Picturing Quantum Processes Aleks Kissinger QTFT, V axj o 2015 - PowerPoint PPT Presentation

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