Picturing Quantum processes - A diagrammatic approach

Preface
 1 Introduction
 1.1 The Penguins and the Polar Bear
 1.2 So What’s New?
 1.2.1 A New Attitude to Quantum Theory: ‘Features’
 1.2.2A New Form of Mathematics: ‘Diagrams’
 1.2.3A New Foundation for Physics: ‘Process Theories’
 1.2.4A New Paradigm: ‘Quantum Picturalism’
 1.3Historical Notes and References
 2Guide to Reading This Textbook
 2.1Who Are You and What Do You Want?
 2.2The Menu
 2.2.1How Diagrams Evolve in This Book
 2.2.2Hollywood-Style Trailer
 2.2.3Some Intermediate Symbolic Pollution
 2.2.4Summaries, Historical Notes, References, Epigraphs
 2.2.5Starred Headings and Advanced Material Sections
 2.3FAQ
 3Processes as Diagrams
 3.1From Processes to Diagrams
 3.1.1Processes as Boxes and Systems as Wires
 3.1.2Process Theories
 3.1.3Diagrams Are Mathematics
 3.1.4Process Equations
 3.1.5Diagram Substitution
 3.2Circuit Diagrams
 3.2.1Parallel Composition
 3.2.2Sequential Composition
 3.2.3Two Equivalent Definitions of Circuits
 3.2.4Diagrams Beat Algebra
 3.3Functions and Relations as Processes
 3.3.1Sets
 3.3.2Functions
 3.3.3Relations
 3.3.4Functions versus Relations
 3.4Special Processes
 3.4.1States, Effects, and Numbers
 3.4.2Saying the Impossible: Zero Diagrams
 3.4.3Processes That Are Equal ‘Up to a Number’
 3.4.4Dirac Notation
 3.5Summary: What to Remember
 3.6Advanced Material 3.6.1Abstract Tensor Systems*
 3.6.2Symmetric Monoidal Categories*
 3.6.3General Diagrams versus Circuits*
 3.7Historical Notes and References
 4String Diagrams
 4.1Cups, Caps, and String Diagrams
 4.1.1Separability
 4.1.2Process–State Duality
 4.1.3The Yanking Equations
 4.1.4String Diagrams
 4.2Transposition and Trace
 4.2.1The Transpose
 4.2.2Transposition of Composite Systems
 4.2.3The Trace and Partial Trace
 4.3Reflecting Diagrams
 4.3.1Adjoints
 4.3.2Conjugates
 4.3.3The Inner Product
 4.3.4Unitarity
 4.3.5Positivity
 4.3.6⊗-Positivity
 4.3.7Projectors
 4.4Quantum Features from String Diagrams
 4.4.1A No-Go Theorem for Universal Separability
 4.4.2Two No-Go Theorems for Cloning
 4.4.3As If time Flows Backwards
 4.4.4Teleportation
 4.5Summary: What to Remember
 4.6Advanced Material*
 4.6.1String Diagrams in Abstract Tensor Systems*
 4.6.2Dual Types and Self-Duality*
 4.6.3Dagger Compact Closed Categories*
 4.7Historical Notes and References
 5Hilbert Space from Diagrams
 5.1Bases and Matrices
 5.1.1Basis for a Type
 5.1.2Matrix of a Process
 5.1.3Sums of Processes
 5.1.4Processes from Matrices
 5.1.5Matrices of Isometries and Unitaries
 5.1.6Matrices of Self-Adjoint and Positive Processes
 5.1.7Traces of Matrices
 5.2Matrix Calculus
 5.2.1Sequential Composition of Matrices
 5.2.2Parallel Composition of Matrices
 5.2.3Matrix Form of Cups and Caps
 5.2.4String Diagrams of Matrices
 5.2.5Matrices as Process Theories
 5.3Hilbert Spaces
 5.3.1Linear Maps and Hilbert Spaces from Diagrams 5.3.2Positivity from Conjugation
 5.3.3Why Mathematicians Love Complex Numbers
 5.3.4Classical Logic Gates as Linear Maps
 5.3.5The X-Basis and the Hadamard Linear Map
 5.3.6Bell Basis and Bell Maps
 5.4Hilbert Spaces versus Diagrams
 5.4.1String Diagrams Are Complete for Linear Maps
 5.4.2The Set-Theoretic Definition of Hilbert Spaces
 5.5Summary: What to Remember
