Domains of commutative C*-subalgebras Chris Heunen 1 / 26
Domains of commutative C*-subalgebras Chris Heunen and Bert Lindenhovius Logic in Computer Science 2015 1 / 26
Measurement unit vector x in C n State: Measurement: in basis e 1 , . . . , e n gives outcome i with probability � e i | x � 2 / 26
Measurement unit vector x in C n State: Measurement: hermitian matrix e in M n with eigenvectors e i given by | i � �→ | e i �� e i | gives outcome i with probability � e i | x � 2 / 26
Measurement unit vector x in C n State: Measurement: hermitian matrix e in M n given by | i � �→ | e i �� e i | gives outcome i with probability tr( | e i �� e i | x ) 2 / 26
Measurement unit vector x in C n State: function e : C n → M n such that Measurement: • e linear • e (1 , . . . , 1) = 1 • e ( x 1 y 1 , . . . , x n y n ) = e ( x ) e ( y ) • e ( x 1 , . . . , x n ) = e ( x ) ∗ gives outcome i with probability tr( e | i � x ) 2 / 26
Measurement unit vector x in C n State: unital ∗ -homomorphism e : C n → M n Measurement: gives outcome i with probability tr( e | i � x ) 2 / 26
Measurement unit vector x in C n State: unital ∗ -homomorphism e : C m → M n Measurement: gives outcome i with probability tr( e | i � x ) 2 / 26
Measurement State: unit vector x in Hilbert space H unital ∗ -homomorphism e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � x ) 2 / 26
Measurement State: unit vector x in Hilbert space H unital ∗ -homomorphism e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � x ) “projection-valued measure” (PVM) “sharp measurement” 2 / 26
Compatible measurements PVMs e, f : C m → B ( H ) are jointly measurable when each e | i � and f | j � commute. 3 / 26
Compatible measurements PVMs e, f : C m → B ( H ) are jointly measurable when each e | i � and f | j � commute. (In)compatibilities form graph: r s p q t 3 / 26
Compatible measurements PVMs e, f : C m → B ( H ) are jointly measurable when each e | i � and f | j � commute. (In)compatibilities form graph: r s p q t Theorem : Any graph can be realised as PVMs on a Hilbert space. “Quantum theory realises all joint measurability graphs” Physical Review A 89(3):032121, 2014 3 / 26
Probabilistic measurement State: unit vector x in Hilbert space H function e : C m → B ( H ) such that Measurement: • e linear • e (1 , . . . , 1) = 1 • e ( x ) ≥ 0 if all x i ≥ 0 gives outcome i with probability tr( e | i � x ) 4 / 26
Probabilistic measurement State: unit vector x in Hilbert space H function e : C m → B ( H ) such that Measurement: • e linear • e (1 , . . . , 1) = 1 • e ( x ∗ 1 x 1 , . . . , x ∗ n x n ) = a ∗ a for some a in B ( H ) gives outcome i with probability tr( e | i � x ) 4 / 26
Probabilistic measurement State: unit vector x in Hilbert space H unital (completely) positive linear e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � x ) 4 / 26
Probabilistic measurement State: unit vector x in Hilbert space H unital (completely) positive linear e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � x ) “positive-operator valued measure” (POVM) “unsharp measurement” 4 / 26
Compatible probabilistic measurements POVMs e, f : C m → B ( H ) are jointly measurable when there exists POVM g : C m 2 → B ( H ) such that e | i � = � j g | ij � and f | j � = � i g | ij � ( e, f are marginals of g ) 5 / 26
Compatible probabilistic measurements POVMs e, f : C m → B ( H ) are jointly measurable when there exists POVM g : C m 2 → B ( H ) such that e | i � = � j g | ij � and f | j � = � i g | ij � ( e, f are marginals of g ) (In)compatibilities form hypergraph: r r s s p p q q t t 5 / 26
Compatible probabilistic measurements POVMs e, f : C m → B ( H ) are jointly measurable when there exists POVM g : C m 2 → B ( H ) such that e | i � = � j g | ij � and f | j � = � i g | ij � ( e, f are marginals of g ) (In)compatibilities form abstract simplicial complex: r r s s p p q q t t Theorem : Any abstract simplicial complex can be realised as POVMs on a Hilbert space. “All joint measurability structures are quantum realizable” Physical Review A 89(5):052126, 2014 5 / 26
States State: unit vector x in Hilbert space H unital (completely) positive linear e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � x ) 6 / 26
States State: ensemble of unit vectors x in Hilbert space H unital (completely) positive linear e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � x ) 6 / 26
