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A Bestiary of Sets and Relations arXiv:1506.05025 Stefano Gogioso - - PowerPoint PPT Presentation

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality A Bestiary of Sets and Relations arXiv:1506.05025 Stefano Gogioso Quantum Group University of Oxford 17 July 2015 Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality

A Bestiary of Sets and Relations

arXiv:1506.05025 Stefano Gogioso

Quantum Group University of Oxford

17 July 2015

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality

Introduction

Today, in this talk: a veritable bestiary of sets and relations.

Credit: Aberdeen Bestiary Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Section 1 Pure State Quantum Mechanics

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Pure State Quantum Mechanics in fRel

Looks like fdHilb, but something is not quite right...

Credit: Chimera, Giovannag, DeviantArt Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

†-SMC Structure

Objects = finite sets

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

†-SMC Structure

Objects = finite sets Morphisms X → Y = relations R ⊆ X × Y

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

†-SMC Structure

Objects = finite sets Morphisms X → Y = relations R ⊆ X × Y Non-cartesian symmetric monoidal structure (fRel, ×, 1)

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

†-SMC Structure

Objects = finite sets Morphisms X → Y = relations R ⊆ X × Y Non-cartesian symmetric monoidal structure (fRel, ×, 1) States 1 → X in fRel ← → subsets of X

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

†-SMC Structure

Objects = finite sets Morphisms X → Y = relations R ⊆ X × Y Non-cartesian symmetric monoidal structure (fRel, ×, 1) States 1 → X in fRel ← → subsets of X Dagger R† := {(y, x)|(x, y) ∈ R}

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

†-SMC Structure

Objects = finite sets Morphisms X → Y = relations R ⊆ X × Y Non-cartesian symmetric monoidal structure (fRel, ×, 1) States 1 → X in fRel ← → subsets of X Dagger R† := {(y, x)|(x, y) ∈ R} Superposition operation = relational union ∨ (distributive enrichment over finite commutative monoids)

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

†-SMC Structure

Objects = finite sets Morphisms X → Y = relations R ⊆ X × Y Non-cartesian symmetric monoidal structure (fRel, ×, 1) States 1 → X in fRel ← → subsets of X Dagger R† := {(y, x)|(x, y) ∈ R} Superposition operation = relational union ∨ (distributive enrichment over finite commutative monoids) Scalars form a semiring ({∅, id1}, ∨, ×) ∼ = B

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Classical Structures

[Pavlovic 2009] If ( , , , ) is a classical structure in fRel on a set X, then there is a unique abelian groupoid ⊕λ∈ΛGλ such that:

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Classical Structures

[Pavlovic 2009] If ( , , , ) is a classical structure in fRel on a set X, then there is a unique abelian groupoid ⊕λ∈ΛGλ such that: The groupoid multiplication is given by the partial function: = (gλ, g′

λ′) →

  • gλ +λ g′

λ if λ = λ′

undefined otherwise

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Classical Structures

[Pavlovic 2009] If ( , , , ) is a classical structure in fRel on a set X, then there is a unique abelian groupoid ⊕λ∈ΛGλ such that: The groupoid multiplication is given by the partial function: = (gλ, g′

λ′) →

  • gλ +λ g′

λ if λ = λ′

undefined otherwise The set of the groupoid units forms the state: = {0λ|λ ∈ Λ}

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Classical Structures

[Pavlovic 2009] If ( , , , ) is a classical structure in fRel on a set X, then there is a unique abelian groupoid ⊕λ∈ΛGλ such that: The groupoid multiplication is given by the partial function: = (gλ, g′

λ′) →

  • gλ +λ g′

λ if λ = λ′

undefined otherwise The set of the groupoid units forms the state: = {0λ|λ ∈ Λ} The classical points are the states |Gλ : 1 → X

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Classical Computation

  • 1. Morphisms of classical structures are used to embed partial

functions (and thus classical computation) in fdHilb: Rf :=

  • λ∈dom f

|f (λ)λ|

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Classical Computation

  • 1. Morphisms of classical structures are used to embed partial

functions (and thus classical computation) in fdHilb: Rf :=

  • λ∈dom f

|f (λ)λ|

  • 2. Morphisms of classical structures ⊕λ∈ΛGλ → ⊕γ∈ΓHγ can be

used to embed all partial functions f : Λ ⇀ Γ in fRel: Rf :=

  • λ∈dom f

|Hf (λ)Gλ|

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Classical Computation

  • 3. However, the correspondence in fRel is not 1-to-1.

