A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 1 / 54
Categorical Quantum Mechanics: The
“Monoidal” Approach
Samson Abramsky
Introduction
Introduction• CategoricalApproaches to Physics
• Some distinctivefeatures
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach
Categorical Approaches to Physics
Introduction• CategoricalApproaches to Physics
• Some distinctivefeatures
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 3 / 54
Categorical Approaches to Physics
Introduction• CategoricalApproaches to Physics
• Some distinctivefeatures
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 3 / 54
• Crane, Baez, Dolan et al. Higher-dimensional categories, TQFT’s,
categorification, etc.
Categorical Approaches to Physics
Introduction• CategoricalApproaches to Physics
• Some distinctivefeatures
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 3 / 54
• Crane, Baez, Dolan et al. Higher-dimensional categories, TQFT’s,
categorification, etc.
• Abramsky, Coecke et al. Dagger compact monoidal categories etc.
Categorical Approaches to Physics
Introduction• CategoricalApproaches to Physics
• Some distinctivefeatures
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 3 / 54
• Crane, Baez, Dolan et al. Higher-dimensional categories, TQFT’s,
categorification, etc.
• Abramsky, Coecke et al. Dagger compact monoidal categories etc.
• Isham, Doring, Butterfield et al. The topos-theoretic approach.
Categorical Approaches to Physics
Introduction• CategoricalApproaches to Physics
• Some distinctivefeatures
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 3 / 54
• Crane, Baez, Dolan et al. Higher-dimensional categories, TQFT’s,
categorification, etc.
• Abramsky, Coecke et al. Dagger compact monoidal categories etc.
• Isham, Doring, Butterfield et al. The topos-theoretic approach.
We aim to give an introduction to the “monoidal” approach, which could
as well be called “linear”.
Some distinctive features
Introduction• CategoricalApproaches to Physics
• Some distinctivefeatures
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 4 / 54
Some distinctive features
Introduction• CategoricalApproaches to Physics
• Some distinctivefeatures
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 4 / 54
Comparison with topos approach:
monoidal vs. cartesian
linear vs. intuitionisticprocesses vs. propositions
geometry of proofs vs. geometric logic
Some distinctive features
Introduction• CategoricalApproaches to Physics
• Some distinctivefeatures
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 4 / 54
Comparison with topos approach:
monoidal vs. cartesian
linear vs. intuitionisticprocesses vs. propositions
geometry of proofs vs. geometric logic
Comparison with n-categories approach. We emphasize:
• operational aspects
• interplay of quantum and classical
• compositionality• open vs. closed systems.
Some distinctive features
Introduction• CategoricalApproaches to Physics
• Some distinctivefeatures
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 4 / 54
Comparison with topos approach:
monoidal vs. cartesian
linear vs. intuitionisticprocesses vs. propositions
geometry of proofs vs. geometric logic
Comparison with n-categories approach. We emphasize:
• operational aspects
• interplay of quantum and classical
• compositionality• open vs. closed systems.
These are important for applications to quantum informatics, but also of
foundational significance.
Quantum Operations/QuantumProcesses
Introduction
QuantumOperations/QuantumProcesses• Bits and Qubits:Classical vs. QuantumOperations• ‘Truth makes anangle with reality’
• QuantumEntanglement
• Towards a generalformalism fordescribing physicalprocesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach
Bits and Qubits: Classical vs. Quantum Operations
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 6 / 54
Bits and Qubits: Classical vs. Quantum Operations
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 6 / 54
Bits :
• have two values 0, 1
Bits and Qubits: Classical vs. Quantum Operations
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 6 / 54
Bits :
• have two values 0, 1• are freely readable and duplicable
Bits and Qubits: Classical vs. Quantum Operations
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 6 / 54
Bits :
• have two values 0, 1• are freely readable and duplicable
• admit arbitrary data transformations
Bits and Qubits: Classical vs. Quantum Operations
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 6 / 54
Bits :
• have two values 0, 1• are freely readable and duplicable
• admit arbitrary data transformations
Qubits :
• have a ‘sphere’ of values spanned by |0〉, |1〉
Bits and Qubits: Classical vs. Quantum Operations
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 6 / 54
Bits :
• have two values 0, 1• are freely readable and duplicable
• admit arbitrary data transformations
Qubits :
• have a ‘sphere’ of values spanned by |0〉, |1〉
|+〉
|−〉
|+〉
|−〉
|Ψ〉
Bits and Qubits: Classical vs. Quantum Operations
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 6 / 54
Bits :
• have two values 0, 1• are freely readable and duplicable
• admit arbitrary data transformations
Qubits :
• have a ‘sphere’ of values spanned by |0〉, |1〉
|+〉
|−〉
|+〉
|−〉
|Ψ〉
• measurements of qubits
• have two outcomes |−〉, |+〉• change the value |ψ〉
Bits and Qubits: Classical vs. Quantum Operations
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 6 / 54
Bits :
• have two values 0, 1• are freely readable and duplicable
• admit arbitrary data transformations
Qubits :
• have a ‘sphere’ of values spanned by |0〉, |1〉
|+〉
|−〉
|+〉
|−〉
|Ψ〉
• measurements of qubits
• have two outcomes |−〉, |+〉• change the value |ψ〉
• admit unitary transformations , i.e. antipodes and angles are preserved.
