Quantum Correlations and Tsirelson s Problem

Transcription

1 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

2 Overview of today s talk Quantum Information Theory Operator Algebras Kirchberg s Conjecture Quantum Measurement Theory Tsirelson s Problem Semidefinite Programming Connes Embedding Conjecture & Positivstellensätze Noncommutative Real Algebraic Geometry Tsirelson s Problem (1993, 2006) Q c = Q s? Kirchberg s Conjecture (1993) C F d max C F d = 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

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

4 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 Xi ). P i = 1 Xi 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 (Ql 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

5 d $d $d $d $d z z z z 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 (Ql 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 i=1 ( ψ, Q l j ψ )m j=1 ψ Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 5 / 17

6 d $d $d $d $d z z z z 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 (Ql 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. α = {Apple, Grape} A = P A P G α = {Red, Green} A = P R P G 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 B +B + B B 2. β = {Hard, Soft} B = Q H Q S β = {Big, Small} B = Q B Q S Does Nature conform this inequality? Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 5 / 17

7 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 C = { Pi k Qj l dµ (P i k ) m i=1, k = 1,..., d, partitions of 1 X, }, (Qj l)m j=1, l = 1,..., d, partitions of 1 X [ Q s = cl{ ψ, (Pi k ] Qj l )ψ [ ] Q c = { ψ, Pi k Qj l ψ [ Θ = { γ ] i γ ψ H K a state (Pi k ) m i=1, k = 1,..., d, PVMs on H, }, (Qj l)m j=1, l = 1,..., d, PVMs on K H a Hilbert space, ψ H a state (Pi k ) m i=1, k = 1,..., d, PVMs on H, (Qj l }, )m j=1, l = 1,..., d, PVMs on H, [Pi k, Qj l] = 0 for all i, j and k, l 0, γ = 1 indep of k, j γ indep of l γ Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 6 / 17 }.

8 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 C = { Pi k Qj l dµ (P i k ) m i=1, k = 1,..., d, partitions of 1 X, }, (Qj l)m j=1, l = 1,..., d, partitions of 1 X [ Q s = cl{ ψ, (Pi k ] Qj l )ψ ψ H K a state (Pi k ) m i=1, k = 1,..., d, PVMs on H, }, (Qj l)m j=1, l = 1,..., d, PVMs on K H a Hilbert space, ψ H a state [ ] Q c = { ψ, Pi k CQ j l ψ Q (Pi k ) m i=1, k = 1,..., d, PVMs on H, s by CHSH (Q l Bell inequality (1969) }, j )m j=1, l = 1,..., d, PVMs on H, A 1 B 1 + A 1 B [P 2 + i k A, Q 2 j l B ] = 1 0 A for 2 B all 2 i, j 2 and k, l [ Θ = { γ ] for commuting variables γ 0, 1 A i, B j 1. γ = 1 i γ indep of k, }. j γ indep of l Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 6 / 17

9 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 C = { Q Pi k Qj l dµ (P k i ) m i=1, k = 1,..., d, partitions of 1 c Θ by Cirel son s Quantum X, (Qj l)m j=1, l = 1, Bell },..., inequality d, partitions (1980) of 1 X A 1 B 1 + A 1 B 2 + A 2 B 1 A 2 B [ for operators 1] A i, B j 1 ψwith H[A i, BK j ] a= state 0. Q s = cl{ ψ, (Pi k Qj l )ψ (Pi k ) m i=1, k = 1,..., d, PVMs on H, }, (Qj l)m j=1, l = 1,..., d, PVMs on K [ ] Q c = { ψ, Pi k Qj l ψ [ Θ = { γ ] i γ H a Hilbert space, ψ H a state (Pi k ) m i=1, k = 1,..., d, PVMs on H, (Qj l }, )m j=1, l = 1,..., d, PVMs on H, [Pi k, Qj l] = 0 for all i, j and k, l 0, γ = 1 indep of k, j γ indep of l γ Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 6 / 17 }.