For example, consider a family (Φλ : Gλ → Hf (λ))λ∈Λ of isomorphisms of abelian groups and embed f : Λ ⇀ Γ as: R′

f := gλ → Φλ(gλ)

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Classical Computation

  • 3. However, the correspondence in fRel is not 1-to-1.

For example, consider a family (Φλ : Gλ → Hf (λ))λ∈Λ of isomorphisms of abelian groups and embed f : Λ ⇀ Γ as: R′

f := gλ → Φλ(gλ)

  • 4. Non-uniqueness is a consequence of the fact that most

classical structures don’t have enough classical points.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Classical Computation

  • 3. However, the correspondence in fRel is not 1-to-1.

For example, consider a family (Φλ : Gλ → Hf (λ))λ∈Λ of isomorphisms of abelian groups and embed f : Λ ⇀ Γ as: R′

f := gλ → Φλ(gλ)

  • 4. Non-uniqueness is a consequence of the fact that most

classical structures don’t have enough classical points.

  • 5. These additional degrees of freedom could be related to

microscopic degrees of freedom in computation using the groupoid framework of [Bar&Vicary (2014)].

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Discrete structures

On each finite set X, the discrete structure is given by: = (x, y) → δxyx (partial function) = 1 × X = x → (x, x) (total function) = x → ⋆ (total function)

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Discrete structures

The discrete structure on X corresponds to groupoid ⊕x∈X0x.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Discrete structures

The discrete structure on X corresponds to groupoid ⊕x∈X0x. It has the singletons {x} as its classical points.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Discrete structures

The discrete structure on X corresponds to groupoid ⊕x∈X0x. It has the singletons {x} as its classical points. It is the only classical structure with enough classical points.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Discrete structures

The discrete structure on X corresponds to groupoid ⊕x∈X0x. It has the singletons {x} as its classical points. It is the only classical structure with enough classical points. It gives the usual 1-to-1 embedding of partial functions: Rf := {(x, f (x))|x ∈ dom f }

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Isometries and Unitaries

A morphism F : X → Y in fRel is an isometry iff it is in the form, for some classical structure ⊕γ∈ΓHγ on Y F =

  • x∈X

|Hf (x){x}| where f : X → Γ is a total injection.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Isometries and Unitaries

A morphism F : X → Y in fRel is an isometry iff it is in the form, for some classical structure ⊕γ∈ΓHγ on Y F =

  • x∈X

|Hf (x){x}| where f : X → Γ is a total injection. Isometries are in the form F = Rf for some total injection f

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Isometries and Unitaries

A morphism F : X → Y in fRel is an isometry iff it is in the form, for some classical structure ⊕γ∈ΓHγ on Y F =

  • x∈X

|Hf (x){x}| where f : X → Γ is a total injection. Isometries are in the form F = Rf for some total injection f Forces discrete structure on X ⇒ more restrictive than fdHilb

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality †-SMC Structure Classical Structures Isometries and Unitaries

Isometries and Unitaries

A morphism F : X → Y in fRel is an isometry iff it is in the form, for some classical structure ⊕γ∈ΓHγ on Y F =

  • x∈X

|Hf (x){x}| where f : X → Γ is a total injection. Isometries are in the form F = Rf for some total injection f Forces discrete structure on X ⇒ more restrictive than fdHilb Indeed this forces unitaries = bijections

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Section 2 CPM and Decoherence

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Decoherence in fRel

One look at it and things turns to stone. Very classical stone.

Credit: Medusa, Miragenathalen, DeviantArt Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

The category CPM[fRel]

Morphisms in CPM[fRel] take the usual doubled-up form:

f f ⋆

where the compact-closed structure on fRel is given by: ∩X := (x, y) → δxy : X × X → 1 ∪X := ∆X : 1 → X × X We call ∩X the discarding map X CPM − → 1, and we will say that a CPM morphism R : X CPM − → Y is causal iff ∩Y · R = ∩X .