States State: ensemble of projections | x �� x | onto vectors in Hilbert space H unital (completely) positive linear e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � | x �� x | ) 6 / 26
States State: ensemble of rank one projections p 2 = p = p ∗ in B ( H ) unital (completely) positive linear e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � | x �� x | ) 6 / 26
States State: positive operator ρ in B ( H ) of norm 1 unital (completely) positive linear e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � ρ ) 6 / 26
States State: linear function ρ : B ( H ) → C such that ρ ( a ) ≥ 0 if a ≥ 0, and ρ (1) = 1 unital (completely) positive linear e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � ρ ) 6 / 26
States State: unital (completely) positive linear ρ : B ( H ) → C “density matrix” unital (completely) positive linear e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � ρ ) 6 / 26
States State: unital (completely) positive linear ρ : B ( H ) → C “density matrix” unital (completely) positive linear e : C m → B ( H ) Measurement: gives outcome i with probability tr( e | i � ρ ) So really only the set B ( H ) matters. It is a C*-algebra. 6 / 26
States State: unital (completely) positive linear ρ : A → C “density matrix” unital (completely) positive linear e : C m → A Measurement: gives outcome i with probability tr( e | i � ρ ) So really only the set B ( H ) matters. It is a C*-algebra. The above works for any C*-algebra A : can formulate measurements, and derive states in terms of A alone 6 / 26
States State: unital (completely) positive linear ρ : A → C “density matrix” unital (completely) positive linear e : C m → A Measurement: gives outcome i with probability tr( e | i � ρ ) So really only the set B ( H ) matters. It is a noncommutative C*-algebra. The above works for any C*-algebra A : can formulate measurements, and derive states in terms of A alone 6 / 26
Continuous measurement State: unital (completely) positive linear ρ : A → C Measurement: with m discrete outcomes unital (completely) positive linear e : C m → A 7 / 26
Continuous measurement State: unital (completely) positive linear ρ : A → C Measurement: with outcomes in compact Hausdorff space X unital (completely) positive linear e : C ( X ) → A Here, C ( X ) = { f : X → C continuous } is a commutative C*-algebra. 7 / 26
Continuous measurement State: unital (completely) positive linear ρ : A → C Measurement: with outcomes in compact Hausdorff space X unital (completely) positive linear e : C ( X ) → A Here, C ( X ) = { f : X → C continuous } is a commutative C*-algebra. Theorem : Every commutative C*-algebra is of the form C ( X ). “On normed rings” Doklady Akademii Nauk SSSR 23:430–432, 1939 7 / 26
Classical data Unsharp measurement: unital positive linear e : C ( X ) → A Sharp measurement: unital ∗ -homomorphism e : C ( X ) → A Measurement: only way to get (classical) data from quantum system 8 / 26
Classical data Unsharp measurement: unital positive linear e : C ( X ) → A Sharp measurement: unital ∗ -homomorphism e : C ( X ) → A Measurement: only way to get (classical) data from quantum system Theorem : ‘unsharp measurements can be dilated to sharp ones’: any POVM e : C ( X ) → B ( H ) allows a PVM f : C ( X ) → B ( K ) and isometry v : H → K such that e ( − ) = v ∗ ◦ f ( − ) ◦ v . Sharp measurements give all (accessible) data about quantum system “Positive functions on C*-algebras” Proceedings of the American Mathematical Society, 6(2):211–216, 1955 8 / 26
Classical data Unsharp measurement: unital positive linear e : C ( X ) → A Sharp measurement: unital ∗ -homomorphism e : C ( X ) → A Measurement: only way to get (classical) data from quantum system Theorem : ‘unsharp measurements can be dilated to sharp ones’: any POVM e : C ( X ) → B ( H ) allows a PVM f : C ( X ) → B ( K ) and isometry v : H → K such that e ( − ) = v ∗ ◦ f ( − ) ◦ v . Sharp measurements give all (accessible) data about quantum system Lemma : the image of a unital ∗ -homomorphism e : C ( X ) → A is a (unital) commutative C*-subalgebra of A . Commutative C*-subalgebras record all data of quantum system “Positive functions on C*-algebras” Proceedings of the American Mathematical Society, 6(2):211–216, 1955 8 / 26
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