 5.6Advanced Material*
 5.6.1Beyond Finite Dimensions*
 5.6.2Categories with Sums and Bases*
 5.6.3Sums in Knot Theory*
 5.6.4Equivalence of Symmetric Monoidal Categories*
 5.7Historical Notes and References
 6Quantum Processes
 6.1Pure Quantum Maps from Doubling
 6.1.1Doubling Generates Probabilities
 6.1.2Doubling Eliminates Global Phases
 6.1.3The Process Theory of Pure Quantum Maps
 6.1.4Things Preserved by Doubling
 6.1.5Things Not Preserved by Doubling
 6.2Quantum Maps from Discarding
 6.2.1Discarding
 6.2.2Impurity
 6.2.3Weight and Causality for Quantum States
 6.2.4The Process Theory of Quantum Maps
 6.2.5Causality for Quantum Maps
 6.2.6Isometry and Unitarity from Causality
 6.2.7Kraus Decomposition and Mixing
 6.2.8The No-Broadcasting Theorem
 6.3Relativity in Process Theories
 6.3.1Causal Structure
 6.3.2Causality Implies Non-signalling
 6.3.3Causality and Covariance
 6.4Quantum Processes
 6.4.1Non-deterministic Quantum Processes
 6.4.2Non-deterministic Realisation of All Quantum Maps
 6.4.3Purification of Quantum Processes
 6.4.4Teleportation Needs Classical Communication
 6.4.5Controlled Processes
 6.4.6Quantum Teleportation in Detail
 6.5Summary: What to Remember
 6.6Advanced Material*
 6.6.1Doubling General Process Theories*
 6.6.2Axiomatizing Doubling*
 6.6.3And Now for Something Completely Different*
 6.7Historical Notes and References
 7Quantum Measurement
 7.1ONB Measurements  7.1.1A Dodo’s Introduction to Measurement Devices
 7.1.2Demolition ONB Measurements
 7.1.3Non-demolition ONB Measurements
 7.1.4Superposition and Interference
 7.1.5The Next Best Thing to Observation
 7.2Measurement Dynamics and Quantum Protocols
 7.2.1Measurement-Induced Dynamics I: Backaction
 7.2.2Example: Gate Teleportation
 7.2.3Measurement-Induced Dynamics II: Collapse
 7.2.4Example: Entanglement Swapping
 7.3More General Species of Measurement
 7.3.1Von Neumann Measurements
 7.3.2Von Neumann’s Quantum Formalism
 7.3.3POVM Measurements
 7.3.4Naimark and Ozawa Dilation
 7.4Tomography
 7.4.1State Tomography
 7.4.2Informationally Complete Measurements
 7.4.3Local Tomography = Process Tomography
 7.5Summary: What to Remember
 7.6Advanced Material*
 7.6.1Do Quantum Measurements Even Exist?*
 7.6.2Projectors and Quantum Logic*
 7.6.3Failure of Local Tomography*
 7.7Historical Notes and References
 8Picturing Classical-Quantum Processes
 8.1Classical Systems as Wires
 8.1.1Double versus Single Wires
 8.1.2Example: Dense Coding
 8.1.3Measurement and Encoding
 8.1.4Classical-Quantum Maps
 8.1.5Deleting and Causality
 8.2Classical Maps from Spiders
 8.2.1Classical Maps
 8.2.2Copying and Deleting
 8.2.3Spiders
 8.2.4If It behaves like a Spider It Is One
 8.2.5All Linear Maps as Spiders + Isometries
 8.2.6Spider Diagrams and Completeness
 8.3Quantum Maps from Spiders
 8.3.1Measuring and Encoding as Spiders
 8.3.2Decoherence
 8.3.3Classical, Quantum, and Bastard Spiders
 8.3.4Mixing with Spiders
 8.3.5Entanglement for Impure States
 8.4Measurements and Protocols with Spiders
 8.4.1ONB Measurements
 8.4.2Controlled Unitaries
 8.4.3Teleportation
 8.4.4Dense coding
 8.4.5Entanglement Swapping 8.4.6Von Neumann Measurements
 8.4.7POVMs and Naimark Dilation
 8.5Summary: What to Remember
 8.6Advanced Material*