10 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 (X, µ) s if we restrict the Hilbert a (finite) prob space spaces H C = { Pi k appearing Qj l in the dµ (P k i ) m definition i=1, k = 1, of.. Q., c d, to partitions fin-dim ones. of 1 X, }, (Qj l)m j=1, l = 1,..., d, partitions of 1 X [ Q s = cl{ ψ, (Pi k ] Qj l )ψ [ ] Q c = { ψ, Pi k Qj l ψ γ ψ H K a state (Pi k ) m i=1, k = 1,..., d, PVMs on H, }, (Qj l)m j=1, l = 1,..., d, PVMs on K H a Hilbert space, ψ H a state (Pi k ) m i=1, k = 1,..., d, PVMs on H, (Qj l }, )m j=1, l = 1,..., d, PVMs on H, [Pi k, Qj l] = 0 for all i, j and k, l [ ] γ 0, Θ = { γ = 1 Tsirelson s Problem i γ indep of k, Q c = Q }. j γ indep s? of l Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 6 / 17

11 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 ei k the standard basis of projections in the k-th copy of l m. Also ei k = ei k 1 and fj l = 1 ej l in C (Γ) C (Γ). Then, one has [ ] Q c = { φ(ei k f l φ a state on C (Γ) max C (Γ)} and j ) [ ] Q s = { φ(ei k fj l ) φ a state on C (Γ) min C (Γ)}. 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. Connes s Embedding Conjecture M R ω for for every II 1 factor M. Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 7 / 17

12 Some quotes [A. Connes; Classification of injective factors. Ann. of Math. 104 (1976)] We now construct an approximate imbedding of N in R. Apparently such an imbedding ought to exist for all II 1 factors because it does for the regular representation of free groups. [M. Navascués, T. Cooney, D. Pérez-García, N. Villanueva; A physical approach to Tsirelson s problem. Found. Phys. 42 (2012)] Tsirelson s problem has recently gained popularity in the Quantum Information community, not only due to the practical consequences of its resolution, but also for the aforementioned connections with sophisticated areas of mathematics. Looking at it from the outside, though, that Tsirelson s problem is related to a conjecture which hundreds of mathematicians have attempted to solve over the course of 35 years is not good news at all. It implies that a physicist aiming at solving it should study advanced mathematics for several years in order to find himself as lost as the very brilliant mathematical minds who are currently trying to solve Connes embedding conjecture. Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 8 / 17

13 Slightly interacting systems We consider the quantum correlation of slightly interacting systems. When Alice and Bob conduct m ment of a state ψ at the same time, the probability of the outcome (i, j) is given by ψ, (P i Q j )ψ, where P Q = (PQP + QPQ)/2. Thus we consider Q! [ Q ε = cl{ ψ, (Pi k ] Qj l )ψ dim H < +, ψ H a state (Pi k ) m i=1, k = 1,..., d, PVMs on H, (Qj l }. )m j=1, l = 1,..., d, PVMs on H, [Pi k, Qj l] ε for all i, j and k, l Surprisingly, it makes no difference if we allow the Hilbert spaces H to be infinite-dimensional, thanks to the fact C (Γ Γ) is quasi-diagonal. Theorem (Oz. 2012) ε>0 Q ε = Q c. Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 9 / 17

14 C -algebras C -algebras Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 10 / 17

15 Group C -algebras Recall Γ = Z d m, where m, d {2, 3,..., } s.t. m + d > 4, and C (Γ Γ) = C (Γ) max C (Γ) = C ((ei k)m i=1, (f j l)m j=1 [ ] ), Q c = { φ(e k i f l j ) φ a state on C (Γ Γ)}. The C -algebra C (Γ) is RFD (Residually Finite Dimensional), i.e. every unitary rep n is weakly contained in the closure of the finite-dim rep ns. (NB! Γ being residually finite doesn t mean C (Γ) being RFD.) Kirchberg s conjecture C (Γ Γ) is RFD. π In fact, for Γ Γ 0 Λ, the unitary rep n of Γ Γ on l2 (Γ π Γ) is weakly contained in the closure of the finite-dim rep ns iff the group vn algebra vn(λ) satisfies the Connes Embedding Conjecture vn(λ) R ω. Note If Λ is sofic, then vn(λ) Rω. Is the converse possibly true...??? Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 11 / 17

