Quantum Correlations and Tsirelsons Problem NaruTaka OZAWA { - PowerPoint PPT Presentation

Quantum Correlations and Tsirelson’s Problem NaruTaka OZAWA � � ¤ { Ø � Research Institute for Mathematical Sciences, Kyoto University C ∗ -Algebras and Noncommutative Dynamics March 2013 About the Connes Embedding Conjecture—algebraic approaches—. Jpn. J. Math., 8 (2013). Tsirelson’s problem and asymptotically commuting unitary matrices. J. Math. Phys., 54 (2013). Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 1 / 17

Overview of today’s talk Operator Algebras Kirchberg’s Conjecture Connes Embedding Conjecture & Quantum Positivstellens¨ atze Measurement Theory Tsirelson’s Tsirelson’s Problem (1993, 2006) Problem Q c = Q s ? Quantum Noncommutative Kirchberg’s Conjecture (1993) Information Real Algebraic C ∗ F d ⊗ max C ∗ F d Semidefinite Theory Programming Geometry = C ∗ F d ⊗ min C ∗ F d ? Connes Embedding Conjecture (1976) ∀ M M ֒ → R ω ? Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 2 / 17

Quantum information theory Quantum information theory Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 3 / 17

Quantum measurement (von Neumann measurement) In probability theory, a trial with m outcomes is described by a probability space ( X , µ ) and a partition X = � m i =1 X i . When one obtains i as an outcome, the ambient probability space changes to ( X i , µ ( X i ) − 1 µ | X i ). � P i = 1 X i are orthogonal projections on L 2 ( X , µ ) with � m i =1 P i = 1. In quantum theory, a PVM (Projection Valued Measure) with m outcomes is an m -tuple ( P i ) m i =1 of orth projections on a Hilbert space H such that � m i =1 P i = 1, and the outcome of a m’ment of a (pure) state ψ ∈ H , a unit vector, is probabilistic: ( � ψ, P i ψ � ) m i =1 ∈ Prob([1 , . . . , m ]). When one obtains i as an outcome, the state ψ collapses to � P i ψ � − 1 P i ψ . Suppose Alice and Bob have d -PVMs respectively and a shared state: ( P k i ) m i =1 , k = 1 , . . . , d and ( Q l j ) m j =1 , l = 1 , . . . , d , and ψ . Each of them conducts a m’ment of ψ by using one of PVMs they have. What are the possibilities? Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 4 / 17

✿ ✿③ ✿③ ✿③ ✿③ EPR Paradox and Bell Test (CHSH Bell inequality) Suppose Alice and Bob have d -PVMs respectively and a shared state: ( P k i ) m i =1 , k = 1 , . . . , d and ( Q l j ) m j =1 , l = 1 , . . . , d , and ψ . Each of them conducts a m’ment of ψ by using one of PVMs they have. ( � ψ, P k i ψ � ) m ( � ψ, Q l j ψ � ) m i =1 ❞ ❞✩ ❞✩ ❞✩ ❞✩ j =1 ψ ψ ψ Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 5 / 17

✿③ ✿③ ✿③ ✿③ ✿ EPR Paradox and Bell Test (CHSH Bell inequality) Suppose Alice and Bob have d -PVMs respectively and a shared state: ( P k i ) m i =1 , k = 1 , . . . , d and ( Q l j ) m j =1 , l = 1 , . . . , d , and ψ . Each of them conducts a m’ment of ψ by using one of PVMs they have. In the classical setting, | E αβ ( AB ) + E αβ ′ ( AB ′ ) + E α ′ β ( A ′ B ) − E α ′ β ′ ( A ′ B ′ ) | ≤ 2, because | AB + AB ′ + A ′ B − A ′ B ′ | α = { Apple, Grape } β = { Hard, Soft } ≤ | B + B ′ | + | B − B ′ | ≤ 2. ❞ ❞✩ ❞✩ ❞✩ ❞✩ A = P A − P G B = Q H − Q S α ′ = { Red, Green } β ′ = { Big, Small } A ′ = P R − P G B ′ = Q B − Q S Does Nature conform this inequality? Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 5 / 17

