Polynomial Optimzation in Quantum Information Theory Sabine Burgdorf University of Konstanz ICERM - 2018 Real Algebraic Geometry and Optimization 1
Warm Up ◮ Entanglement is one of the key features in Quantum Information ◮ Bell ’64: Quantum Q Classical C ◮ How to distinguish C and Q ? ◮ What is the correct definition for Q ? Does it matter? ◮ Can Polynomial Optimization help to understand these sets? 2
RAG and POP basics Polynomial Optimization ◮ f ∈ R [ X ] polynomial in commuting variables ◮ g 0 = 1 , g 1 , . . . , g r ∈ R [ X ] defining a semi-algebraic set: K = { a ∈ R n | g 0 ( a ) ≥ 0 , . . . , g r ( a ) ≥ 0 } ◮ Want to minimize f over K f ∗ = inf f ( a ) s.t. a ∈ K = sup a ∈ R s.t. f − a ≥ 0 on K ◮ NP-hard 3
RAG and POP basics RAG helps f ∗ = sup a ∈ R s.t. f − a ≥ 0 on K NP-hard j h 2 ◮ M ( g ) := { p = � j g i j for some h i ∈ R [ X ] } ◮ sos relaxation f sos = sup a ∈ R s.t. f − a ∈ M ( g ) "SDP" 4
RAG and POP basics RAG helps f ∗ = sup a ∈ R s.t. f − a ≥ 0 on K NP-hard j h 2 ◮ M ( g ) := { p = � j g i j for some h i ∈ R [ X ] } ◮ sos relaxation f sos = sup a ∈ R s.t. f − a ∈ M ( g ) "SDP" ◮ f sos is always a lower bound but might be strict ◮ If M ( g ) is archimedean: f ∗ = f sos x 4 1 x 2 2 + x 2 1 x 4 2 − 3 x 2 1 x 2 2 + 1 4
RAG and POP basics SOS hierarchy j h 2 ◮ M ( g ) t := { p = � j g i j for some h i ∈ R [ X ] t } ◮ sos hierarchy f t = sup a ∈ R s.t. f − a ∈ M ( g ) t SDP ◮ We have ◮ f t ≤ f t + 1 ≤ f ∗ ◮ f t converges to f sos as t → ∞ ◮ If M ( g ) is archimedean: f sos = f ∗ 5
RAG and POP basics SOS hierarchy j h 2 ◮ M ( g ) t := { p = � j g i j for some h i ∈ R [ X ] t } ◮ sos hierarchy f t = sup a ∈ R s.t. f − a ∈ M ( g ) t SDP ◮ We have ◮ f t ≤ f t + 1 ≤ f ∗ ◮ f t converges to f sos as t → ∞ ◮ If M ( g ) is archimedean: f sos = f ∗ ◮ Certificate of exactness: ◮ Flatness of dual solution ◮ Allows extraction of optimizers 5
NC-RAG and NC-POP NC Polynomials ◮ Want to replace scalar variables by matrices/operators ◮ Free algebra R � X � with noncommuting variables X 1 , . . . , X n ◮ Polynomial � f = f w w w ◮ Let A ∈ ( S d ) n : f ( A ) = f 1 I d + f X 1 A 1 + f X 2 X 1 A 2 A 1 . . . 6
NC-RAG and NC-POP NC Polynomials ◮ Want to replace scalar variables by matrices/operators ◮ Free algebra R � X � with noncommuting variables X 1 , . . . , X n ◮ Polynomial � f = f w w w ◮ Let A ∈ ( S d ) n : f ( A ) = f 1 I d + f X 1 A 1 + f X 2 X 1 A 2 A 1 . . . ◮ Add involution ∗ on R � X � ◮ fixes R and { X 1 , . . . , X n } pointwise ◮ X ∗ i = X i ◮ Consequence f ∗ f ( A ) = f ( A ) T f ( A ) � 0 6
NC-RAG and NC-POP NC Polynomial Optimization ◮ Let f ∈ R � X � ◮ g 0 = 1 , g 1 , . . . , g r ∈ R � X � defining a semi-algebraic set: K = { A | g 0 ( A ) � 0 , . . . , g r ( A ) � 0 } ◮ Want to minimize f over K f ∗ = sup a ∈ R s.t. f − a ≥ 0 on K 7
NC-RAG and NC-POP Eigenvalue optimization ◮ Let f ∈ R � X � f nc = sup a ∈ R s.t. f − a � 0 on K NP-hard ◮ Observation: Checking if f = � i h ∗ i h i is an SDP j h ∗ so as well checking f = � j g i j h j (with degree bounds) 8
NC-RAG and NC-POP Eigenvalue optimization ◮ Let f ∈ R � X � f nc = sup a ∈ R s.t. f − a � 0 on K NP-hard ◮ Observation: Checking if f = � i h ∗ i h i is an SDP j h ∗ so as well checking f = � j g i j h j (with degree bounds) ◮ sos relaxation j h ∗ M nc ( g ) := { p = � j g i j h j for some h i ∈ R � X �} f sos = sup a ∈ R s.t. f − a ∈ M nc ( g ) ◮ Fact: f sos ≤ f nc ◮ Theorem (Helton et al.): If M nc ( g ) is archimedean, then f sos = f nc . 8