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graphs for CPM

[Marsden 2015] A clever graph-theoretic formalism for CPM[fRel]: States ρ : 1 CPM − → X in CPM[fRel] correspond to subgraphs Gρ

  • f the complete graph KX on X.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graphs for CPM

[Marsden 2015] A clever graph-theoretic formalism for CPM[fRel]: States ρ : 1 CPM − → X in CPM[fRel] correspond to subgraphs Gρ

  • f the complete graph KX on X.

Morphisms R : X CPM − → Y correspond to subgraphs GR of the complete graph KX×Y .

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graphs for CPM

[Marsden 2015] A clever graph-theoretic formalism for CPM[fRel]: States ρ : 1 CPM − → X in CPM[fRel] correspond to subgraphs Gρ

  • f the complete graph KX on X.

Morphisms R : X CPM − → Y correspond to subgraphs GR of the complete graph KX×Y . Composition is done by lifting and projecting edges.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graphs of CPM states (example)

·

  • ·

·

  • ·

· · ·

  • ·

a pure state in a 12 element set

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graphs of CPM states (example)

·

  • ·

·

  • ·
  • ·
  • a non-pure state in a 12 element set

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graphs of CPM states (example)

·

  • ·

·

  • ·
  • ·
  • a discrete state in a 12 element set

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graphs of CPM maps (example)

identity on 4 element set X as a graph on X × X 1 2 3 4 1 2 3 4

  • ·

· · ·

  • ·

· · ·

  • ·

· · ·

  • ∩X on 4 element set X

as a graph on X × 1 1 2 3 4 ⋆

  • Stefano Gogioso

A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graph composition (example)

Computing the image Rρ : 1 CPM − → Y of a CPM state ρ : 1 CPM − → X under a CPM map R : X CPM − → Y using the associated graphs.

· ·

  • ·

X Gρ · · · · · · · · · · Y

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graph composition (example)

Computing the image Rρ : 1 CPM − → Y of a CPM state ρ : 1 CPM − → X under a CPM map R : X CPM − → Y using the associated graphs.

· ·

  • ·

X Gρ · · · · · · · · · · Y GR

  • Stefano Gogioso

A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graph composition (example)

Computing the image Rρ : 1 CPM − → Y of a CPM state ρ : 1 CPM − → X under a CPM map R : X CPM − → Y using the associated graphs.

· ·

  • ·

X Gρ · · · · · · · · · · Y GR

  • Stefano Gogioso

A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graph composition (example)

Computing the image Rρ : 1 CPM − → Y of a CPM state ρ : 1 CPM − → X under a CPM map R : X CPM − → Y using the associated graphs.

· ·

  • ·

X Gρ · · · · · · · · · · Y

  • Stefano Gogioso

A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graph composition (example)

Computing the image Rρ : 1 CPM − → Y of a CPM state ρ : 1 CPM − → X under a CPM map R : X CPM − → Y using the associated graphs.

· ·

  • ·

X Gρ · · · · · · · · · · Y

  • Stefano Gogioso

A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Graph composition (example)

Computing the image Rρ : 1 CPM − → Y of a CPM state ρ : 1 CPM − → X under a CPM map R : X CPM − → Y using the associated graphs.

· ·

  • ·

X Gρ · · · · · · · · · · Y GRρ

  • •
  • Stefano Gogioso

A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Purity of states

A CPM state ρ is pure if and only if Gρ is a clique.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Purity of states

A CPM state ρ is pure if and only if Gρ is a clique. Define a relative purity partial order on states ρ, σ : 1 CPM − → X by setting ρ σ iff Gρ is a subgraph of Gσ, covering all nodes.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Purity of states

A CPM state ρ is pure if and only if Gρ is a clique. Define a relative purity partial order on states ρ, σ : 1 CPM − → X by setting ρ σ iff Gρ is a subgraph of Gσ, covering all nodes. Pure states are the maxima of .