‘Truth makes an angle with reality’
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 7 / 54
|+〉
|−〉
|Ψ〉
θ+
θ− We have partial constant maps Q → Q on thesphere Q
P+ : |ψ〉 7→ |+〉 P− : |ψ〉 7→ |−〉
which have chance prob(θ♯) for ♯ ∈ {+,−}.We know the value after the measurement, but not
what the qubit was before the measurement!
So measurements change the state, and apparently lose information. This seems
like bad news! However . . .
Quantum Entanglement
Introduction
QuantumOperations/QuantumProcesses• Bits and Qubits:Classical vs. QuantumOperations• ‘Truth makes anangle with reality’
• QuantumEntanglement
• Towards a generalformalism fordescribing physicalprocesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 8 / 54
Bell state:|00〉+ |11〉
EPR state:|01〉+ |10〉
Quantum Entanglement
Introduction
QuantumOperations/QuantumProcesses• Bits and Qubits:Classical vs. QuantumOperations• ‘Truth makes anangle with reality’
• QuantumEntanglement
• Towards a generalformalism fordescribing physicalprocesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 8 / 54
Bell state:|00〉+ |11〉
EPR state:|01〉+ |10〉
Compound systems are represented by tensor product : H1 ⊗H2.
Typical element:∑
i
λi · φi ⊗ ψi
Superposition encodes correlation . Einstein’s ‘spooky action at a
distance’. Even if the particles are spatially separated, measuring one
has an effect on the state of the other.
Bell’s theorem: QM is essentially non-local .
Towards a general formalism for describing physical proces ses
Introduction
QuantumOperations/QuantumProcesses• Bits and Qubits:Classical vs. QuantumOperations• ‘Truth makes anangle with reality’
• QuantumEntanglement
• Towards a generalformalism fordescribing physicalprocesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 9 / 54
Towards a general formalism for describing physical proces ses
Introduction
QuantumOperations/QuantumProcesses• Bits and Qubits:Classical vs. QuantumOperations• ‘Truth makes anangle with reality’
• QuantumEntanglement
• Towards a generalformalism fordescribing physicalprocesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 9 / 54
Desiderata:
Towards a general formalism for describing physical proces ses
Introduction
QuantumOperations/QuantumProcesses• Bits and Qubits:Classical vs. QuantumOperations• ‘Truth makes anangle with reality’
• QuantumEntanglement
• Towards a generalformalism fordescribing physicalprocesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 9 / 54
Desiderata:
• Following operational/process philosophy, the structure should be in
the operations/actions, not in the “elements”.
Towards a general formalism for describing physical proces ses
Introduction
QuantumOperations/QuantumProcesses• Bits and Qubits:Classical vs. QuantumOperations• ‘Truth makes anangle with reality’
• QuantumEntanglement
• Towards a generalformalism fordescribing physicalprocesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 9 / 54
Desiderata:
• Following operational/process philosophy, the structure should be in
the operations/actions, not in the “elements”.
• Operations should be typed .
Towards a general formalism for describing physical proces ses
Introduction
QuantumOperations/QuantumProcesses• Bits and Qubits:Classical vs. QuantumOperations• ‘Truth makes anangle with reality’
• QuantumEntanglement
• Towards a generalformalism fordescribing physicalprocesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 9 / 54
Desiderata:
• Following operational/process philosophy, the structure should be in
the operations/actions, not in the “elements”.
• Operations should be typed .
• Basic reflection of time: must have sequential composition of
operations.
Towards a general formalism for describing physical proces ses
Introduction
QuantumOperations/QuantumProcesses• Bits and Qubits:Classical vs. QuantumOperations• ‘Truth makes anangle with reality’
• QuantumEntanglement
• Towards a generalformalism fordescribing physicalprocesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 9 / 54
Desiderata:
• Following operational/process philosophy, the structure should be in
the operations/actions, not in the “elements”.
• Operations should be typed .
• Basic reflection of time: must have sequential composition of
operations.
• Basic reflection of space: must be able to describe compoundsystems , operations localized to part of a compound system, and
operations performed independently on different parts of a compoundsystem — parallel composition .
Towards a general formalism for describing physical proces ses
Introduction
QuantumOperations/QuantumProcesses• Bits and Qubits:Classical vs. QuantumOperations• ‘Truth makes anangle with reality’
• QuantumEntanglement
• Towards a generalformalism fordescribing physicalprocesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 9 / 54
Desiderata:
• Following operational/process philosophy, the structure should be in
the operations/actions, not in the “elements”.