NC-RAG and NC-POP Eigenvalue optimization ◮ Let f ∈ R � X � f nc = sup a ∈ R s.t. f − a � 0 on K NP-hard ◮ M nc ( g ) t := { p = � j h ∗ j g i j h j for some h j ∈ R � X � t } ◮ sos hierarchy f t = sup a ∈ R s.t. f − a ∈ M nc ( g ) t SDP ◮ f t ≤ f t + 1 ≤ f nc but inequalities might be strict ◮ f t converges to f sos as t → ∞ ◮ If M nc ( g ) is archimedean: f sos = f nc and hence f t → f nc as t → ∞ 9
NC-RAG and NC-POP Trace optimization ◮ Let f ∈ R � X � f tr = sup a ∈ R s.t. Tr ( f − a ) ≥ 0 on K NP-hard ◮ K contains only operators, for which a trace is defined 10
NC-RAG and NC-POP Trace optimization ◮ Let f ∈ R � X � f tr = sup a ∈ R s.t. Tr ( f − a ) ≥ 0 on K NP-hard ◮ K contains only operators, for which a trace is defined ◮ If f = � j h ∗ j g i j h j + � k [ p k , q k ] then Tr ( f ( A )) ≥ 0 for all A ∈ K ◮ sos relaxation j h ∗ M tr ( g ) := { � j g i j h j for some h i ∈ R � X �} + [ R � X � , R � X � ] f sos = sup a ∈ R s.t. f − a ∈ M tr ( g ) ◮ Fact: f sos ≤ f tr ◮ Theorem (B.,Klep et al.): If M tr ( g ) is archimedean, then f sos = f tr . 10
NC-RAG and NC-POP Trace optimization ◮ Let f ∈ R � X � f tr = sup a ∈ R s.t. Tr ( f − a ) ≥ 0 on K NP-hard ◮ M tr ( g ) t := { � j h ∗ j g i j h j for some h j ∈ R � X � t } + � [ R � X � , R � X � ] ◮ sos hierarchy f t = sup a ∈ R s.t. f − a ∈ M tr ( g ) t SDP ◮ f t ≤ f t + 1 ≤ f tr but inequalities might be strict ◮ f t converges to f sos as t → ∞ ◮ If M tr ( g ) is archimedean: f sos = f tr and hence f t → f tr as t → ∞ 11
Back to Quantum Information ◮ Entanglement is one of the key features in Quantum Information ◮ Bell ’64: Quantum Q Classical C ◮ How to distinguish C and Q ? ◮ What is the correct definition for Q ? Does it matter? ◮ Can Polynomial Optimization help to understand these sets? 12
Basics of quantum theory ◮ A quantum system corresponds to a Hilbert space H ◮ Its states are unit vectors on H 13
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 13
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. 13
Nonlocal bipartite correlations ◮ Question sets S , T , Answer sets A , B ◮ No (classical) communication s t b a ◮ Which correlations p ( a , b | s , t ) are possible? 14
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 15
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 ◮ Nonlocality: ( E a s ⊗ 1 )( 1 ⊗ F b t ) = ( 1 ⊗ F b t )( E a s ⊗ 1 ) ◮ If ψ = ψ A ⊗ ψ B then we have classical correlation 15
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 16
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 16
Tsirelson’s problem Fact C ⊆ Q ⊆ Q ⊆ Q c ◮ Bell: C � = Q ◮ closure conjecture [Slofstra ’16]: Q � = Q ◮ weak Tsirelson [Slofstra ’16]: Q � = Q c ◮ Dykema et al. ’17: Concrete example in a decent subset of Q ◮ strong Tsirelson (open): Is Q = Q c ? ◮ strong Tsirelson is equivalent to Connes embedding problem 17
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 } 18
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) � � ω = sup π ( s , t ) V ( a , b ; s , t ) p ( a , b | s , t ) p s ∈ S , t ∈ T a ∈ A , b ∈ B � = sup f abst p ( a , b | s , t ) p a , b , s , t ◮ optimize over correlations p ∈ {C , Q , Q c } 18
SOS relaxation over C � f abst p a s · p b ω C = sup t p a , b , s , t 19
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