purity in euclidean jordan algebras
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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


  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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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