Tsirelson’s Problem on quantum correlations We consider the convex sets C ⊂ Q s ⊂ Q c ⊂ Θ ⊂ M md ( R ≥ 0 ) of the classical and quantum correlation matrices for two separated systems: ( X , µ ) a (finite) prob space �� � P k i Q l ( P k i ) m C = { j d µ : i =1 , k = 1 , . . . , d , partitions of 1 X , } , k , l ( Q l j ) m j =1 , l = 1 , . . . , d , partitions of 1 X i , j ψ ∈ H ⊗ K a state � � � ψ, ( P k i ⊗ Q l ( P k i ) m i =1 , k = 1 , . . . , d , PVMs on H , Q s = cl { j ) ψ � : } , k , l ( Q l j ) m j =1 , l = 1 , . . . , d , PVMs on K i , j H a Hilbert space, ψ ∈ H a state ( P k i ) m i =1 , k = 1 , . . . , d , PVMs on H , � � � ψ, P k i Q l Q c = { j ψ � : } , ( Q l j ) m k , l j =1 , l = 1 , . . . , d , PVMs on H , i , j [ P k i , Q l j ] = 0 for all i , j and k , l γ k , l i , j γ k , l i , j ≥ 0, � i , j = 1 � � γ k , l Θ = { : i , j indep of l } . i γ k , l j γ k , l i , j k , l � i , j indep of k , � i , j Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 6 / 17

Tsirelson’s Problem on quantum correlations We consider the convex sets C ⊂ Q s ⊂ Q c ⊂ Θ ⊂ M md ( R ≥ 0 ) of the classical and quantum correlation matrices for two separated systems: ( X , µ ) a (finite) prob space �� � P k i Q l ( P k i ) m C = { j d µ : i =1 , k = 1 , . . . , d , partitions of 1 X , } , k , l ( Q l j ) m j =1 , l = 1 , . . . , d , partitions of 1 X i , j ψ ∈ H ⊗ K a state � � � ψ, ( P k i ⊗ Q l ( P k i ) m i =1 , k = 1 , . . . , d , PVMs on H , Q s = cl { j ) ψ � : } , k , l ( Q l j ) m j =1 , l = 1 , . . . , d , PVMs on K i , j H a Hilbert space, ψ ∈ H a state ( P k i ) m i =1 , k = 1 , . . . , d , PVMs on H , � � � ψ, P k i Q l C � = Q s by CHSH Bell inequality (1969) Q c = { j ψ � : } , ( Q l j ) m k , l j =1 , l = 1 , . . . , d , PVMs on H , | A 1 B 1 + A 1 B 2 + A 2 B 1 − A 2 B 2 | ≤ 2 i , j [ P k i , Q l j ] = 0 for all i , j and k , l for commuting variables − 1 ≤ A i , B j ≤ 1. γ k , l i , j γ k , l i , j ≥ 0, � i , j = 1 � � γ k , l Θ = { : i , j indep of l } . i γ k , l j γ k , l i , j k , l � i , j indep of k , � i , j Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 6 / 17

