Moments in Quantum Information Theory Sabine Burgdorf University of - PowerPoint PPT Presentation

Moments in Quantum Information Theory Sabine Burgdorf University of Konstanz EWM GM 2018 - Graz Real Algebraic Geometry in Action 1

What is this talk about? Moment problem in Action: Quantum Information ◮ Entanglement: key feature of Quantum Mechanics ◮ Nonlocal games ◮ Quantum correlations ◮ Relation to the moment problem 2

Entanglement ◮ Entanglement is one of the most striking features of QM ◮ 2 particles – split up and send to Alice & Bob ◮ 2 possible features – randomly distributed ◮ 2 ways to learn the feature (measurements) ◮ Alice checks by method 1 ◮ Bob checks by method 2: anything can happen ◮ Bob checks by same method: ALWAYS the opposite answer Alice Bob 1 2 1 2 3

Basics of quantum theory ◮ A quantum system corresponds to a Hilbert space H ◮ Its states are unit vectors on H ◮ A state on a composite system is a unit vector ψ on a tensor Hilbert space, e.g. H A ⊗ H B ◮ ψ is entangled if it is not a product state ψ A ⊗ ψ B with ψ A ∈ H A , ψ B ∈ H B ◮ A state ψ ∈ H can be measured ◮ outcomes a ∈ A ◮ POVM: a family { E a } a ∈ A ⊆ B ( H ) with E a � 0 and � a ∈ A E a = 1 ◮ probablity of getting outcome a is p ( a ) = ψ T E a ψ. ◮ Entanglement can be studied via nonlocal games 4

One nonlocal game ◮ Two players: Alice and Bob ◮ During the game they are not allowed to communicate ◮ Alice gets 1 picture or ◮ Bob gets 1 picture or ◮ They both answer 0 or 1 ◮ Winning: – If both get Graz, their answers must agree – Otherwise their answers must differ ◮ Classical strategy: winning probability 0.75 ◮ Quantum strategy: winning probability cos ( π/ 8 ) 2 ≈ 0 . 85 5

Nonlocal games ◮ Characterized by ◮ 2 sets of questions S , T , asked with probability distribution π ◮ 2 sets of answers A , B ◮ A winning predicate V : A × B × S × T → { 0 , 1 } ◮ Winning probability (value of the game) � � V ( a , b ; s , t ) p ( a , b | s , t ) ω = sup π ( s , t ) p s ∈ S , t ∈ T a ∈ A , b ∈ B ◮ optimize over a set of correlations p = ( p ( a , b | s , t )) a , b , s , t ◮ ω depends on the chosen set of allowed correlations 6

Correlations Classical strategy C Independent probability distributions { p a s } a and { p b t } b : p ( a , b | s , t ) = p a s · p b t shared randomness: allow convex combinations Quantum strategy Q POVMs { E a s } a and { F b t } b on Hilbert spaces H A , H B , ψ ∈ H A ⊗ H B : p ( a , b | s , t ) = ψ T ( E a s ⊗ F b t ) ψ Theorem [Bell ’64] There exist games such that ω C < ω Q . 7

More correlations Quantum strategy Q POVMs { E a s } a and { F b t } b on Hilbert spaces H A , H B , ψ ∈ H A ⊗ H B : p ( a , b | s , t ) = ψ T ( E a s ⊗ F b t ) ψ Quantum strategy Q c POVMs { E a s } a and { F b t } b on a joint Hilbert space, but [ E a x , F b y ] = 0: p ( a , b | s , t ) = ψ T ( E a s · F b t ) ψ Fact C ⊆ Q ⊆ Q ⊆ Q c 8

Tsirelson’s problem Fact C ⊆ Q ⊆ Q ⊆ Q c ◮ Bell: C � = Q ◮ weak Tsirelson [Slofstra ’16]: Q � = Q c ◮ strong Tsirelson (open): Is Q = Q c ? ◮ strong Tsirelson is equivalent to Connes embedding problem ◮ Goal: Understand correlations via their values of a game ◮ Usually hard to compute... ◮ Brute force: lower bounds for ω C or ω Q ◮ What about upper bounds? 9

NC moment problems 1 Classical moment problem Let L : R [ x ] → R be linear, L ( 1 ) = 1. Does there exist a probability measure µ (with supp µ ⊆ K ) such that for all f ∈ R [ x ] : � L ( f ) = f ( a ) d µ ( a )? (psd) NC moment problem Let L : R � X � → R be linear, L ( 1 ) = 1. Does there exist a Hilbert space H , a unit vector ψ ∈ H and a ∗ -representation π on B ( H ) such that for all f ∈ R � X � : L ( f ) = � ψπ ( f ) , ψ � ? 1 B., Klep, Povh: Optimization of polynomials in non-commuting variables 10

