Reconstruction of Quantum Theory from Universal Filters John van de - - PowerPoint PPT Presentation

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Reconstruction of Quantum Theory from Universal Filters

John van de Wetering

wetering@cs.ru.nl

Institute for Computing and Information Sciences Radboud University Nijmegen

Foundations 2018 13th of July

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 1 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Why Quantum Theory?

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 2 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Why Quantum Theory?

Its mathematical description is not particularly compelling:

• Systems are described by C∗-algebras.
• States are density matrices.
• Dynamics are completely positive maps.
• Measurement outcomes are governed by the trace rule.
• Composite systems are formed using the tensor product.
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 2 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Why Quantum Theory?

Its mathematical description is not particularly compelling:

• Systems are described by C∗-algebras.
• States are density matrices.
• Dynamics are completely positive maps.
• Measurement outcomes are governed by the trace rule.
• Composite systems are formed using the tensor product.

Not clear at all why this describes nature so well.

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 2 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Why Quantum Theory?

A way to answer this: Find intuitively sensible requirements on nature from which it follows.

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 3 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Why Quantum Theory?

A way to answer this: Find intuitively sensible requirements on nature from which it follows. If successful, we can say: Quantum theory describes nature because “it couldn’t have been any other way”

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 3 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Why Quantum Theory?

A way to answer this: Find intuitively sensible requirements on nature from which it follows. If successful, we can say: Quantum theory describes nature because “it couldn’t have been any other way” (without nature being that much weirder)

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 3 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Intuitive Postulates of Quantum Theory

What are “intuitively sensible requirements on nature”?

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 4 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Intuitive Postulates of Quantum Theory

What are “intuitively sensible requirements on nature”? Ideally they:

• can be described in a few words,
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 4 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Intuitive Postulates of Quantum Theory

What are “intuitively sensible requirements on nature”? Ideally they:

• can be described in a few words,
• are “natural”,
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 4 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Intuitive Postulates of Quantum Theory

What are “intuitively sensible requirements on nature”? Ideally they:

• can be described in a few words,
• are “natural”,
• can be confirmed by a direct experiment,
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 4 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Intuitive Postulates of Quantum Theory

What are “intuitively sensible requirements on nature”? Ideally they:

• can be described in a few words,
• are “natural”,
• can be confirmed by a direct experiment,
• and they seem intuitively true.
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 4 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

The reconstruction approach

• Pick a general framework.
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 5 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

The reconstruction approach

• Pick a general framework.
• Pick some nice postulates and requirements.
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 5 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

The reconstruction approach

• Pick a general framework.
• Pick some nice postulates and requirements.
• Translate them into mathematical statements.
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 5 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

The reconstruction approach

• Pick a general framework.
• Pick some nice postulates and requirements.
• Translate them into mathematical statements.
• Do some math.
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 5 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

The reconstruction approach

• Pick a general framework.
• Pick some nice postulates and requirements.
• Translate them into mathematical statements.
• Do some math.
• Conclude that theories satisfying the requirements are (close

to) quantum theory.

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 5 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Modern reconstructions

• Hardy (2001): First modern reconstructions. 5 axioms.
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 6 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Modern reconstructions

• Hardy (2001): First modern reconstructions. 5 axioms.
• Barrett (2007): Generalised Probabilistic Theories.
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 6 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Modern reconstructions

• Hardy (2001): First modern reconstructions. 5 axioms.
• Barrett (2007): Generalised Probabilistic Theories.
• Daki´

c and Brukner (2009): Local tomography. Strong axioms.

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 6 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Modern reconstructions

• Hardy (2001): First modern reconstructions. 5 axioms.
• Barrett (2007): Generalised Probabilistic Theories.
• Daki´

c and Brukner (2009): Local tomography. Strong axioms.

• Chiribella, D’Ariano, Perinotti (2011): Informational axioms.
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 6 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Modern reconstructions

• Hardy (2001): First modern reconstructions. 5 axioms.
• Barrett (2007): Generalised Probabilistic Theories.
• Daki´

c and Brukner (2009): Local tomography. Strong axioms.