 8.6.1Spiders Are Frobenius Algebras*
 8.6.2Non-commutative Spiders*
 8.6.3Hairy Spiders*
 8.6.4Spiders as Words*
 8.7Historical Notes and References
 9Picturing Phases and Complementarity
 9.1Decorated Spiders
 9.1.1Unbiasedness and Phase States
 9.1.2Phase Spiders
 9.1.3Phase Spider Fusion
 9.1.4The Phase Group
 9.1.5Phase Gates
 9.2Multicoloured Spiders
 9.2.1Complementary Spiders
 9.2.2Complementarity and Unbiasedness
 9.2.3The CNOT-Gate from Complementarity
 9.2.4‘Colours’ of Classical Data
 9.2.5Complementary Measurements
 9.2.6Quantum Key Distribution
 9.2.7Teleportation with Complementary Measurements
 9.3Strong Complementarity
 9.3.1The Missing Rules
 9.3.2Monogamy of Strong Complementarity
 9.3.3Faces of Strong Complementarity
 9.3.4The Classical Subgroup
 9.3.5Parity Maps from Spiders
 9.3.6Classifying Strong Complementarity
 9.4ZX-Calculus
 9.4.1ZX-Diagrams Are Universal
 9.4.2ZX-Calculus for Clifford Diagrams
 9.4.3ZX for Dodos: Just Diagrams, Nothing Else
 9.4.4ZX for Pros: Build Your Own Calculus
 9.4.5ZX for the God(esse)s: Completeness
 9.4.6Where We Stand with Full ZX-Calculus
 9.5Summary: What to Remember
 9.6Advanced Material*
 9.6.1Strongly Complementary Spiders Are Hopf Algebras*
 9.6.2Strong Complementarity and Normal Forms*
 9.7Historical Notes and References
 10Quantum Theory: The Full Picture
 10.1The Diagrams
 10.1.1Circuit Diagrams
 10.1.2String Diagrams
 10.1.3Doubled Diagrams
 10.1.4Spider Diagrams
 10.1.5ZX-Diagrams 10.2The Processes
 10.2.1Causality
 10.2.2Process Decomposition and No-Broadcasting
 10.2.3Examples
 10.3The Laws
 10.3.1Complementarity
 10.3.2Strong Complementarity
 10.3.3ZX-Calculus
 10.4Historical Notes and References
 11Quantum Foundations
 11.1Quantum Non-locality
 11.1.1Refinements of Quantum Theory
 11.1.2GHZ-Mermin Scenarios
 11.1.3Drawing a Contradiction
 11.2Quantum-like Process Theories
 11.2.1Complementarity in relations
 11.2.2Spekkens’ Toy Quantum Theory
 11.2.3Phases in spek
 11.2.4ZX-Calculus for spek
 11.2.5Non-locality in spek?
 11.3Summary: What to Remember
 11.4Historical Notes and References
 12Quantum Computation
 12.1The Circuit Model
 12.1.1Quantum Computing as ZX-Diagrams
 12.1.2Building Quantum Gates as ZX-Diagrams
 12.1.3Circuit Universality
 12.2Quantum Algorithms
 12.2.1A Quantum Oracle’s (False?) Magic
 12.2.2The Deutsch–Jozsa Algorithm
 12.2.3Quantum Search
 12.2.4The Hidden Subgroup Problem
 12.3Measurement-Based Quantum Computation
 12.3.1Graph States and Cluster States
 12.3.2Measuring Graph States
 12.3.3Feed-Forward
 12.3.4Feed-Forward with Classical Wires
 12.3.5Universality
 12.4Summary: What to Remember
 12.5Historical Notes and References
 13Quantum Resources
 13.1Resource Theories
 13.1.1Free Processes
 13.1.2Comparing Resources
 13.1.3Measuring Resources
 13.2Purity Theory
 13.2.1Comparing Purity
 13.2.2Measuring (Im)purity
 13.3Entanglement Theory 3.3.1LOCC Entanglement
 13.3.2SLOCC Entanglement
 13.3.3Exploding Spiders
 13.3.4Back to Basics: Arithmetic
 13.4Summary: What to Remember
 13.5Historical Notes and References
 14Quantomatic
 14.1Taking Quantomatic for a Spin
 14.2!-Boxes: Replacing the ‘Dot, Dot, Dot’
 14.3Synthesising Physical Theories
 14.4Historical Notes and References
 Appendix Some Notations
 References
 Index