• Operations should be typed .
• Basic reflection of time: must have sequential composition of
operations.
• Basic reflection of space: must be able to describe compoundsystems , operations localized to part of a compound system, and
operations performed independently on different parts of a compoundsystem — parallel composition .
So we want a general setting in which we can describe processes (of
whatever kind) closed under sequential and parallel composition.
Basic Setting: SymmetricMonoidal Categories
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
• Categories
• Symmetric MonoidalCategories
• The Logic of TensorProduct• TowardsDiagrammatics
• Boxes and Wires:Typed Operations
• Series and ParallelComposition
• Geometry absorbsFunctoriality, Naturality
• Bras, Kets andScalars
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammatic
A Survey of Categorical QM: the Monoidal Approach
Categories
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
• Categories
• Symmetric MonoidalCategories
• The Logic of TensorProduct• TowardsDiagrammatics
• Boxes and Wires:Typed Operations
• Series and ParallelComposition
• Geometry absorbsFunctoriality, Naturality
• Bras, Kets andScalars
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammatic
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 11 / 54
A category C has objects (types) A, B, C, . . . , and for each pair of
objects A, B a set of morphisms C(A,B). (Notation: f : A→ B). It
also has identities idA : A→ A, and composition g ◦ f when types
match:
Af−→ B
g−→ C
Categories allow us to deal explictly with typed processes , e.g.
Logic Programming ComputationPropositions Data Types States
Proofs Programs Transitions
For QM:
Types of system (e.g. qubits Q) objects
Processes/Operations (e.g. measurements) morphisms
Symmetric Monoidal Categories
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 12 / 54
A symmetric monoidal category comes equipped with an associative operationwhich acts on both objects and morphisms — a bifunctor :
A⊗B f1 ⊗ f2 : A1 ⊗A2 −→ B1 ⊗B2
There is also a symmetry operation
σA,B : A⊗B −→ B ⊗A
which satisfies some ‘obvious’ rules, e.g. naturality:
A1 ⊗A2f1⊗f2- B1 ⊗B2
A2 ⊗A1
σA1,A2
?f1⊗f2- B2 ⊗B1
σB1,B2
?
The Logic of Tensor Product
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 13 / 54
Tensor can express independent or concurrent actions (mathematically:
bifunctoriality):
A1 ⊗A2f1⊗id- B1 ⊗A2
A1 ⊗B2
id⊗f2
?
f1⊗id
- B1 ⊗B2
id⊗f2
?
But tensor is not a cartesian product, in the sense that we cannot reconstruct an‘element’ of the tensor from its components .This turns out to comprise the absence of
A∆−→ A⊗A A1 ⊗A2
πi−→ AiCf. A ⊢ A ∧A A1 ∧A2 ⊢ Ai
Hence monoidal categories provide a setting for resource-sensitive logics such as
Linear Logic.
The Logic of Tensor Product
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 13 / 54
Tensor can express independent or concurrent actions (mathematically:
bifunctoriality):
A1 ⊗A2f1⊗id- B1 ⊗A2
A1 ⊗B2
id⊗f2
?
f1⊗id
- B1 ⊗B2
id⊗f2
?
But tensor is not a cartesian product, in the sense that we cannot reconstruct an‘element’ of the tensor from its components .This turns out to comprise the absence of
A∆−→ A⊗A A1 ⊗A2
πi−→ AiCf. A ⊢ A ∧A A1 ∧A2 ⊢ Ai
Hence monoidal categories provide a setting for resource-sensitive logics such as
Linear Logic.
No-Cloning and No-Deleting are built in!
Towards Diagrammatics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 14 / 54
Towards Diagrammatics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 14 / 54
We have seen that any symmetric monoidal category can be viewed as a settingfor describing processes in a resource sensitive way, close d under sequentialand parallel composition
Towards Diagrammatics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 14 / 54
We have seen that any symmetric monoidal category can be viewed as a settingfor describing processes in a resource sensitive way, close d under sequentialand parallel composition
There is a natural objection to this, that this is too abstract, and we lose all grip of
what is going on.
Towards Diagrammatics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 14 / 54
We have seen that any symmetric monoidal category can be viewed as a settingfor describing processes in a resource sensitive way, close d under sequentialand parallel composition
There is a natural objection to this, that this is too abstract, and we lose all grip of
what is going on.
But this objection does not really hold water! Monoidal categories, quite generally,
admit a beautiful graphical or diagrammatic calculus (Joyal, Street et al.) whichmakes equational proofs perspicuous, and is sound and complete for equational
reasoning in monoidal categories. It also supports links with Logic (e.g. Proof Nets)
and with Geometry (Knots, Braids, Temperley-Lieb algebra etc.)
