Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Purity in Euclidean Jordan algebras Joint work with Bram and Bas Westerbaan { bram,bas } @westerbaan.name John van de Wetering wetering@cs.ru.nl Institute for Computing and Information Sciences Radboud University Nijmegen QPL2018 4th of June 2018 (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 1 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Overview • Pure maps • Effectus theory, Corners and Filters • Purity and Euclidean Jordan algebras (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 2 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Pure maps in quantum theory • What are the pure states in quantum theory? Everyone agrees. (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 3 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Pure maps in quantum theory • What are the pure states in quantum theory? Everyone agrees. • What are the pure maps? There is disagreement. (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 3 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Pure maps in quantum theory • What are the pure states in quantum theory? Everyone agrees. • What are the pure maps? There is disagreement. For f : M n ( C ) → M n ( C ) it is clear: f pure when ∃ V ∈ M n ( C ): f ( A ) = VAV ∗ ∀ A ∈ M n ( C ) These maps are called Kraus rank 1 . (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 3 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Pure maps in quantum theory • What are the pure states in quantum theory? Everyone agrees. • What are the pure maps? There is disagreement. For f : M n ( C ) → M n ( C ) it is clear: f pure when ∃ V ∈ M n ( C ): f ( A ) = VAV ∗ ∀ A ∈ M n ( C ) These maps are called Kraus rank 1 . But what about f : A → B where A and B are C ∗ -algebras? (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 3 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Pure maps in quantum theory • What are the pure states in quantum theory? Everyone agrees. • What are the pure maps? There is disagreement. For f : M n ( C ) → M n ( C ) it is clear: f pure when ∃ V ∈ M n ( C ): f ( A ) = VAV ∗ ∀ A ∈ M n ( C ) These maps are called Kraus rank 1 . But what about f : A → B where A and B are C ∗ -algebras? = ⇒ Different definitions give different results (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 3 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Which maps should be pure? (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 4 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Which maps should be pure? • Isomorphisms. (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 4 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Which maps should be pure? • Isomorphisms. • f ◦ g when f and g are pure. (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 4 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Which maps should be pure? • Isomorphisms. • f ◦ g when f and g are pure. • f † when f is pure. (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 4 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Which maps should be pure? • Isomorphisms. • f ◦ g when f and g are pure. • f † when f is pure. • Kraus rank 1 maps A �→ VAV ∗ . (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 4 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Which maps should be pure? • Isomorphisms. • f ◦ g when f and g are pure. • f † when f is pure. • Kraus rank 1 maps A �→ VAV ∗ . • ’Corner maps’: � A � B �→ A C D � A � 0 A �→ 0 0 (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 4 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Which maps should be pure? • Isomorphisms. • f ◦ g when f and g are pure. • f † when f is pure. • Kraus rank 1 maps A �→ VAV ∗ . • ’Corner maps’: � A � B �→ A C D � A � 0 A �→ 0 0 Q: How do we generalise this to more general theories? (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 4 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Effectus Theory Enter effectus theory: • K. Cho, B. Jacobs, B. Westerbaan & A. Westerbaan (2015): Introduction to effectus theory . • B. Westerbaan (2018): Dagger and dilations in von Neumann algebras . (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 5 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Effectus Theory Enter effectus theory: • K. Cho, B. Jacobs, B. Westerbaan & A. Westerbaan (2015): Introduction to effectus theory . • B. Westerbaan (2018): Dagger and dilations in von Neumann algebras . An effectus ≈ ’generalised generalised probabilistic theory’ real numbers ⇒ effect monoids vector spaces ⇒ effect algebras. (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 5 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Effectus Theory Enter effectus theory: • K. Cho, B. Jacobs, B. Westerbaan & A. Westerbaan (2015): Introduction to effectus theory . • B. Westerbaan (2018): Dagger and dilations in von Neumann algebras . An effectus ≈ ’generalised generalised probabilistic theory’ real numbers ⇒ effect monoids vector spaces ⇒ effect algebras. Important notions in effectus theory: quotient and comprehension . (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 5 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Quotient and Comprehension 1 On ordered vector spaces: • Quotient �→ Filter • Comprehension �→ Corner (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 6 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Quotient and Comprehension 1 On ordered vector spaces: • Quotient �→ Filter • Comprehension �→ Corner An effect q is an element in an ordered vector space with 0 ≤ q ≤ 1. (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 6 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Quotient and Comprehension 1 On ordered vector spaces: • Quotient �→ Filter • Comprehension �→ Corner An effect q is an element in an ordered vector space with 0 ≤ q ≤ 1. A filter or corner for an effect is a map with a certain universal property. (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 6 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Quotient and Comprehension 1 On ordered vector spaces: • Quotient �→ Filter • Comprehension �→ Corner An effect q is an element in an ordered vector space with 0 ≤ q ≤ 1. A filter or corner for an effect is a map with a certain universal property. Example: Let q = � i λ i q i . Define ⌈ q ⌉ = � i q i . ⌊ q ⌋ = � i ; λ i =1 q i . (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 6 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Quotient and Comprehension 1 On ordered vector spaces: • Quotient �→ Filter • Comprehension �→ Corner An effect q is an element in an ordered vector space with 0 ≤ q ≤ 1. A filter or corner for an effect is a map with a certain universal property. Example: Let q = � i λ i q i . Define ⌈ q ⌉ = � i q i . ⌊ q ⌋ = � i ; λ i =1 q i . ξ : ⌈ q ⌉A⌈ q ⌉ → A by ξ ( p ) = √ qp √ q is a filter The projection π : A → ⌊ q ⌋A⌊ q ⌋ is a corner. (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 6 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Quotient and Comprehension 2 A filter for an effect q is a positive map ξ : V q → V with ξ (1) ≤ q , (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 7 / 12
Pure maps Corners and Filters Radboud University Nijmegen Purity and EJAs Conclusion and Discussion Quotient and Comprehension 2 A filter for an effect q is a positive map ξ : V q → V with ξ (1) ≤ q , such that for all f : W → V with f (1) ≤ q : ξ V q V f f W (Westerbaan) 2 & van de Wetering QPL2018 Purity in EJAs 7 / 12
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