Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Why do we need more reconstructions? The axioms are still not satisfactory. • They are often quite strong. • They are handpicked to work (they are very “convenient”). • They are not focussed on a single aspect of the theory. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 7 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Overview of this talk In this talk: two approaches. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 8 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Overview of this talk In this talk: two approaches. • Reconstruction based on pure maps and filters. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 8 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Overview of this talk In this talk: two approaches. • Reconstruction based on pure maps and filters. “Any theory with well-behaved pure maps is quantum theory” J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 8 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Overview of this talk In this talk: two approaches. • Reconstruction based on pure maps and filters. “Any theory with well-behaved pure maps is quantum theory” • Reconstruction based on sequential measurement. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 8 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Overview of this talk In this talk: two approaches. • Reconstruction based on pure maps and filters. “Any theory with well-behaved pure maps is quantum theory” • Reconstruction based on sequential measurement. “Any theory with well-behaved seq. meas. is quantum theory” J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 8 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The framework: Generalised Probabilistic Theories • We have physical systems A , B , . . . . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 9 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The framework: Generalised Probabilistic Theories • We have physical systems A , B , . . . . • These can be prepared in states ω ∈ St( A ), J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 9 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The framework: Generalised Probabilistic Theories • We have physical systems A , B , . . . . • These can be prepared in states ω ∈ St( A ), • Transformed by transformations f ∈ Trans( A , B ), J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 9 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The framework: Generalised Probabilistic Theories • We have physical systems A , B , . . . . • These can be prepared in states ω ∈ St( A ), • Transformed by transformations f ∈ Trans( A , B ), • And measured by effects e ∈ Eff( A ). J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 9 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The framework: Generalised Probabilistic Theories • We have physical systems A , B , . . . . • These can be prepared in states ω ∈ St( A ), • Transformed by transformations f ∈ Trans( A , B ), • And measured by effects e ∈ Eff( A ). • Measuring a state gives a probability ω ( e ) ∈ [0 , 1]. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 9 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The framework: Generalised Probabilistic Theories • We have physical systems A , B , . . . . • These can be prepared in states ω ∈ St( A ), • Transformed by transformations f ∈ Trans( A , B ), • And measured by effects e ∈ Eff( A ). • Measuring a state gives a probability ω ( e ) ∈ [0 , 1]. • Causality: Measurements do not effect state preparations. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 9 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The framework: Generalised Probabilistic Theories • We have physical systems A , B , . . . . • These can be prepared in states ω ∈ St( A ), • Transformed by transformations f ∈ Trans( A , B ), • And measured by effects e ∈ Eff( A ). • Measuring a state gives a probability ω ( e ) ∈ [0 , 1]. • Causality: Measurements do not effect state preparations. • No infinitesimal effects: ω ( e ) = 0 for all ω , then e = 0. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 9 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The framework: Generalised Probabilistic Theories • We have physical systems A , B , . . . . • These can be prepared in states ω ∈ St( A ), • Transformed by transformations f ∈ Trans( A , B ), • And measured by effects e ∈ Eff( A ). • Measuring a state gives a probability ω ( e ) ∈ [0 , 1]. • Causality: Measurements do not effect state preparations. • No infinitesimal effects: ω ( e ) = 0 for all ω , then e = 0. • Classical probabilistic combinations of states/effects are allowed: ( t ω 1 + (1 − t ) ω 2 )( e ) = t ω 1 ( e ) + (1 − t ) ω 2 ( e ). J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 9 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The framework: Generalised Probabilistic Theories • We have physical systems A , B , . . . . • These can be prepared in states ω ∈ St( A ), • Transformed by transformations f ∈ Trans( A , B ), • And measured by effects e ∈ Eff( A ). • Measuring a state gives a probability ω ( e ) ∈ [0 , 1]. • Causality: Measurements do not effect state preparations. • No infinitesimal effects: ω ( e ) = 0 for all ω , then e = 0. • Classical probabilistic combinations of states/effects are allowed: ( t ω 1 + (1 − t ) ω 2 )( e ) = t ω 1 ( e ) + (1 − t ) ω 2 ( e ). As a result, A is described by an order unit space space V A with Eff( A ) ⊂ [0 , 1] V A and St( A ) ⊂ V ∗ A . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 9 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The first approach Quantum Theory from pure maps and filters J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 10 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Pure maps in quantum theory • Isomorphisms. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 11 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Pure maps in quantum theory • Isomorphisms. • Kraus rank 1 maps A �→ VAV ∗ . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 11 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Pure maps in quantum theory • Isomorphisms. • Kraus rank 1 maps A �→ VAV ∗ . • Filters (’Corner maps’): � A � B �→ A C D � A 0 � �→ A 0 0 J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 11 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Pure maps in quantum theory • Isomorphisms. • Kraus rank 1 maps A �→ VAV ∗ . • Filters (’Corner maps’): � A � B �→ A C D � A 0 � �→ A 0 0 • Compositions of pure maps are pure. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 11 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Pure maps in quantum theory • Isomorphisms. • Kraus rank 1 maps A �→ VAV ∗ . • Filters (’Corner maps’): � A � B �→ A C D � A 0 � �→ A 0 0 • Compositions of pure maps are pure. • Adjoint of pure maps are pure. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 11 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Pure maps in quantum theory • Isomorphisms. • Kraus rank 1 maps A �→ VAV ∗ . • Filters (’Corner maps’): � A � B �→ A C D � A 0 � �→ A 0 0 • Compositions of pure maps are pure. • Adjoint of pure maps are pure. Q: How do we generalise this to more general theories? J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 11 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions 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 . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 12 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions 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. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 12 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions 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 . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 12 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Quotient and Comprehension 1 On ordered vector spaces: • Quotient �→ Filter • Comprehension �→ Corner J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 13 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Quotient and Comprehension 1 On ordered vector spaces: • Quotient �→ Filter • Comprehension �→ Corner Before the definition, lets first look at a concrete example J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 13 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Quotient and Comprehension 1 On ordered vector spaces: • Quotient �→ Filter • Comprehension �→ Corner Before the definition, lets first look at a concrete example In Quantum Theory: Let q = � i λ i q i be an arbitrary effect. Define ⌈ q ⌉ = � i q i . ⌊ q ⌋ = � i ; λ i =1 q i . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 13 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Quotient and Comprehension 1 On ordered vector spaces: • Quotient �→ Filter • Comprehension �→ Corner Before the definition, lets first look at a concrete example In Quantum Theory: Let q = � i λ i q i be an arbitrary effect. 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. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 13 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Quotient and Comprehension 2 A filter for an effect q is a positive map ξ : V q → V with ξ (1) ≤ q , J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 14 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions 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 J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 14 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions 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 A corner for an effect q is a positive map π : V → { V | q } with π (1) = π ( q ), J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 14 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions 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 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 ): π { V | q } V f f W J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 14 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Images and sharpness Definition The image of a transformation f is the smallest effect q such that f (1) = f ( q ). J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 15 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Images and sharpness Definition The image of a transformation f is the smallest effect q such that f (1) = f ( q ). An effect is sharp when it is the image of a transformation. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 15 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Images and sharpness Definition The image of a transformation f is the smallest effect q such that f (1) = f ( q ). An effect is sharp when it is the image of a transformation. Related: The kernel of a transformation is the largest effect p such that f ( p ) = 0. We have image = 1 − kernel. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 15 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Images and sharpness Definition The image of a transformation f is the smallest effect q such that f (1) = f ( q ). An effect is sharp when it is the image of a transformation. Related: The kernel of a transformation is the largest effect p such that f ( p ) = 0. We have image = 1 − kernel. Example: In quantum theory, the sharp effects are the projections. For sharp effect: π ◦ ξ = id. “First forgetting a predicate is true and then forcing its truthiness is equivalent to doing nothing” J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 15 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions First Axioms Axiom 1: All transformations have an image. Axiom 2: Every effect has a filter and a corner. Furthermore, for sharp effects: π ◦ ξ = id. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 16 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions First Axioms Axiom 1: All transformations have an image. Axiom 2: Every effect has a filter and a corner. Furthermore, for sharp effects: π ◦ ξ = id. Interpretation: For each measurement we can find a subsystem where the measurement outcome is always positive, and the operations to and from this subsystem are physical. Going back and forth from the subsystem is a reversible operation. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 16 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Results Spectral Theorem For any effect q in a GPT satisfying Axiom 1 and 2 we can find numbers λ i ≥ 0 and orthogonal sharp effects q i such that q = � i λ i q i . This decomposition is essentially unique. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 17 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Results Spectral Theorem For any effect q in a GPT satisfying Axiom 1 and 2 we can find numbers λ i ≥ 0 and orthogonal sharp effects q i such that q = � i λ i q i . This decomposition is essentially unique. Corollary: You can define thermodynamic quantities like entropy in such theories. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 17 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Results Spectral Theorem For any effect q in a GPT satisfying Axiom 1 and 2 we can find numbers λ i ≥ 0 and orthogonal sharp effects q i such that q = � i λ i q i . This decomposition is essentially unique. Corollary: You can define thermodynamic quantities like entropy in such theories. Duality For any pure effect q in a GPT satisfying Axiom 1 and 2 we can find a unique pure state ω q such that ω q ( q ) = 1. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 17 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Additional axioms Recall that filters and corners are pure maps in quantum theory. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 18 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Additional axioms Recall that filters and corners are pure maps in quantum theory. Definition A map f is pure when it is a composition of a filter and a corner: f = ξ ◦ π J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 18 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Additional axioms Recall that filters and corners are pure maps in quantum theory. Definition A map f is pure when it is a composition of a filter and a corner: f = ξ ◦ π Axiom 3: Pure maps are closed under composition. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 18 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Additional axioms Recall that filters and corners are pure maps in quantum theory. Definition A map f is pure when it is a composition of a filter and a corner: f = ξ ◦ π Axiom 3: Pure maps are closed under composition. Also true in quantum theory: Let q be sharp, then ξ † = π where ξ a filter and π a corner for q . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 18 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Additional axioms Recall that filters and corners are pure maps in quantum theory. Definition A map f is pure when it is a composition of a filter and a corner: f = ξ ◦ π Axiom 3: Pure maps are closed under composition. Also true in quantum theory: Let q be sharp, then ξ † = π where ξ a filter and π a corner for q . Axiom 4: Pure maps have a dagger. Filters and corners are dual. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 18 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Additional axioms Recall that filters and corners are pure maps in quantum theory. Definition A map f is pure when it is a composition of a filter and a corner: f = ξ ◦ π Axiom 3: Pure maps are closed under composition. Also true in quantum theory: Let q be sharp, then ξ † = π where ξ a filter and π a corner for q . Axiom 4: Pure maps have a dagger. Filters and corners are dual. “Pure maps are time-reversible. The inverse of forgetting is remembering.” J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 18 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The first characterisation Theorem The order unit space V A associated to a system A of a GPT satisfying Axioms 1-4 is isomorphic to a Euclidean Jordan algebra. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 19 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The first characterisation Theorem The order unit space V A associated to a system A of a GPT satisfying Axioms 1-4 is isomorphic to a Euclidean Jordan algebra. What is a Euclidean Jordan algebra? J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 19 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The first characterisation Theorem The order unit space V A associated to a system A of a GPT satisfying Axioms 1-4 is isomorphic to a Euclidean Jordan algebra. What is a Euclidean Jordan algebra? It’s a “Nearly quantum” system J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 19 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions From Jordan algebras to quantum theory Q: How are Jordan algebras different from C ∗ -algebras? J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 20 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions From Jordan algebras to quantum theory Q: How are Jordan algebras different from C ∗ -algebras? A: Jordan algebras really don’t like tensor products. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 20 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions From Jordan algebras to quantum theory Q: How are Jordan algebras different from C ∗ -algebras? A: Jordan algebras really don’t like tensor products. Axiom 5: Composite systems are locally tomographic . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 20 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions From Jordan algebras to quantum theory Q: How are Jordan algebras different from C ∗ -algebras? A: Jordan algebras really don’t like tensor products. Axiom 5: Composite systems are locally tomographic . “A bipartite state can be completely characterised using only local measurements.” J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 20 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions From Jordan algebras to quantum theory Q: How are Jordan algebras different from C ∗ -algebras? A: Jordan algebras really don’t like tensor products. Axiom 5: Composite systems are locally tomographic . “A bipartite state can be completely characterised using only local measurements.” Theorem The order unit space V A associated to a system A of a GPT satisfying Axioms 1-5 is isomorphic to a C ∗ -algebra. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 20 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Quantum theory is the unique locally tomographic theory where the pure maps form a dagger category such that the filters and corners of sharp effects are isometries. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 21 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions The second approach Quantum Theory from sequential measurement J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 22 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Measurement • Suppose we have a physical black-box system on which we can do binary measurements (effects). J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 23 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Measurement • Suppose we have a physical black-box system on which we can do binary measurements (effects). • New effect from effects a and b : a & b := “First measure a and then measure b ” J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 23 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Measurement • Suppose we have a physical black-box system on which we can do binary measurements (effects). • New effect from effects a and b : a & b := “First measure a and then measure b ” • a & b is ‘successful’ when both a and b are. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 23 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Measurement • Suppose we have a physical black-box system on which we can do binary measurements (effects). • New effect from effects a and b : a & b := “First measure a and then measure b ” • a & b is ‘successful’ when both a and b are. Of course, in general a & b � = b & a J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 23 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Measurement • Suppose we have a physical black-box system on which we can do binary measurements (effects). • New effect from effects a and b : a & b := “First measure a and then measure b ” • a & b is ‘successful’ when both a and b are. Of course, in general a & b � = b & a Definition We call a and b compatible when a & b = b & a . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 23 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Measurement • Suppose we have a physical black-box system on which we can do binary measurements (effects). • New effect from effects a and b : a & b := “First measure a and then measure b ” • a & b is ‘successful’ when both a and b are. Of course, in general a & b � = b & a Definition We call a and b compatible when a & b = b & a . In that case we write a | b . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 23 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Measurement • Suppose we have a physical black-box system on which we can do binary measurements (effects). • New effect from effects a and b : a & b := “First measure a and then measure b ” • a & b is ‘successful’ when both a and b are. Of course, in general a & b � = b & a Definition We call a and b compatible when a & b = b & a . In that case we write a | b . (In quantum theory a & b := √ ab √ a and a | b when ab = ba ) J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 23 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Additivity • Start with an ensemble of systems. Measure a on all of them. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 24 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Additivity • Start with an ensemble of systems. Measure a on all of them. • Split in two sets. Measure b on first set, c on the second. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 24 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Additivity • Start with an ensemble of systems. Measure a on all of them. • Split in two sets. Measure b on first set, c on the second. • Probability b or c is successful is then a & ( b + c ). J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 24 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Additivity • Start with an ensemble of systems. Measure a on all of them. • Split in two sets. Measure b on first set, c on the second. • Probability b or c is successful is then a & ( b + c ). Same situation, different description: • Split in two groups. Then measure a & b and a & c on these. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 24 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Additivity • Start with an ensemble of systems. Measure a on all of them. • Split in two sets. Measure b on first set, c on the second. • Probability b or c is successful is then a & ( b + c ). Same situation, different description: • Split in two groups. Then measure a & b and a & c on these. • The disjunctive probability is ( a & b ) + ( a & c ). J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 24 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Additivity • Start with an ensemble of systems. Measure a on all of them. • Split in two sets. Measure b on first set, c on the second. • Probability b or c is successful is then a & ( b + c ). Same situation, different description: • Split in two groups. Then measure a & b and a & c on these. • The disjunctive probability is ( a & b ) + ( a & c ). = ⇒ a & ( b + c ) = a & b + a & c . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 24 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Additivity • Start with an ensemble of systems. Measure a on all of them. • Split in two sets. Measure b on first set, c on the second. • Probability b or c is successful is then a & ( b + c ). Same situation, different description: • Split in two groups. Then measure a & b and a & c on these. • The disjunctive probability is ( a & b ) + ( a & c ). = ⇒ a & ( b + c ) = a & b + a & c . NOTE: In general ( a + b ) & c � = a & c + b & c . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 24 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Additivity • Start with an ensemble of systems. Measure a on all of them. • Split in two sets. Measure b on first set, c on the second. • Probability b or c is successful is then a & ( b + c ). Same situation, different description: • Split in two groups. Then measure a & b and a & c on these. • The disjunctive probability is ( a & b ) + ( a & c ). = ⇒ a & ( b + c ) = a & b + a & c . NOTE: In general ( a + b ) & c � = a & c + b & c . Resistance to noise: a �→ a & b is continuous. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 24 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Compatibility When a | b , the measurement order is irrelevant. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 25 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Compatibility When a | b , the measurement order is irrelevant. So we can see them as measured at the same time. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 25 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Compatibility When a | b , the measurement order is irrelevant. So we can see them as measured at the same time. a | b = ⇒ a & ( b & c ) = ( a & b ) & c J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 25 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Compatibility When a | b , the measurement order is irrelevant. So we can see them as measured at the same time. a | b = ⇒ a & ( b & c ) = ( a & b ) & c The effect 1 is trivial: always successful. So of course 1 & a = a . J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 25 / 30
Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Radboud University Nijmegen Quantum from Sequential Measurement Conclusions Sequential Product: Compatibility When a | b , the measurement order is irrelevant. So we can see them as measured at the same time. a | b = ⇒ a & ( b & c ) = ( a & b ) & c The effect 1 is trivial: always successful. So of course 1 & a = a . Let a ⊥ := 1 − a = ‘measure a and invert the outcome’. J. van de Wetering Foundations 2018 Reconstructions of Quantum Theory 25 / 30
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