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Purity of states

A CPM state ρ is pure if and only if Gρ is a clique. Define a relative purity partial order on states ρ, σ : 1 CPM − → X by setting ρ σ iff Gρ is a subgraph of Gσ, covering all nodes. Pure states are the maxima of . The lower set ρ ↓ of any pure state ρ is an atomic semilattice under union ∨ of graphs (i.e. convex combination of states).

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Purity of states

A CPM state ρ is pure if and only if Gρ is a clique. Define a relative purity partial order on states ρ, σ : 1 CPM − → X by setting ρ σ iff Gρ is a subgraph of Gσ, covering all nodes. Pure states are the maxima of . The lower set ρ ↓ of any pure state ρ is an atomic semilattice under union ∨ of graphs (i.e. convex combination of states). Therefore every pure state ρ (clique Gρ) can be expressed as a convex combination of non-pure states (the atoms of ρ ↓).

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Purity of states (example)

·

  • ·

·

  • ·

· · · ·

  • ·

Purity lower-set of pure state {2, 5, 11} in {1, 2, ..., 12} pure state ·

  • ·

·

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· · · ·

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·

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·

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·

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·

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·

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·

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·

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·

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· · · ·

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·

  • ·

·

  • ·

· · · ·

  • ·

atoms ·

  • ·

·

  • ·

· · · ·

  • ·

discrete state

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Decoherence Maps in †-SMCs

Let be a classical structure on some object X of a compact closed †-SMC. The

  • decoherence map dec( ) is the following

causal CPM morphism X CPM − → X:

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Decoherence Maps in fdHilb

In fdHilb, the decoherence map sends any (causal) CPM state to a (probabilistic) convex combination of

  • classical points:

dec( )ρ =

  • z

z|ρ|z |zz|

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Decoherence Maps in fdHilb

In fdHilb, the decoherence map sends any (causal) CPM state to a (probabilistic) convex combination of

  • classical points:

dec( )ρ =

  • z

z|ρ|z |zz| This also justifies the following quantum-classical notation when all

  • perations after the decoherence are
  • classical:

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Decoherence Maps in fRel

This convex combination assumption fails in fRel: The result of decohering a CPM state ρ to dec( )ρ is not in general a convex combination of

  • classical points.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Decoherence Maps in fRel

This convex combination assumption fails in fRel: The result of decohering a CPM state ρ to dec( )ρ is not in general a convex combination of

  • classical points.

Unless is the discrete structure, no causal CPM map exists preserving

  • classical points and always resulting in a convex

combination of

  • classical points.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Decoherence Maps in fRel

This convex combination assumption fails in fRel: The result of decohering a CPM state ρ to dec( )ρ is not in general a convex combination of

  • classical points.

Unless is the discrete structure, no causal CPM map exists preserving

  • classical points and always resulting in a convex

combination of

  • classical points.

In the case of fRel, the CPM category cannot be interpreted as a category of mixed states in the usual sense.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Decoherence Maps in fRel (example)

Let X be a 5 element set, and the classical structure of groupoid Z2 ⊕ Z3. Then Gdec(

) is the following subgraph of KX×X:

0Z2 1Z2 0Z3 1Z3 2Z3 0Z2 1Z2 0Z3 1Z3 2Z3

  • ·

· ·

  • ·

· · · ·

  • ·

·

  • ·

·

  • Tq

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Decoherence Maps in fRel (example)

Let X be a 5 element set, and the classical structure of groupoid Z2 ⊕ Z3. Then Gdec(

) is the following subgraph of KX×X:

0Z2 1Z2 0Z3 1Z3 2Z3 0Z2 1Z2 0Z3 1Z3 2Z3

  • ·

· ·

  • ·

· · · ·

  • ·

·

  • ·

·

  • Z2

Z3

The cliques on the boxed sets of nodes are the

  • classical states.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Graphs for CPM Purity of states Decoherence Maps

Decoherence Maps in fRel (example)

However, the decoherence of the discrete structure always yields a convex combination of singletons (i.e. it eliminates all edges):

0Z2 1Z2 0Z3 1Z3 2Z3 0Z2 1Z2 0Z3 1Z3 2Z3

  • ·

· · · ·

  • ·

· · · ·

  • ·

· · · ·

  • ·

· · · ·

  • This is because the discrete structure has enough classical points.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Non-Demolition Measurements Demolition Measurements Empirical Models and Locality

Section 3 Measurements and Locality

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Measurements and Locality in fRel

The riddle with no apparent answer. We should ask Oedipus.

