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Purity in Euclidean Jordan algebras Joint work with Bram and Bas - - PowerPoint PPT Presentation

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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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Overview

  • Pure maps
  • Effectus theory, Corners and Filters
  • Purity and Euclidean Jordan algebras

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 2 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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 : Mn(C) → Mn(C) it is clear: f pure when ∃V ∈ Mn(C): f (A) = VAV ∗ ∀A ∈ Mn(C) These maps are called Kraus rank 1.

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 3 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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 : Mn(C) → Mn(C) it is clear: f pure when ∃V ∈ Mn(C): f (A) = VAV ∗ ∀A ∈ Mn(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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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 : Mn(C) → Mn(C) it is clear: f pure when ∃V ∈ Mn(C): f (A) = VAV ∗ ∀A ∈ Mn(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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

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Which maps should be pure?

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 4 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

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Which maps should be pure?

  • Isomorphisms.

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 4 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

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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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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 C D

  • →

A A → A

  • (Westerbaan)2 & van de Wetering

QPL2018 Purity in EJAs 4 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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 C D

  • →

A A → A

  • Q: How do we generalise this to more general theories?

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 4 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Quotient and Comprehension 1

On ordered vector spaces:

  • Quotient → Filter
  • Comprehension → Corner

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 6 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

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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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

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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

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

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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 λiqi.

Define ⌈q⌉ =

i qi. ⌊q⌋ = i;λi=1 qi.

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 6 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

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 λiqi.

Define ⌈q⌉ =

i qi. ⌊q⌋ = i;λi=1 qi.

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

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Quotient and Comprehension 2

A filter for an effect q is a positive map ξ : Vq → V with ξ(1) ≤ q,

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 7 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Quotient and Comprehension 2

A filter for an effect q is a positive map ξ : Vq → V with ξ(1) ≤ q, such that for all f : W → V with f (1) ≤ q: Vq V W

ξ f f

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 7 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

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Quotient and Comprehension 2

A filter for an effect q is a positive map ξ : Vq → V with ξ(1) ≤ q, such that for all f : W → V with f (1) ≤ q: Vq V W

ξ f f

A corner for an effect q is a positive map π : V → {V |q} with π(1) = π(q),

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 7 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Quotient and Comprehension 2

A filter for an effect q is a positive map ξ : Vq → V with ξ(1) ≤ q, such that for all f : W → V with f (1) ≤ q: Vq V W

ξ f f

A corner for an effect q is a positive map π : V → {V |q} with π(1) = π(q), such that for all f : V → W with f (1) = f (q): {V |q} V W

f π f

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 7 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

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Purity

Definition

A map f is pure when it is a composition of a filter and a corner: f = ξ ◦ π

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 8 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

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Purity

Definition

A map f is pure when it is a composition of a filter and a corner: f = ξ ◦ π

  • Why this definition?

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 8 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Purity

Definition

A map f is pure when it is a composition of a filter and a corner: f = ξ ◦ π

  • Why this definition?

⇒ Because it is definable in a very general setting. ⇒ Because the right maps turn out to be pure. ⇒ Because it gives rise to interesting structure.

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 8 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Purity

Definition

A map f is pure when it is a composition of a filter and a corner: f = ξ ◦ π

  • Why this definition?

⇒ Because it is definable in a very general setting. ⇒ Because the right maps turn out to be pure. ⇒ Because it gives rise to interesting structure.

  • Is this even closed under composition?

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 8 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Purity

Definition

A map f is pure when it is a composition of a filter and a corner: f = ξ ◦ π

  • Why this definition?

⇒ Because it is definable in a very general setting. ⇒ Because the right maps turn out to be pure. ⇒ Because it gives rise to interesting structure.

  • Is this even closed under composition?

⇒ In the right categories, yes.

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 8 / 12

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Euclidean Jordan algebras

Definition

A Euclidean Jordan algebra (EJA) (E, ·, ·, ∗, 1) is a real Hilbert space with a product that satisfies: a∗1 = a a∗b = b∗a a∗(b∗a2) = (a∗b)∗a2 a∗b, c = b, a∗c

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 9 / 12

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Euclidean Jordan algebras

Definition

A Euclidean Jordan algebra (EJA) (E, ·, ·, ∗, 1) is a real Hilbert space with a product that satisfies: a∗1 = a a∗b = b∗a a∗(b∗a2) = (a∗b)∗a2 a∗b, c = b, a∗c Example: Mn(F)sa — self-adjoint matrices over F = R, C, H with A ∗ B := 1

2(AB + BA) and A, B := tr(AB).