Tsirelson’s Problem on quantum correlations We consider the convex sets C ⊂ Q s ⊂ Q c ⊂ Θ ⊂ M md ( R ≥ 0 ) of the classical and quantum correlation matrices for two separated systems: ( X , µ ) a (finite) prob space �� � P k i Q l ( P k i ) m C = { j d µ : i =1 , k = 1 , . . . , d , partitions of 1 X , } , Q c � = Θ by Cirel’son’s Quantum Bell inequality (1980) k , l √ ( Q l j ) m j =1 , l = 1 , . . . , d , partitions of 1 X i , j | A 1 B 1 + A 1 B 2 + A 2 B 1 − A 2 B 2 | ≤ 2 2 ψ ∈ H ⊗ K a state for operators − 1 ≤ A i , B j ≤ 1 with [ A i , B j ] = 0. � � � ψ, ( P k i ⊗ Q l ( P k i ) m i =1 , k = 1 , . . . , d , PVMs on H , Q s = cl { j ) ψ � : } , k , l ( Q l j ) m j =1 , l = 1 , . . . , d , PVMs on K i , j H a Hilbert space, ψ ∈ H a state ( P k i ) m i =1 , k = 1 , . . . , d , PVMs on H , � � � ψ, P k i Q l Q c = { j ψ � : } , ( Q l j ) m k , l j =1 , l = 1 , . . . , d , PVMs on H , i , j [ P k i , Q l j ] = 0 for all i , j and k , l γ k , l i , j γ k , l i , j ≥ 0, � i , j = 1 � � γ k , l Θ = { : i , j indep of l } . i γ k , l j γ k , l i , j k , l � i , j indep of k , � i , j Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 6 / 17

Tsirelson’s Problem on quantum correlations We consider the convex sets C ⊂ Q s ⊂ Q c ⊂ Θ ⊂ M md ( R ≥ 0 ) of the classical and quantum correlation matrices for two separated systems: Note that Q c becomes same as Q s if we restrict the Hilbert ( X , µ ) a (finite) prob space �� � spaces H appearing in the definition of Q c to fin-dim ones. P k i Q l ( P k i ) m C = { j d µ : i =1 , k = 1 , . . . , d , partitions of 1 X , } , k , l ( Q l j ) m j =1 , l = 1 , . . . , d , partitions of 1 X i , j ψ ∈ H ⊗ K a state � � � ψ, ( P k i ⊗ Q l ( P k i ) m i =1 , k = 1 , . . . , d , PVMs on H , Q s = cl { j ) ψ � : } , k , l ( Q l j ) m j =1 , l = 1 , . . . , d , PVMs on K i , j H a Hilbert space, ψ ∈ H a state ( P k i ) m i =1 , k = 1 , . . . , d , PVMs on H , � � � ψ, P k i Q l Q c = { j ψ � : } , ( Q l j ) m k , l j =1 , l = 1 , . . . , d , PVMs on H , i , j [ P k i , Q l j ] = 0 for all i , j and k , l γ k , l i , j γ k , l i , j ≥ 0, � i , j = 1 � � γ k , l Tsirelson’s Problem: Q c = Q s ? Θ = { : i , j indep of l } . i γ k , l j γ k , l i , j k , l � i , j indep of k , � i , j Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 6 / 17

Quantum correlation and C ∗ -algebras Quantum correlation matrices are related to the C ∗ -algebra ℓ m ∞ ∗ · · · ∗ ℓ m ( d -fold unital full free product) , ∞ which is isomorphic to the full group C ∗ -algebra C ∗ (Γ) of Γ = Z ∗ d m . Denote by e k i the standard basis of projections in the k -th copy of ℓ m ∞ . j in C ∗ (Γ) ⊗ C ∗ (Γ). Then, one has Also e k i := e k i ⊗ 1 and f l j := 1 ⊗ e l � � φ ( e k i f l : φ a state on C ∗ (Γ) ⊗ max C ∗ (Γ) } Q c = { j ) k , l i , j and � � φ ( e k i f l : φ a state on C ∗ (Γ) ⊗ min C ∗ (Γ) } . Q s = { j ) k , l i , j Theorem (Kirchberg 1993, Fritz and Junge et al. 2010, Oz. 2012) The following conjectures are equivalent. Tsirelson’s problem has an affirmative answer: Q c = Q s for all m , d . Kirchberg’s Conjecture: C ∗ (Γ) ⊗ max C ∗ (Γ) = C ∗ (Γ) ⊗ min C ∗ (Γ) holds. → R ω for for every II 1 factor M . Connes’s Embedding Conjecture: M ֒ Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 7 / 17