• Chiribella, D’Ariano, Perinotti (2011): Informational axioms.
• Lot of others since then.
• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 6 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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: (tω1 + (1 − t)ω2)(e) = tω1(e) + (1 − t)ω2(e).

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 9 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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: (tω1 + (1 − t)ω2)(e) = tω1(e) + (1 − t)ω2(e). As a result, A is described by an order unit space space VA with Eff(A) ⊂ [0, 1]VA 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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Pure maps in quantum theory

• Isomorphisms.
• Kraus rank 1 maps A → VAV ∗.
• Filters (’Corner maps’):

A B C D

• →

A A → A

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 11 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Pure maps in quantum theory

• Isomorphisms.
• Kraus rank 1 maps A → VAV ∗.
• Filters (’Corner maps’):

A B C D

• →

A A → A

• 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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Pure maps in quantum theory

• Isomorphisms.
• Kraus rank 1 maps A → VAV ∗.
• Filters (’Corner maps’):

A B C D

• →

A A → A

• 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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Pure maps in quantum theory

• Isomorphisms.
• Kraus rank 1 maps A → VAV ∗.
• Filters (’Corner maps’):

A B C D

• →

A A → A

• 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 Quantum from Sequential Measurement Conclusions

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.

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 12 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

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.

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 12 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

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.

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 12 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 λiqi be an arbitrary effect.

Define ⌈q⌉ =

i qi. ⌊q⌋ = i;λi=1 qi.

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 13 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 λiqi be an arbitrary effect.

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.

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 13 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Quotient and Comprehension 2

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

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 14 / 30

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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

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 14 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

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),

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 14 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

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

• J. van de Wetering

Foundations 2018 Reconstructions of Quantum Theory 14 / 30

Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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

• perations 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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Results

Spectral Theorem

For any effect q in a GPT satisfying Axiom 1 and 2 we can find numbers λi ≥ 0 and orthogonal sharp effects qi such that q =

i λiqi. 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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Results

Spectral Theorem

For any effect q in a GPT satisfying Axiom 1 and 2 we can find numbers λi ≥ 0 and orthogonal sharp effects qi such that q =

i λiqi. 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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

Results

Spectral Theorem

For any effect q in a GPT satisfying Axiom 1 and 2 we can find numbers λi ≥ 0 and orthogonal sharp effects qi such that q =

i λiqi. 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 ωq such that ωq(q) = 1.

• J. van de Wetering

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Why Quantum Theory? The reconstruction approach Quantum from Pure Maps Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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 Quantum from Sequential Measurement Conclusions

Radboud University Nijmegen

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

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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.”

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The first characterisation

Theorem

The order unit space VA associated to a system A of a GPT satisfying Axioms 1-4 is isomorphic to a Euclidean Jordan algebra.

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The first characterisation

Theorem

The order unit space VA 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?

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The first characterisation

Theorem

The order unit space VA 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

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From Jordan algebras to quantum theory

Q: How are Jordan algebras different from C∗-algebras?

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From Jordan algebras to quantum theory

Q: How are Jordan algebras different from C∗-algebras? A: Jordan algebras really don’t like tensor products.

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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.

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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.”

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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 VA associated to a system A of a GPT satisfying Axioms 1-5 is isomorphic to a C∗-algebra.

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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.

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The second approach

Quantum Theory from sequential measurement

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Sequential Measurement

• Suppose we have a physical black-box system
• n which we can do binary measurements (effects).
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Sequential Measurement

• Suppose we have a physical black-box system
• n which we can do binary measurements (effects).
• New effect from effects a and b:

a & b := “First measure a and then measure b”

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Sequential Measurement

• Suppose we have a physical black-box system
• n 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.
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Sequential Measurement

• Suppose we have a physical black-box system
• n 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

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Sequential Measurement

• Suppose we have a physical black-box system
• n 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.

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Sequential Measurement

• Suppose we have a physical black-box system
• n 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.

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Sequential Measurement

• Suppose we have a physical black-box system
• n 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)

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Sequential Product: Additivity

• Start with an ensemble of systems. Measure a on all of them.
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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.
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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).
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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.
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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).
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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.

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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.

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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.

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Sequential Product: Compatibility

When a | b, the measurement order is irrelevant.