Boxes and Wires: Typed Operations
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
• Categories
• Symmetric MonoidalCategories
• The Logic of TensorProduct• TowardsDiagrammatics
• Boxes and Wires:Typed Operations
• Series and ParallelComposition
• Geometry absorbsFunctoriality, Naturality
• Bras, Kets andScalars
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammatic
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 15 / 54
f
B
A
g
· · ·B1 Bm
· · ·A1 An
f : A −→ B g : A1 ⊗ · · · ⊗An −→ B1 ⊗ · · · ⊗Bm.
Series and Parallel Composition
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
• Categories
• Symmetric MonoidalCategories
• The Logic of TensorProduct• TowardsDiagrammatics
• Boxes and Wires:Typed Operations
• Series and ParallelComposition
• Geometry absorbsFunctoriality, Naturality
• Bras, Kets andScalars
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammatic
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 16 / 54
g
· · ·C1 Cr
f
· · ·A1 An
B1 Bm· · · f
· · ·B1 Bm
· · ·A1 An
h
· · ·D1 Ds
· · ·C1 Cr
g ◦ f
f ⊗ h
Geometry absorbs Functoriality, Naturality
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
• Categories
• Symmetric MonoidalCategories
• The Logic of TensorProduct• TowardsDiagrammatics
• Boxes and Wires:Typed Operations
• Series and ParallelComposition
• Geometry absorbsFunctoriality, Naturality
• Bras, Kets andScalars
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammatic
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 17 / 54
f
g
=
f
g
(f ⊗ 1) ◦ (1⊗ g) = f ⊗ g = (1⊗ g) ◦ (f ⊗ 1)
Bras, Kets and Scalars
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 18 / 54
φ
· · ·A1 An
ψ
· · ·A1 An
s
φ : A1 ⊗ · · · ⊗An −→ I ψ : I −→ A1 ⊗ · · · ⊗An s : I −→ I.
Bras: no outputs
Kets: no inputs
Scalars: no inputs or outputs.
Bras, Kets and Scalars
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 18 / 54
φ
· · ·A1 An
ψ
· · ·A1 An
s
φ : A1 ⊗ · · · ⊗An −→ I ψ : I −→ A1 ⊗ · · · ⊗An s : I −→ I.
Bras: no outputs
Kets: no inputs
Scalars: no inputs or outputs.
Two-dimensional generalization of Dirac notation !
〈φ | | ψ〉 〈φ | ψ〉
Interlude: The Miracle of Scalars
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars• Scalars in monoidalcategories
• Scalar Product
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach
Scalars in monoidal categories
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars• Scalars in monoidalcategories
• Scalar Product
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 20 / 54
A scalar in a monoidal category is a morphism s : I → I .
Examples : (FdVecK,⊗), (Rel,×).
Scalars in monoidal categories
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars• Scalars in monoidalcategories
• Scalar Product
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 20 / 54
A scalar in a monoidal category is a morphism s : I → I .
Examples : (FdVecK,⊗), (Rel,×).
(1) C(I, I) is a commutative monoid
Iρ−1
I - I ⊗ I ==== I ⊗ IλI - I
I
s
6
ρ−1
I - I ⊗ I
s⊗1
6
s⊗t- I ⊗ I
1⊗t
?λI - I
t
?
I
t
?
λ−1
I
- I ⊗ I
1⊗t
?==== I ⊗ I
s⊗1
6
ρI
- I
s
6
using the coherence equation λI = ρI .
Scalar Product
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars• Scalars in monoidalcategories
• Scalar Product
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 21 / 54
(2) Each scalar s : I → I induces a natural transformation
sA : A≃- I ⊗A
s⊗1A- I ⊗A≃- A .
AsA - A
B
f
?
sB
- B
f
?
We write s • f for f ◦ sA = sB ◦ f . Note that
s • (t • f) = (s ◦ t) • f(s • g) ◦ (r • f) = (s ◦ r) • (g ◦ f)(s • f)⊗ (t • g) = (s ◦ t) • (f ⊗ g)
Dagger Monoidal Categories
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
• Adjoint Arrows —reflection in the x-axis
• Unitaries
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach
Adjoint Arrows — reflection in the x-axis
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 23 / 54
f
· · ·A1 An
· · ·B1 Bm
f †
· · ·B1 Bm
· · ·A1 An
f : A→ B
f † : B → Af †† = f (g ◦ f)† = f † ◦ g†
Adjoint Arrows — reflection in the x-axis
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 23 / 54
f
· · ·A1 An
· · ·B1 Bm
f †
· · ·B1 Bm
· · ·A1 An
f : A→ B
f † : B → Af †† = f (g ◦ f)† = f † ◦ g†
We can turn kets into bras and vice versa — full scale Dirac notation! Givenφ, ψ : I −→ A,
〈φ | ψ〉 = φ† ◦ ψ : I −→ I
which is indeed a scalar!
