with probability 1) Entanglement requirements in non-local games William Slofstra
Quantum solutions of Ax = b Theorem (Cleve-Mittal, Cleve-Liu-S): Can play linear system game perfectly with a quantum strategy iff: there are observables X j such that 1 X 2 j = I for all j j =1 X A ij 2 � n = ( − I ) b i for all i j 3 If A ij , A ik � = 0, then X j X k = X k X j (We’ve written linear equations multiplicatively) Entanglement requirements in non-local games William Slofstra
Quantum solutions of Ax = b Theorem (Cleve-Mittal, Cleve-Liu-S): Can play linear system game perfectly with a quantum strategy iff: there are observables X j such that 1 X 2 j = I for all j j =1 X A ij 2 � n = ( − I ) b i for all i j 3 If A ij , A ik � = 0, then X j X k = X k X j (We’ve written linear equations multiplicatively) If this happens, say that Ax = b has a quantum solution (Warning: there are some big footnotes here) Entanglement requirements in non-local games William Slofstra
Connection with group theory The solution group Γ of Ax = b is the group generated by X 1 , . . . , X n , J such that j = [ X j , J ] = J 2 = e for all j 1 X 2 j =1 X A ij 2 � n = J b i for all i j 3 If A ij , A ik � = 0, then [ X j , X k ] = e where [ a , b ] = aba − 1 b − 1 , e = group identity Theorem (Cleve-Mittal) Let G be the game for linear system Ax = b. Then G has a perfect (tensor-product) strategy if and only if J is non-trivial in some finite-dimensional representation of the solution group Γ . Entanglement requirements in non-local games William Slofstra
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