16 Quasi-diagonality Recall Γ = Z d m, where m, d {2, 3,..., } s.t. md > 4, and Kirchberg s conjecture C (Γ Γ) is RFD. Theorem (Brown Oz. 2008) The C -algebra C (Γ Γ) is QD (quasi-diagonal). A C -algebra A is QD if there are unital completely positive maps θ n A M k(n) (C) such that θ n (a)θ n (b) θ n (ab) 0 and θ n (a) a for all a, b A. Proof. The theorem follows from homotopy invariance of quasi-diagonality (Voiculescu 1991) and the fact that every unitary rep n of Γ Γ can be dilated to a unitary rep n which is homotopic to a finite-dim rep n. The proof does not provide finite-dim approximants explicitly... Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 12 / 17

17 Noncommutative real algebraic geometry Noncommutative real algebraic geometry Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 13 / 17

18 Positivstellensätze A linear functional φ C[Γ] C is called a state if φ(f f ) 0 and φ(1) = 1. It is tracial if it moreover satisfies τ(f g) = τ(g f ). Theorem (Hahn Banach + GNS) Let Γ be a discrete group and f C[Γ]. Then, f 0 in C (Γ), i.e. π(f ) 0 for every unitary rep n π. 2 f + ε1 { i g i g i g i C[Γ]} for every ε > 0. 3 φ(f ) 0 for every tracial state φ on C[Γ]. 4 f + ε1 { i g i g i g i C[Γ]} + commutators, for every ε > 0. When Γ = F d, C (Γ) is RFD and it s enough to consider fin-dim π s in 1. Theorem (Klep Schweighofer 2008) Tsirelson s Problem has an affirmative answer iff 3 for Γ = F d is equiv to 5 Tr(π(f )) 0 for every finite-dimensional unitary rep n π of F d. Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 14 / 17

19 Strict Positivstellensätze Theorem (Hahn Banach + GNS) Let Γ be a discrete group and f C[Γ]. Then, f 0 in C (Γ), i.e. π(f ) 0 for every unitary rep n π. 2 f + ε1 { i g i g i g i C[Γ]} for every ε > 0. Theorem (Riesz Fejér, Schmüdgen, Bakonyi Timotin 2007) Let f C[F d ] be s.t. supp f EE 1 for a conn. subset 1 E F d. Then, TFAE. f 0 in C (F d ), i.e. π(f ) 0 for every finite (dim) unitary rep n π. f { i g i gi g i C[Γ], supp g i E}. Deeper results from real algebraic geometry Scheiderer (2006) +ε1 isn t necessary for Γ = Z 2, but it s instable. Scheiderer (2009) +ε1 is necessary for Γ Z 3. How about F d F d? Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 15 / 17

20 Semidefinite programming Recall φ C[Γ] C corresponds to φ Γ C by φ(f ) = φ(x)f (x), and φ is a state on C[Γ] (or C (Γ)) φ is of positive type and φ(1) = 1. Here, a function φ Γ C is of positive type on E Γ if [φ(xy 1 )] x,y E is positive semidefinite. Hence, States on C[Γ] = {φ φ positive type on E}. [ ] Therefore, Q c = { φ(ei k fj l ) E finite φ a state on C[Z d m Z d m ]} is determined by infinitely many explicit inequalities, and instability of Γ Γ probably means infinitely many inequality are necessary, i.e. Q c is very likely not semi-algebraic, unless (m, d) (2, 2). Something similar for Q s, too. Also, infinitely many Bell type inequalities. Taka OZAWA (RIMS) QC & TP Sde Boker, 13/03/14 16 / 17

21 Thank you for your attention! Quantum Information Theory Operator Algebras Kirchberg s Conjecture Quantum Measurement Theory Semidefinite Programming Connes Embedding Conjecture & Positivstellensätze Tsirelson s Problem Tsirelson s Problem (1993, 2006) Q c = Q s? Noncommutative Real Algebraic Geometry Kirchberg s Conjecture (1993) C F d max C F d = 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 17 / 17