Unitaries
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
• Adjoint Arrows —reflection in the x-axis
• Unitaries
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 24 / 54
We can also define unitaries . An isomorphism U : A∼=−→ B is unitary if
U−1 = U †.
Unitaries
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
• Adjoint Arrows —reflection in the x-axis
• Unitaries
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 24 / 54
We can also define unitaries . An isomorphism U : A∼=−→ B is unitary if
U−1 = U †.
A dagger monoidal category is one in which the canonical isomorphisms
for the monoidal structure are unitary.
Unitaries
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
• Adjoint Arrows —reflection in the x-axis
• Unitaries
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 24 / 54
We can also define unitaries . An isomorphism U : A∼=−→ B is unitary if
U−1 = U †.
A dagger monoidal category is one in which the canonical isomorphisms
for the monoidal structure are unitary.
Examples: Hilb , Rel.
Entanglement, Bell States andCompact Categories
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
• From ‘paradox’ to‘feature’: Teleportation• Entangled states aslinear maps
• Some notation forprojectors
• On the trail ofstructure• Teleportation: basiccase• Teleportation:general case
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow in
A Survey of Categorical QM: the Monoidal Approach
From ‘paradox’ to ‘feature’: Teleportation
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
• From ‘paradox’ to‘feature’: Teleportation• Entangled states aslinear maps
• Some notation forprojectors
• On the trail ofstructure• Teleportation: basiccase• Teleportation:general case
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow in
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 26 / 54
MBell
Ux
|00〉+ |11〉
x ∈ B2
|φ〉
|φ〉
Entangled states as linear maps
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 27 / 54
Indeed,H1 ⊗H2 is spanned by
|11〉 · · · |1m〉...
. . ....
|n1〉 · · · |nm〉
hence
∑
i,j
αij |ij〉 ←→
α11 · · · α1m...
. . ....
αn1 · · · αnm
←→ |i〉 7→
∑
j
αij |j〉
Pairs |ψ1, ψ2〉 are a special case — |ij〉 in a well-chosen basis.
This is Map-State Duality .
Some notation for projectors
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
• From ‘paradox’ to‘feature’: Teleportation• Entangled states aslinear maps
• Some notation forprojectors
• On the trail ofstructure• Teleportation: basiccase• Teleportation:general case
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow in
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 28 / 54
A projector onto the 1-dimensional subspace spanned by a vector |ψ〉 will
be written Pψ. It is essentially (up to scalar multiples) a “partial constant
map”
Pψ : |φ〉 7→ |ψ〉.
This will correspond e.g. to a branch of a (projective, non-degenerate)measurement, or to a preparation.
Some notation for projectors
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
• From ‘paradox’ to‘feature’: Teleportation• Entangled states aslinear maps
• Some notation forprojectors
• On the trail ofstructure• Teleportation: basiccase• Teleportation:general case
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow in
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 28 / 54
A projector onto the 1-dimensional subspace spanned by a vector |ψ〉 will
be written Pψ. It is essentially (up to scalar multiples) a “partial constant
map”
Pψ : |φ〉 7→ |ψ〉.
This will correspond e.g. to a branch of a (projective, non-degenerate)measurement, or to a preparation.
We combine this notation with Map-State Duality: we write a projector Pψon a tensor product spaceH1 ⊗H2 as Pf , where f is the linear map
H1 →H2 associated to ψ under Map-State Duality.
On the trail of structure
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 29 / 54
The identity map
|id : Q→ Q〉 ∈ Q⊗Q |11〉+ · · ·+ |nn〉 ←→ |i〉 7→ |i〉
is the Bell state .
A measurement of Q⊗Q has four outcomes
|f1〉, |f2〉, |f3〉, |f4〉 (cf. |00〉, |01〉, |10〉, |11〉)
and corresponding projectors
Pf : Q⊗Q→ Q⊗Q :: |g〉 7→ |f〉
E.g. the Bell state is produced by
Pid : Q⊗Q→ Q⊗Q :: |g〉 7→ |id〉
Key Question: Do entangled states qua functions compose (somehow)?
Teleportation: basic case
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
• From ‘paradox’ to‘feature’: Teleportation• Entangled states aslinear maps
• Some notation forprojectors
• On the trail ofstructure• Teleportation: basiccase• Teleportation:general case
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow in
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 30 / 54
id
id
id ◦ id = id
Teleportation: general case
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
• From ‘paradox’ to‘feature’: Teleportation• Entangled states aslinear maps
• Some notation forprojectors
• On the trail ofstructure• Teleportation: basiccase• Teleportation:general case
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow in
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 31 / 54
βi
id
β−1i
β−1i ◦ id ◦ βi = id, 1 ≤ i ≤ 4
Categorical Axiomatics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
• Axiomatizing BellStates
• Cups and Caps
• Graphical Calculusfor Information Flow• Names andConames in theGraphical Calculus
• Definition of Duality
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements and
A Survey of Categorical QM: the Monoidal Approach
Axiomatizing Bell States
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 33 / 54
Axiomatizing Bell States
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 33 / 54
We can axiomatize Bell states and costates in a very direct and simple form in the
setting of dagger monoidal categories, yielding all the structure needed todescribe and reason about (bipartite) entanglement .