Credit: Sphynx, Snaketoast, DeviantArt Stefano Gogioso A Bestiary of Sets and Relations

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Testing against classical states

Testing against

  • classical points yields a more familiar

scenario for decoherence:

z1 z2 ρ = if z1 = z2

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Testing against classical states

Testing against

  • classical points yields a more familiar

scenario for decoherence:

z1 z2 ρ = if z1 = z2

However quotienting by equivalence in testing against states trivializes the CPM construction entirely.

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Non-Demolition Measurements Demolition Measurements Empirical Models and Locality

Non-Demolition Measurements

Let be a classical structure in a compact-closed †-SMC on some

  • bject Z . A
  • valued non-demolition measurement on some
  • bject X is a causal CPM morphism M : X CPM

− → X ⊗ Z taking the following form, and which is

  • idempotent and
  • self-adjoint:

P P Causality is equivalent to P : X → X ⊗ Z being an isometry.

Stefano Gogioso A Bestiary of Sets and Relations

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Non-Demolition Measurements: idempotence

The required

  • idempotence is defined by the following equation:

P P P P = P P

Stefano Gogioso A Bestiary of Sets and Relations

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Non-Demolition Measurements: self-adjointness

The required

  • self-adjointness is defined by the following equation:

P P = P P

Stefano Gogioso A Bestiary of Sets and Relations

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Demolition Measurements

If M : X CPM − → X ⊗ Z is a non-demolition measurement, the demolition measurement ¯ M is defined by discarding X: ¯ M := (∩X ⊗ idZ) · M : X CPM − → Z

Stefano Gogioso A Bestiary of Sets and Relations

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Demolition Measurements

If M : X CPM − → X ⊗ Z is a non-demolition measurement, the demolition measurement ¯ M is defined by discarding X: ¯ M := (∩X ⊗ idZ) · M : X CPM − → Z Because of the convex-combination issues with decoherence in fRel, we are forced to test the demolition measurement ¯ M against classical points of to get the classical outcomes:

  • ¯

Mλ := ρ†

Gλ · ¯

M

  • λ∈Λ

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Non-Demolition Measurements Demolition Measurements Empirical Models and Locality

Demolition Measurements

Testing against classical points makes things a bit boring: The same classical outcomes of a

  • valued demolition

measurement ¯ M : X CPM − → Z can be obtained by using a decoherence dec( ) on X, followed by a classical map:

  • γ∈Γ |Gf (γ)Hγ| :

→

Stefano Gogioso A Bestiary of Sets and Relations

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Pure State Quantum Mechanics CPM and Decoherence Measurements and Locality Non-Demolition Measurements Demolition Measurements Empirical Models and Locality

Demolition Measurements

Testing against classical points makes things a bit boring: The same classical outcomes of a

  • valued demolition

measurement ¯ M : X CPM − → Z can be obtained by using a decoherence dec( ) on X, followed by a classical map:

  • γ∈Γ |Gf (γ)Hγ| :

→ On the plus side, we only need to consider empirical models coming from decoherences in our proof of locality. This leads to the simplified definition of empirical model that follows.

Stefano Gogioso A Bestiary of Sets and Relations

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Possibilistic Empirical Models

Let ρ be a mixed state in X1 × ... × XN

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Possibilistic Empirical Models

Let ρ be a mixed state in X1 × ... × XN Let (

m j )m=1,...,M be a family of classical structures on Xj

Stefano Gogioso A Bestiary of Sets and Relations

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Possibilistic Empirical Models

Let ρ be a mixed state in X1 × ... × XN Let (

m j )m=1,...,M be a family of classical structures on Xj

Let (Λm

j )jm be the sets indexing the classical points

Stefano Gogioso A Bestiary of Sets and Relations

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Possibilistic Empirical Models

