Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Picturing Quantum Processes Aleks Kissinger QTFT, V¨ axj¨ o 2015 June 10, 2015
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Quantum Picturalism: what it is , what it isn’t • ‘QPism’ ☺ is a methodology for expressing , teaching , and reasoning about quantum processes
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Quantum Picturalism: what it is , what it isn’t • ‘QPism’ ☺ is a methodology for expressing , teaching , and reasoning about quantum processes • Diagrams live at the centre, thus composition and interaction
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Quantum Picturalism: what it is , what it isn’t • ‘QPism’ ☺ is a methodology for expressing , teaching , and reasoning about quantum processes • Diagrams live at the centre, thus composition and interaction • QP is not a reconstruction , but some ideas from operational reconstructions play a major role, e.g. ρ ′ ρ ρ ′ = Φ Φ � f µ ρ purification local/process tomography
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Quantum Picturalism: what it is , what it isn’t • ‘QPism’ ☺ is a methodology for expressing , teaching , and reasoning about quantum processes • Diagrams live at the centre, thus composition and interaction • QP is not a reconstruction , but some ideas from operational reconstructions play a major role, e.g. ρ ′ ρ ρ ′ = Φ Φ � f µ ρ purification local/process tomography • ...and relationship between operational setups and theoretical models : � � � � � U � � � � = � � � � � � � � � U � � ρ
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Picturing Quantum Processes A first course in quantum theory and diagrammatic reasoning Bob Coecke & Aleks Kissinger CUP 2015
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Outline Picturing Quantum Processes chapters 4-9 (roughly)
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Outline Picturing Quantum Processes chapters 4-9 (roughly) 1. Process theory of linear maps
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Outline Picturing Quantum Processes chapters 4-9 (roughly) 1. Process theory of linear maps 2. quantum maps via ‘doubling’ construction
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Outline Picturing Quantum Processes chapters 4-9 (roughly) 1. Process theory of linear maps 2. quantum maps via ‘doubling’ construction 3. Consequences: purification, causality, no-signalling, no-broadcasting
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Outline Picturing Quantum Processes chapters 4-9 (roughly) 1. Process theory of linear maps 2. quantum maps via ‘doubling’ construction 3. Consequences: purification, causality, no-signalling, no-broadcasting 4. Classical/quantum interaction
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Outline Picturing Quantum Processes chapters 4-9 (roughly) 1. Process theory of linear maps 2. quantum maps via ‘doubling’ construction 3. Consequences: purification, causality, no-signalling, no-broadcasting 4. Classical/quantum interaction 5. Complementarity
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Recap • Wires represent systems , boxes represent processes lists breakfast noise poo cooking baby quicksort lists eggs bacon food love
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Recap • Wires represent systems , boxes represent processes lists breakfast noise poo cooking baby quicksort lists eggs bacon food love • The world is organised into process theories , collections of processes that make sense to combine into diagrams A B C g systems A D ψ h A processes
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Recap • Certain processes play a special role: φ states: effects: numbers: ψ λ
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Recap • Certain processes play a special role: φ states: effects: numbers: ψ λ • State + effect = number, interpreted as: test φ probability ψ state this is called the Born rule .
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity linear maps In the process theory of linear maps :
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity linear maps In the process theory of linear maps : (L1) Every type has a (finite) basis : g f g for all : = = ⇒ = f i i i
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity linear maps In the process theory of linear maps : (L1) Every type has a (finite) basis : g f g for all : = = ⇒ = f i i i (L2) Processes can be summed : g g � � � � � where f i h i = h i f f i i i g g
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity linear maps In the process theory of linear maps : (L1) Every type has a (finite) basis : g f g for all : = = ⇒ = f i i i (L2) Processes can be summed : g g � � � � � where f i h i = h i f f i i i g g (L3) Numbers are the complex numbers : ∈ C λ
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Bases ⇔ process tomography Theorem j j g g : = = ⇒ = for all , j i f f i i
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Bases ⇔ process tomography Theorem j j g g for all , j : = = ⇒ = f f i i i Proof. j j g = f i i
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Bases ⇔ process tomography Theorem j j g g for all , j : = = ⇒ = f f i i i Proof. j j = g f
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Bases ⇔ process tomography Theorem j j g g for all , j : = = ⇒ = f f i i i Proof. g f = j j
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Bases ⇔ process tomography Theorem j j g g for all , j : = = ⇒ = f f i i i Proof. g = f
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Bases ⇔ process tomography Theorem j j g g for all , j : = = ⇒ = f f i i i Proof. g = f
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Bases ⇔ process tomography Theorem j j g g for all , j : = = ⇒ = f f i i i • In other words, f is uniquely fixed by its matrix : f 1 f 1 f 1 · · · 1 2 m j f 2 f 2 f 2 · · · 1 2 m f j := where f . . . ... i . . . . . . i f n f n f n · · · 1 2 m
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity What about the Born rule? test φ probability ψ state
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity The Born rule for relations test φ B := { 0 , 1 } ψ state
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity The Born rule for relations possibility test φ B := { 0 , 1 } ψ state
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity The Born rule for linear maps ??? test φ C ψ state
Introduction 1. Linear maps 2. Quantum maps 3. Consequences 4. Classical 5. Complementarity Fixing the problem probability test φ φ R ≥ 0 ψ ψ state
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