Axiomatizing Bell States
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 33 / 54
We can axiomatize Bell states and costates in a very direct and simple form in the
setting of dagger monoidal categories, yielding all the structure needed todescribe and reason about (bipartite) entanglement .
Moreover, this structure is very canonical mathematically:
Axiomatizing Bell States
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 33 / 54
We can axiomatize Bell states and costates in a very direct and simple form in the
setting of dagger monoidal categories, yielding all the structure needed todescribe and reason about (bipartite) entanglement .
Moreover, this structure is very canonical mathematically:
• Algebraically, it gives exactly the structure of compact categories ; the
fundamental laws are the triangular identities for adjunctions.
Axiomatizing Bell States
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 33 / 54
We can axiomatize Bell states and costates in a very direct and simple form in the
setting of dagger monoidal categories, yielding all the structure needed todescribe and reason about (bipartite) entanglement .
Moreover, this structure is very canonical mathematically:
• Algebraically, it gives exactly the structure of compact categories ; the
fundamental laws are the triangular identities for adjunctions.
• Logically, Bell states correspnd to axiom links and costates to cut links , and we
can track entangled quantum information flows essentially by Cut-elimination .
Axiomatizing Bell States
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 33 / 54
We can axiomatize Bell states and costates in a very direct and simple form in the
setting of dagger monoidal categories, yielding all the structure needed todescribe and reason about (bipartite) entanglement .
Moreover, this structure is very canonical mathematically:
• Algebraically, it gives exactly the structure of compact categories ; the
fundamental laws are the triangular identities for adjunctions.
• Logically, Bell states correspnd to axiom links and costates to cut links , and we
can track entangled quantum information flows essentially by Cut-elimination .
• Diagrammatically, these are basic geometric simplifications: planar versions
yield the Temperley-Lieb algebra .
Axiomatizing Bell States
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 33 / 54
We can axiomatize Bell states and costates in a very direct and simple form in the
setting of dagger monoidal categories, yielding all the structure needed todescribe and reason about (bipartite) entanglement .
Moreover, this structure is very canonical mathematically:
• Algebraically, it gives exactly the structure of compact categories ; the
fundamental laws are the triangular identities for adjunctions.
• Logically, Bell states correspnd to axiom links and costates to cut links , and we
can track entangled quantum information flows essentially by Cut-elimination .
• Diagrammatically, these are basic geometric simplifications: planar versions
yield the Temperley-Lieb algebra .
• From this structure we can define the trace and partial trace with all the key
algebraic properties.
Axiomatizing Bell States
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 33 / 54
We can axiomatize Bell states and costates in a very direct and simple form in the
setting of dagger monoidal categories, yielding all the structure needed todescribe and reason about (bipartite) entanglement .
Moreover, this structure is very canonical mathematically:
• Algebraically, it gives exactly the structure of compact categories ; the
fundamental laws are the triangular identities for adjunctions.
• Logically, Bell states correspnd to axiom links and costates to cut links , and we
can track entangled quantum information flows essentially by Cut-elimination .
• Diagrammatically, these are basic geometric simplifications: planar versions
yield the Temperley-Lieb algebra .
• From this structure we can define the trace and partial trace with all the key
algebraic properties.
• Diagrammatically, the (partial) trace closes (part of) the system; when we closethe whole system we get loops — i.e. scalars!
Cups and Caps
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
• Axiomatizing BellStates
• Cups and Caps
• Graphical Calculusfor Information Flow• Names andConames in theGraphical Calculus
• Definition of Duality
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements and
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 34 / 54
A∗A
A∗ A
ǫA : A⊗A∗ −→ I ηA : I −→ A∗ ⊗A.
Caps = Bell States; Cups = Bell Tests.
Graphical Calculus for Information Flow
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 35 / 54
Compact Closure : The basic algebraic laws for units and counits.
= =
(ǫA ⊗ 1A) ◦ (1A ⊗ ηA) = 1A (1A∗ ⊗ ǫA) ◦ (ηA ⊗ 1A∗) = 1A∗
For coherence with the dagger structure, we require that ǫA = η†A.
Names and Conames in the Graphical Calculus
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
• Axiomatizing BellStates
• Cups and Caps
• Graphical Calculusfor Information Flow• Names andConames in theGraphical Calculus
• Definition of Duality
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements and
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 36 / 54
f
g
xfy : A⊗B∗ → I pfq : I → A∗ ⊗B
C(A⊗B∗, I) ≃ C(A,B) ≃ C(I, A∗ ⊗B).