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Sequential Product: Compatibility

When a | b, the measurement order is irrelevant. So we can see them as measured at the same time.

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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

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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.

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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’.

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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’. a | b = ⇒ a⊥ | b

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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’. a | b = ⇒ a⊥ | b Similarly we require: a | b, a | c = ⇒ a | b + c, a | b & c

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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’. a | b = ⇒ a⊥ | b Similarly we require: a | b, a | c = ⇒ a | b + c, a | b & c Suppose a & b = 0.

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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’. a | b = ⇒ a⊥ | b Similarly we require: a | b, a | c = ⇒ a | b + c, a | b & c Suppose a & b = 0. ‘a and b are never true at the same time.’

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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’. a | b = ⇒ a⊥ | b Similarly we require: a | b, a | c = ⇒ a | b + c, a | b & c Suppose a & b = 0. ‘a and b are never true at the same time.’ a & b = 0 = ⇒ b & a = 0.

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Summary of framework

Definition

A sequential effect space is an order unit space V with E = {a ∈ V ; 0 ≤ a ≤ 1} and & : E × E → E satisfying:

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Summary of framework

Definition

A sequential effect space is an order unit space V with E = {a ∈ V ; 0 ≤ a ≤ 1} and & : E × E → E satisfying:

• a & (b + c) = a & b + a & c
• 1 & a = a and if a & b = 0 then also b & a = 0.
• If a | b then a & (b & c) = (a & b) & c.
• If a | b then a | b⊥, and if also a | c then a | (b + c) & a | (b & c)
• a → a & b is continuous.
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Summary of framework

Definition

A sequential effect space is an order unit space V with E = {a ∈ V ; 0 ≤ a ≤ 1} and & : E × E → E satisfying:

• a & (b + c) = a & b + a & c
• 1 & a = a and if a & b = 0 then also b & a = 0.
• If a | b then a & (b & c) = (a & b) & c.
• If a | b then a | b⊥, and if also a | c then a | (b + c) & a | (b & c)
• a → a & b is continuous.

– Not a random collection of properties! Sequential effect algebra (Gudder, Greechie 2002)

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Summary of framework

Definition

A sequential effect space is an order unit space V with E = {a ∈ V ; 0 ≤ a ≤ 1} and & : E × E → E satisfying:

• a & (b + c) = a & b + a & c
• 1 & a = a and if a & b = 0 then also b & a = 0.
• If a | b then a & (b & c) = (a & b) & c.
• If a | b then a | b⊥, and if also a | c then a | (b + c) & a | (b & c)
• a → a & b is continuous.

– Not a random collection of properties! Sequential effect algebra (Gudder, Greechie 2002) – Not enough to characterise b → √ab√a.

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The second Characterisation

Theorem

A sequential effect space is a Euclidean Jordan algebra.

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The second Characterisation

Theorem

A sequential effect space is a Euclidean Jordan algebra. And again, we get to quantum theory using local tomography:

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The second Characterisation

Theorem

A sequential effect space is a Euclidean Jordan algebra. And again, we get to quantum theory using local tomography:

Theorem

A locally tomographic GPT of sequential effect spaces consists of C∗-algebras.

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Quantum theory is the unique locally tomographic GPT where the sequential measurement of compatible measurements acts classically.

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Conclusion

• Quantum Theory can be axiomatised,
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Conclusion

• Quantum Theory can be axiomatised,
• using (semi-)intuitive axioms,
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Conclusion

• Quantum Theory can be axiomatised,
• using (semi-)intuitive axioms,
• focussing on small parts of the theory,
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Conclusion

• Quantum Theory can be axiomatised,
• using (semi-)intuitive axioms,
• focussing on small parts of the theory,
• in many different ways.
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Conclusion

• Quantum Theory can be axiomatised,
• using (semi-)intuitive axioms,
• focussing on small parts of the theory,
• in many different ways.

Instead of quantum theory being weird, it is in fact particularly nice!

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vdW (2018): Reconstruction of Quantum Theory from Universal Filters

arXiv:1801.05798 vdW (2018): Sequential Measurement Characterises Quantum Theory arXiv:1803.11139

Thank you for your attention

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