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 9 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

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Euclidean Jordan algebras

Definition

A Euclidean Jordan algebra (EJA) (E, ·, ·, ∗, 1) is a real Hilbert space with a product that satisfies: a∗1 = a a∗b = b∗a a∗(b∗a2) = (a∗b)∗a2 a∗b, c = b, a∗c Example: Mn(F)sa — self-adjoint matrices over F = R, C, H with A ∗ B := 1

2(AB + BA) and A, B := tr(AB).

Q: Why care about EJAs?

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 9 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Euclidean Jordan algebras

Definition

A Euclidean Jordan algebra (EJA) (E, ·, ·, ∗, 1) is a real Hilbert space with a product that satisfies: a∗1 = a a∗b = b∗a a∗(b∗a2) = (a∗b)∗a2 a∗b, c = b, a∗c Example: Mn(F)sa — self-adjoint matrices over F = R, C, H with A ∗ B := 1

2(AB + BA) and A, B := tr(AB).

Q: Why care about EJAs?

Koecher-Vinberg Theorem

Every homogeneous and self-dual ordered vector space is an EJA. As a result they crop up in a lot of places.

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 9 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Purity and Euclidean Jordan algebras

Our paper shows the following for EJAs:

  • Every effect has a filter and a corner.

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 10 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Purity and Euclidean Jordan algebras

Our paper shows the following for EJAs:

  • Every effect has a filter and a corner.
  • Pure maps are closed under composition.

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 10 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Purity and Euclidean Jordan algebras

Our paper shows the following for EJAs:

  • Every effect has a filter and a corner.
  • Pure maps are closed under composition.
  • Pure maps closed under dagger

⇒ Pure maps form a dagger category.

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 10 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Purity and Euclidean Jordan algebras

Our paper shows the following for EJAs:

  • Every effect has a filter and a corner.
  • Pure maps are closed under composition.
  • Pure maps closed under dagger

⇒ Pure maps form a dagger category.

  • Pure dagger-positive maps f ◦ f † are determined by their

action at 1.

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 10 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Purity and Euclidean Jordan algebras

Our paper shows the following for EJAs:

  • Every effect has a filter and a corner.
  • Pure maps are closed under composition.
  • Pure maps closed under dagger

⇒ Pure maps form a dagger category.

  • Pure dagger-positive maps f ◦ f † are determined by their

action at 1. Why show this for EJAs?

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 10 / 12

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Pure maps Corners and Filters Purity and EJAs Conclusion and Discussion

Radboud University Nijmegen

Purity and Euclidean Jordan algebras

Our paper shows the following for EJAs:

  • Every effect has a filter and a corner.
  • Pure maps are closed under composition.
  • Pure maps closed under dagger

⇒ Pure maps form a dagger category.

  • Pure dagger-positive maps f ◦ f † are determined by their

action at 1. Why show this for EJAs?

Theorem

If a generalised probabilistic theory satisfies the above points, then the systems are EJAs. see vdW (2018): Reconstruction of quantum theory from universal filters

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 10 / 12

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Conclusion and Discussion

  • Effectus definition of purity works on EJAs

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 11 / 12

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Radboud University Nijmegen

Conclusion and Discussion

  • Effectus definition of purity works on EJAs
  • It characterises EJAs.

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 11 / 12

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Radboud University Nijmegen

Conclusion and Discussion

  • Effectus definition of purity works on EJAs
  • It characterises EJAs.
  • Originally defined for von Neumann algebras.

EJA+ vNA = JBW algebras. Does it work there?

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 11 / 12

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Advertisements

The category of von Neumann algebras

  • A. Westerbaan

arXiv:1804.02203 Dagger and dilations in the category of von Neumann algebras

  • B. Westerbaan

arXiv:1803.01911

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 12 / 12

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Advertisements

The category of von Neumann algebras

  • A. Westerbaan

arXiv:1804.02203 Dagger and dilations in the category of von Neumann algebras

  • B. Westerbaan

arXiv:1803.01911 Purity in Euclidean Jordan algebras

  • A. Westerbaan, B.Westerbaan, vdW

arXiv:1805.11496 Reconstruction of Quantum Theory from Universal Filters vdW arXiv:1801.05798

Thank you for your attention

(Westerbaan)2 & van de Wetering QPL2018 Purity in EJAs 12 / 12