This is the general form of Map-State duality.
Definition of Duality
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
• Axiomatizing BellStates
• Cups and Caps
• Graphical Calculusfor Information Flow• Names andConames in theGraphical Calculus
• Definition of Duality
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements and
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 37 / 54
f
A∗
A
B
B∗
f∗ = (1⊗ ǫB) ◦ (1⊗ f ⊗ 1) ◦ (ηA ⊗ 1).
A Little Taste of DiagrammaticReasoning
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
• Duality is Involutive• Moving Boxes roundCups and Caps
• FeedbackDinaturality
• Application:Invariance of TraceUnder CyclicPermutations• Graphical Proof ofFeedback Dinaturality
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach
Duality is Involutive
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
• Duality is Involutive• Moving Boxes roundCups and Caps
• FeedbackDinaturality
• Application:Invariance of TraceUnder CyclicPermutations• Graphical Proof ofFeedback Dinaturality
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 39 / 54
f
A
B
A
B
A∗
B∗
= f
A
B
f∗∗ = f.
Moving Boxes round Cups and Caps
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 40 / 54
f
B
A A∗
= f∗
A
B∗B
f
A
B B∗
= f∗B∗
A A∗
Diagrammatic Proof
f∗B∗
A A∗
∆=
f
AA∗A
B B∗
= f
A
B B∗
Feedback Dinaturality
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
• Duality is Involutive• Moving Boxes roundCups and Caps
• FeedbackDinaturality
• Application:Invariance of TraceUnder CyclicPermutations• Graphical Proof ofFeedback Dinaturality
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 41 / 54
f
g
=
f
g
Application: Invariance of Trace Under Cyclic Permutation s
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
• Duality is Involutive• Moving Boxes roundCups and Caps
• FeedbackDinaturality
• Application:Invariance of TraceUnder CyclicPermutations• Graphical Proof ofFeedback Dinaturality
Illuminating QuantumInformation Flow inEntangled Systems
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 42 / 54
f
g
=
g
f
Graphical Proof of Feedback Dinaturality
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 43 / 54
f
g
=f
g∗
=f
g∗
=f
g∗∗
We use g∗∗ = g to conclude.
Illuminating Quantum InformationFlow in Entangled Systems
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
• Bipartite Projectors
• ProjectorsDecomposed
• Compositionality
• Compositionality ctd
• Compositionality ctd
• Teleportationdiagrammatically
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach
Bipartite Projectors
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
• Bipartite Projectors
• ProjectorsDecomposed
• Compositionality
• Compositionality ctd
• Compositionality ctd
• Teleportationdiagrammatically
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 45 / 54
Information flow in entangled states can be captured mathematically by
the isomorphism
Hom(A,B) ∼= A∗ ⊗B.
This leads to a decomposition of bipartite projectors into “names”(preparations) and “conames” (measurements).
In graphical notation:
f
ff †
f †
Projectors Decomposed
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
• Bipartite Projectors
• ProjectorsDecomposed
• Compositionality
• Compositionality ctd
• Compositionality ctd
• Teleportationdiagrammatically
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 46 / 54
f †
f
BA∗
A∗ B
Compositionality
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
• Bipartite Projectors
• ProjectorsDecomposed
• Compositionality
• Compositionality ctd
• Compositionality ctd
• Teleportationdiagrammatically
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 47 / 54
The key algebraic fact from which teleportation (and many other
protocols) can be derived.
f
g
=
f
g
Compositionality ctd
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
• Bipartite Projectors
• ProjectorsDecomposed
• Compositionality
• Compositionality ctd
• Compositionality ctd
• Teleportationdiagrammatically
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 48 / 54
f
g
=
Compositionality ctd
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
• Bipartite Projectors
• ProjectorsDecomposed
• Compositionality
• Compositionality ctd
• Compositionality ctd
• Teleportationdiagrammatically
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 49 / 54
f
g
=
g
f
Teleportation diagrammatically
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
• Bipartite Projectors
• ProjectorsDecomposed
• Compositionality
• Compositionality ctd
• Compositionality ctd
• Teleportationdiagrammatically
Measurements andClassical Information
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 50 / 54
βi
β−1i
=
βi
β−1i
=
Measurements and ClassicalInformation
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach
First Approach: Biproducts
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 52 / 54
Suppose we assume biproducts (need only assume products: Robin
Houston) in a dagger compact category.
These can be used to represent branching on measurementoutcomes :
AM - ⊕n
i=1Ai
Lni=1
Ui - ⊕ni=1Bi
Propagation of the outcome of a measurement performed on one part ofa compound system to other parts — “classical communication” — can
be expressed using distributivity :
(A1 ⊕A2)⊗B ∼= (A1 ⊗B)⊕ (A2 ⊗B).