NC moment problems Classical moment problem Let L : R [ x ] → R be linear, L ( 1 ) = 1. Does there exist a probability measure µ (with supp µ ⊆ K ) such that for all f ∈ R [ x ] : � L ( f ) = f ( a ) d µ ( a )? tracial moment problem Let L : R � X � → R be linear, L ( 1 ) = 1, L ([ p , q ]) = 0 for all p , q ∈ R � X � . Does there exist a finite von Neumann algebra N with trace τ and a ∗ -representation π on N such that for all f ∈ R � X � : L ( f ) = τ ( π ( f ))? ◮ Von Neumann algebra = ∞ -dim. analog of a matrix algebra ◮ The measure µ is hidden in the von Neumann algebra via direct integral decomposition 11

Moment relaxation of ω C ◮ Reminder ω = sup p � s ∈ S , t ∈ T π ( s , t ) � a ∈ A , b ∈ B V ( a , b ; s , t ) p ( a , b | s , t ) ◮ Let � � V ( a , b ; s , t ) E a s F b f ( E , F ) = π ( s , t ) t s ∈ S , t ∈ T a ∈ A , b ∈ B Then p a s , q b a p a b p b t ≥ 0 , � s = � t = 1 ω C = sup f ( p , q ): = inf λ : f − λ ≥ 0 on K K ◮ Moment relaxation [Lasserre] ω s = sup L ( f ): L ∈ R [ x ] ∨ 2 s , M K ( L ) � 0 , L ( 1 ) = 1 . ◮ We have 2 ω s ≥ ω s + 1 ≥ ω C and lim s →∞ ω s → ω C ◮ If best L for ω s has a moment representation then ω C = ω s 2 essentially [Putinar ’93] 12

Moment relaxation of ω Q c ◮ Reminder ω = sup p � s ∈ S , t ∈ T π ( s , t ) � a ∈ A , b ∈ B V ( a , b ; s , t ) p ( a , b | s , t ) ◮ Let � � V ( a , b ; s , t ) E a s F b f ( E , F ) = π ( s , t ) t s ∈ S , t ∈ T a ∈ A , b ∈ B Then E s , F t POVM , [ E a s , F b t ] = 0 ω Q c = sup ψ T f ( E , F ): = inf λ : f − λ � 0 on K K ◮ Moment relaxation [Pironio, Navascues, Acin] ω Q c , s = sup L ( f ): L ∈ R � X � ∨ 2 s , M K ( L ) � 0 , L ( 1 ) = 1 . ◮ We have 3 ω Q c , s ≥ ω Q c , s + 1 ≥ ω Q c and lim s →∞ ω Q c , s → ω Q c ◮ If best L for ω Q c , s has an NC moment representation then ω Q c = ω Q c , s 3 essentially [Helton ’2000] 13

Moment relaxation of ω Q ◮ For most games we can write p ∈ Q as 4 p ( a , b | s , t ) = Tr (˜ s ˜ E a F b t ) with ˜ s , ˜ a ˜ b ˜ E a F b E a F b t = D with Tr ( D 2 ) = 1 t � 0 , � s = � K ◮ Thus ω Q = sup Tr ( f ( E , F )): ( E , F ) ∈ K = inf λ : Tr ( f − λ ) ≥ 0 on K ◮ Moment relaxation ω Q , s = sup L ( f ): L ∈ R � X � ∨ 2 s , tracial , M K ( L ) � 0 , L ( 1 ) = 1 . ◮ We have ω Q , s ≥ ω Q , s + 1 ≥ ω Q and lim s →∞ ω Q , s → ω Q ◮ If best L for ω Q , s has a tracial moment representation then ω Q = ω Q , s 4 Berta, Fawzi; Sikora,Varvitsiotis; Manˇ cinska,Roberson;... 14

It’s just the beginning... Numerical experiments 5 ◮ improved bounds for quantum graph parameters on specific graphs ◮ disproved a conjecture on quantum graph parameters by additional use of Gröber bases ◮ lower bounds on the needed amount of entanglement for specific games Other relaxations ◮ combinatorial relaxation of the tracial polynomial optimization problem not using moments ◮ better relaxations by adding additional equalities/inequalities ◮ Feasibility criteria to show existence/non-existence of several types of solutions (e.g. projections) ◮ ... 5 with de Laat, Gribling, Laurent, Piovesan, Manˇ cinska, Roberson 15

Final Remarks Comments/Questions ◮ Non-commutative moment problems in combination with polynomial optimization give upper bounds for (quantum) values of nonlocal games ◮ If the optimizer corresponds to a flat matrix, we can even extract (numerically) the best strategy ◮ But flat solution is always finite dimensional: How can we verify exactness without flatness? ◮ Is there a way to compare ω Q c , s with ω Q , s ? ◮ Is there a nonlocal game which does not have a finite dimensional optimizer? Big open question Is there a nonlocal game with ω Q < ω Q c ? Thank you for your attention. 16