Let ρ be a mixed state in X1 × ... × XN Let (

m j )m=1,...,M be a family of classical structures on Xj

Let (Λm

j )jm be the sets indexing the classical points

The empirical model is the family of boolean functions Φm(λm

1 , ..., λm N) : Λm 1 × ... × Λm N → {⊥, ⊤} defined as follows

Φm(λm

1 , ..., λm N)

:= Gλm

1

Gλm

1

... Gλm

N

Gλm

N

ρ

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Locality

Theorem Every empirical model (Φm)m admits a local hidden variable ν: (i) the mixed state ρ is decohered in the discrete structures (ii) the discrete classical data is appropriately copied

ν′ ν D local map Z 1

1 ... Z 1 N

... Z m

j

... Z M

1 ...Z M N

Y 1

1 ...Y K 1

... Y k

j

... Y 1

N ...Y K N

... ... ... ... X1 Xj XN ... ... ρ

Stefano Gogioso A Bestiary of Sets and Relations

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Locality

Key points of the proof (in a nice graph flavour): In a measurement framework where we test against classical points, any CPM state ρ is equivalent to the discrete τ ρ.

Stefano Gogioso A Bestiary of Sets and Relations

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Locality

Key points of the proof (in a nice graph flavour): In a measurement framework where we test against classical points, any CPM state ρ is equivalent to the discrete τ ρ. This is immediate to see from the graph perspective: σ†·ρ = 1 ⇐ ⇒ Gσ∧Gρ = ∅ ⇐ ⇒ Gσ∧Gτ = ∅ ⇐ ⇒ σ†·τ = 1

Stefano Gogioso A Bestiary of Sets and Relations

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Locality

Key points of the proof (in a nice graph flavour): In a measurement framework where we test against classical points, any CPM state ρ is equivalent to the discrete τ ρ. This is immediate to see from the graph perspective: σ†·ρ = 1 ⇐ ⇒ Gσ∧Gρ = ∅ ⇐ ⇒ Gσ∧Gτ = ∅ ⇐ ⇒ σ†·τ = 1 Decoherence in the discrete structure turns any CPM state ρ into the discrete state τ ρ (i.e. removes all edges).

Stefano Gogioso A Bestiary of Sets and Relations

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Locality

Key points of the proof (in a nice graph flavour): In a measurement framework where we test against classical points, any CPM state ρ is equivalent to the discrete τ ρ. This is immediate to see from the graph perspective: σ†·ρ = 1 ⇐ ⇒ Gσ∧Gρ = ∅ ⇐ ⇒ Gσ∧Gτ = ∅ ⇐ ⇒ σ†·τ = 1 Decoherence in the discrete structure turns any CPM state ρ into the discrete state τ ρ (i.e. removes all edges). A discrete state is a convex combination of classical points of the discrete structure , and can be appropriately copied.

Stefano Gogioso A Bestiary of Sets and Relations

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Conclusions

fRel, with all the fundamental ingredients and many exotic features, still provides an excellent sandbox for CQM.

Stefano Gogioso A Bestiary of Sets and Relations

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Conclusions

fRel, with all the fundamental ingredients and many exotic features, still provides an excellent sandbox for CQM. The issues with decoherence invite a deeper reflection on the quantum-classical boundary in CQM, and on the operational interpretation of CPM as a category of mixed states.

Stefano Gogioso A Bestiary of Sets and Relations

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Conclusions

fRel, with all the fundamental ingredients and many exotic features, still provides an excellent sandbox for CQM. The issues with decoherence invite a deeper reflection on the quantum-classical boundary in CQM, and on the operational interpretation of CPM as a category of mixed states. In a framework where decoherence doesn’t return convex combinations, testing against classical points may not be physically sound. Measurements/locality need revisiting.

Stefano Gogioso A Bestiary of Sets and Relations

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Thank You!

Thanks for Your Attention! Any Questions?

[Pavlovic (2009)] Quantum and classical structures in nondeterministic computation [Bar&Vicary (2014)] Groupoid Semantics for Thermal Computing [Marsden (2015)] A Graph Theoretic Perspective on CPM(Rel)

Stefano Gogioso A Bestiary of Sets and Relations