Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra University of Waterloo April 4th, 2018 Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Finite-dimensional models... of groups G = � x 1 , . . . , x n : r 1 , . . . , r m � , a finitely presented group Finite-dimensional model of G : complex matrices X 1 , . . . , X n such that r i ( X 1 , . . . , X n ) = 1 for 1 ≤ i ≤ m a.k.a. a finite-dimensional representation G → GL( C d ) Question: when can we recover G from its finite-dimensional models? Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Question: when can we recover G from its finite-dimensional models? • G is linear : a subgroup of GL( C d ) for some d • G is residually finite-dimensional (RFD) : for every w ∈ G \ { e } , there is a finite-dimensional representation φ with φ ( w ) � = 1. Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Question: when can we recover G from its finite-dimensional models? • G is linear : a subgroup of GL( C d ) for some d G is residually finite-dimensional (RFD) : for every w ∈ G \ { e } , there is a finite-dimensional representation φ with φ ( w ) � = 1. Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Residually finite-dimensional groups G is residually finite-dimensional (RFD) : for every w ∈ G \ { e } , there is a finite-dimensional representation φ with φ ( w ) � = 1. Are there non-RFD groups? Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Residually finite-dimensional groups G is residually finite-dimensional (RFD) : for every w ∈ G \ { e } , there is a finite-dimensional representation φ with φ ( w ) � = 1. Are there non-RFD groups? Yes, lots... • Baumslag-Solitar group: BS (2 , 3) = � x , y : xy 2 x − 1 = y 3 � • Higman’s group: � a , b , c , d : aba − 1 = b 2 , bcb − 1 = c 2 , cdc − 1 = d 2 , dad − 1 = a 2 � (Higman’s group has no non-trivial finite-dimensional reps) Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Quantum mechanics (quantum probability) ... is a framework for working with physical systems Axioms of quantum mechanics • Physical systems = Hilbert space H • State of system = unit vector v ∈ H • Measurement: projections { P a } a ∈ O on H with � a P a = 1. O = set of measurement outcomes Probability of measuring a is v ∗ P a v • ... Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Surprising features of quantum mechanics Uncertainty principle Might not be able to measure two properties simultaneously Can measure { M a } and { N b } simultaneously only if M a N b = N b M a for all a , b Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Surprising features: contextuality Problem: x 1 x 2 x 3 1 assign ± 1 to x 1 , . . . , x 9 so that 1. product across rows is 1 x 4 x 5 x 6 2. product across columns is − 1 1 (this is a linear system over Z 2 x 7 x 8 x 9 1 with 9 variables, 6 equations) − 1 − 1 − 1 Not possible by parity argument Mermin-Peres magic square Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Surprising features: contextuality Imagine a physical system, where x 1 x 2 x 3 1 state of system is set, and then we measure either a row or a x 4 x 5 x 6 column 1 If magic square conditions are x 7 x 8 x 9 1 satisfied, then x i seems to depend on whether it is measured as part of a row or column − 1 − 1 − 1 Mermin-Peres magic square Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Surprising features: contextuality Quantum solution: there are x 1 x 2 x 3 1 unitaries X 1 , . . . , X 9 such that 0. X 2 i = 1 for i = 1 , . . . , 9 x 4 x 5 x 6 1. product across rows is 1 1 2. product across columns is − 1 3. if X i , X j belong to the same x 7 x 8 x 9 1 row or column, then X i X j = X j X i Interpretation: quantum − 1 − 1 − 1 mechanics is contextual! Mermin-Peres magic square Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Quantum solutions of linear systems Ax = b : m × n linear system over Z 2 Quantum solution: Collection of unitaries X 1 , . . . , X n ∈ U ( H ) such that 1. X 2 j = 1 for all j , j =1 X A ij 2. � n = ( − 1) b i for all i = 1 , . . . , n , j 3. X j X k = X k X j if A ij , A ik � = 0 for some i . Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Quantum solutions of linear systems Ax = b : m × n linear system over Z 2 Solution group of Ax = b j = 1 = [ x j , J ] = J 2 for all j Γ( A , b ) = � x 1 , . . . , x n , J : x 2 x A ij � = J b i , i = 1 , . . . , m j j [ x j , x k ] = 1 if A ij , A ik � = 0 , some i � Group commutator: [ a , b ] = aba − 1 b − 1 Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Linear system non-local games Mermin-Peres contextuality is hard to observe in an experiment m × n linear system Ax = b = ⇒ game with two separated players (Aravind, Cleve-Mittal) satisfying assignment equation index A 1 ≤ i ≤ m to variables in equation i variable index B assignment to x j 1 ≤ j ≤ n Inputs chosen at random Players win if Alice’s output is consistent with Bob’s output Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Non-local game Linear system games are examples of non-local games: A x ∈ I A a ∈ O A Win if V ( a , b | x , y ) = 1 B y ∈ I B b ∈ O B Inputs chosen from I A × I B according to some distribution Winning condition: function V : O A × O B × I A × I B → { 0 , 1 } Players know rules of game, want to cooperate to win Cannot communicate once game starts... may not be able to play perfectly Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Linear system non-local games: classical strategies Inputs chosen at random equation index satisfying assignment A 1 ≤ i ≤ m to variables in equation i Players win if Alice’s output is consistent variable index B assignment to x j with Bob’s output 1 ≤ j ≤ n Classical strategies (deterministic or shared randomness): Optimal strategy achieved by a deterministic strategy Players can play perfectly if and only if Ax = b has a solution (Otherwise optimal strategy has success probability p < 1) Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Linear system non-local games: quantum strategies Inputs chosen at random equation index satisfying assignment A 1 ≤ i ≤ m to variables in equation i Players win if Alice’s output is consistent variable index B assignment to x j with Bob’s output 1 ≤ j ≤ n Theorem (Cleve-Mittal) The game associated to Ax = b has a perfect quantum strategy if and only if Ax = b has a finite-dimensional quantum solution. Mermin-Peres square: perfect quantum strategy Best classical strategy succeeds with probability 35 / 36 Bell test: success probability > 35 / 36 = ⇒ non-classicality Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Quantum strategies and quantum correlations A x ∈ I A a ∈ O A B y ∈ I B b ∈ O B Alice and Bob’s behaviour in a non-local game is described by a family of probability distributions { p ( a , b | x , y ) } ⊂ R O A ×O B ×I A ×I B where p ( a , b | x , y ) = probability of output ( a , b ) on input ( x , y ) { p ( a , b | x , y ) } is called a correlation matrix Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Quantum strategies and quantum correlations Which correlations can arise in quantum mechanics? A B Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Quantum strategies and quantum correlations Which correlations can arise in quantum H A ⊗ H B mechanics? H B Axiom for separated subsystems If H A and H B are Hilbert spaces of two separated systems, then joint sys- H B tem has Hilbert space H A ⊗ H B Important note: State of joint system does not have to be a product v ⊗ w We can have entangled states like e 1 ⊗ e 1 + e 2 ⊗ e 2 Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Quantum correlations: formal definition Quantum correlations: Alice and Bob generate output by measuring a shared quantum state Definition A correlation { p ( a , b | x , y ) } ∈ R O A ×O B ×I A ×I B is quantum if there are: • Hilbert spaces H A , H B , • a state v ∈ H A ⊗ H B , • measurements { M x a } a ∈O A on H A for every x ∈ I A , and • measurements { N y b } b ∈O B on H B for every y ∈ I B , a ⊗ N y such that p ( a , b | x , y ) = v ∗ M x b v for all a , b , x , y . Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
Sets of correlations • C c = C c ( O A , O B , I A , I B ) := classical correlations • C q := quantum correlations with finite-dimensional Hilbert spaces • C qs := Quantum correlations with any Hilbert spaces All three sets are convex, C c is closed C q and C qs capture behaviour of entangled states What correlation set matches reality? (what correlations can we generate in nature?) Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra
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