Teleportation categorically
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 53 / 54
Q(1⊗η)
(1)- Q⊗Q⊗Q
〈xUiy〉4
i=1⊗1
(2)- ⊕4
i=1 I ⊗Qdist(3)- ⊕4
i=1Q
L
4
i=1U−1
i
(4)- ⊕4
i=1Q
Q
w
w
w
w
w
w
w
w
〈1〉i=4
i=1
- ⊕4i=1Q
w
w
w
w
w
w
w
(1) Produce EPR pair(2) Perform measurement in Bell-basis(3) Propagate classical information(4) Perform unitary correction .
Teleportation categorically
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 53 / 54
Q(1⊗η)
(1)- Q⊗Q⊗Q
〈xUiy〉4
i=1⊗1
(2)- ⊕4
i=1 I ⊗Qdist(3)- ⊕4
i=1Q
L
4
i=1U−1
i
(4)- ⊕4
i=1Q
Q
w
w
w
w
w
w
w
w
〈1〉i=4
i=1
- ⊕4i=1Q
w
w
w
w
w
w
w
(1) Produce EPR pair(2) Perform measurement in Bell-basis(3) Propagate classical information(4) Perform unitary correction .
N.B. Alternative approach (which brings important new structure to light): classicalstructures (Coecke and Pavlovic), i.e. Frobenius dagger-algebras.
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
• Classical Structures
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
• Classical Structures
• Axiomatics of No-Cloning and No-Deleting
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
• Classical Structures
• Axiomatics of No-Cloning and No-Deleting
• CPM construction (Selinger): density operators and completely
positive maps in the abstract setting
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
• Classical Structures
• Axiomatics of No-Cloning and No-Deleting
• CPM construction (Selinger): density operators and completely
positive maps in the abstract setting
• Connections to Temperley-Lieb algebra etc.
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
• Classical Structures
• Axiomatics of No-Cloning and No-Deleting
• CPM construction (Selinger): density operators and completely
positive maps in the abstract setting
• Connections to Temperley-Lieb algebra etc.
• Categorical Quantum Logic (A and Duncan)
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
• Classical Structures
• Axiomatics of No-Cloning and No-Deleting
• CPM construction (Selinger): density operators and completely
positive maps in the abstract setting
• Connections to Temperley-Lieb algebra etc.
• Categorical Quantum Logic (A and Duncan)
• Fock space (Vicary)
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
• Classical Structures
• Axiomatics of No-Cloning and No-Deleting
• CPM construction (Selinger): density operators and completely
positive maps in the abstract setting
• Connections to Temperley-Lieb algebra etc.
• Categorical Quantum Logic (A and Duncan)
• Fock space (Vicary)
• Representation Theorems
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
• Classical Structures
• Axiomatics of No-Cloning and No-Deleting
• CPM construction (Selinger): density operators and completely
positive maps in the abstract setting
• Connections to Temperley-Lieb algebra etc.
• Categorical Quantum Logic (A and Duncan)
• Fock space (Vicary)
• Representation Theorems
• Planarity, braiding, toplogical quantum computing
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
• Classical Structures
• Axiomatics of No-Cloning and No-Deleting
• CPM construction (Selinger): density operators and completely
positive maps in the abstract setting
• Connections to Temperley-Lieb algebra etc.
• Categorical Quantum Logic (A and Duncan)
• Fock space (Vicary)
• Representation Theorems
• Planarity, braiding, toplogical quantum computing
• Differential Categories?
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
• Classical Structures
• Axiomatics of No-Cloning and No-Deleting
• CPM construction (Selinger): density operators and completely
positive maps in the abstract setting
• Connections to Temperley-Lieb algebra etc.
• Categorical Quantum Logic (A and Duncan)
• Fock space (Vicary)
• Representation Theorems
• Planarity, braiding, toplogical quantum computing
• Differential Categories?
• Types, polarities, LL?
Further Topics
Introduction
QuantumOperations/QuantumProcesses
Basic Setting:Symmetric MonoidalCategories
Interlude: The Miracleof Scalars
Dagger MonoidalCategories
Entanglement, BellStates and CompactCategories
Categorical Axiomatics
A Little Taste ofDiagrammaticReasoning
Illuminating QuantumInformation Flow inEntangled Systems
Measurements andClassical Information• First Approach:Biproducts• Teleportationcategorically
• Further Topics
A Survey of Categorical QM: the Monoidal Approach CLFP Workshop Jan 9 2008 – 54 / 54
• Classical Structures
• Axiomatics of No-Cloning and No-Deleting
• CPM construction (Selinger): density operators and completely
positive maps in the abstract setting
• Connections to Temperley-Lieb algebra etc.
• Categorical Quantum Logic (A and Duncan)
• Fock space (Vicary)
• Representation Theorems
• Planarity, braiding, toplogical quantum computing
• Differential Categories?
• Types, polarities, LL?
• Distributed QM, “discrete QFT”??