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Complete positivity and natural representation of quantum computations QPL15 Mathys Rennela (Radboud University) Sam Staton (Oxford University) 15th July 2015 Page 1 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality
QPL’15 Mathys Rennela (Radboud University) Sam Staton (Oxford University)
15th July 2015
Page 1 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality
Outline
Types for quantum computation How to build representations of completely positive maps Application: Quantum domain theory Concluding remarks
Page 2 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality
Where we are, sofar
Types for quantum computation How to build representations of completely positive maps Application: Quantum domain theory Concluding remarks
Types as C*-algebras
Page 3 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Types as C*-algebras
◮ A type A is interpreted as a C*-algebra A.
Page 3 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Types as C*-algebras
◮ A type A is interpreted as a C*-algebra A.
(measurable quantities of a physical system).
Page 3 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Types as C*-algebras
◮ A type A is interpreted as a C*-algebra A.
(measurable quantities of a physical system). ◮ Bool: bool = C ⊕ C
Page 3 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Types as C*-algebras
◮ A type A is interpreted as a C*-algebra A.
(measurable quantities of a physical system). ◮ Bool: bool = C ⊕ C ◮ Qubit: qubit = M2 = B(C2)
Page 3 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Types as C*-algebras
◮ A type A is interpreted as a C*-algebra A.
(measurable quantities of a physical system). ◮ Bool: bool = C ⊕ C ◮ Qubit: qubit = M2 = B(C2) ◮ Tensor: x : A, y : B = A ⊗ B
Page 3 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Types as C*-algebras
◮ A type A is interpreted as a C*-algebra A.
(measurable quantities of a physical system). ◮ Bool: bool = C ⊕ C ◮ Qubit: qubit = M2 = B(C2) ◮ Tensor: x : A, y : B = A ⊗ B ◮ Void: () = C
Page 3 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Types as C*-algebras
◮ A type A is interpreted as a C*-algebra A.
(measurable quantities of a physical system). ◮ Bool: bool = C ⊕ C ◮ Qubit: qubit = M2 = B(C2) ◮ Tensor: x : A, y : B = A ⊗ B ◮ Void: () = C ◮ Natural numbers: nat = ⊕n∈N C
Page 3 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Programs as completely positive maps
◮ f = x : A ⊢ t : B : B → A (predicate transformer)
Page 4 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Programs as completely positive maps
◮ f = x : A ⊢ t : B : B → A (predicate transformer)
Page 4 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Programs as completely positive maps
◮ f = x : A ⊢ t : B : B → A (predicate transformer)
Page 4 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Programs as completely positive maps
◮ f = x : A ⊢ t : B : B → A (predicate transformer)
◮
positive element: a = x∗x for some x.
◮
Page 4 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Programs as completely positive maps
◮ f = x : A ⊢ t : B : B → A (predicate transformer)
subsystem of a bigger system.
Page 4 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Programs as completely positive maps
◮ f = x : A ⊢ t : B : B → A (predicate transformer)
subsystem of a bigger system.
◮
M2n(f ) : M2n(B) → M2n(A) positive.
Page 4 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Programs as completely positive maps
◮ f = x : A ⊢ t : B : B → A (predicate transformer)
subsystem of a bigger system.
◮
idqubit⊗n ⊗f : qubit⊗n ⊗ B → qubit⊗n ⊗ A positive.
Page 4 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Programs as completely positive maps
◮ f = x : A ⊢ t : B : B → A (predicate transformer)
subsystem of a bigger system.
◮
idqubit⊗n ⊗f : qubit⊗n ⊗ B → qubit⊗n ⊗ A positive. ◮ Complete positivity is at the core of quantum computation
Page 4 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Programs as completely positive maps
◮ f = x : A ⊢ t : B : B → A (predicate transformer)
subsystem of a bigger system.
◮
idqubit⊗n ⊗f : qubit⊗n ⊗ B → qubit⊗n ⊗ A positive. ◮ Complete positivity is at the core of quantum computation ◮ Our contribution: a method to consider complete positive maps as natural families of positive maps.
Page 4 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Types for quantum computation
Where we are, sofar
Types for quantum computation How to build representations of completely positive maps Application: Quantum domain theory Concluding remarks
What is a representation?
Page 5 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
What is a representation?
◮ Representation of C in R
F : C → R
Page 5 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
What is a representation?
◮ Representation of C in R
C(A, B) ∼ = R(F(A), F(B)) for A, B objects in C
Page 5 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
What is a representation?
◮ Representation of C in R
C(A, B) ∼ = R(F(A), F(B)) for A, B objects in C ◮ Biggest advantage: it gives more structure to types without altering the interpretation of programs.
Page 5 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
States and effects duality: the “Nijmegen triangle”
Page 6 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
States and effects duality: the “Nijmegen triangle”
transformers
transformers
Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
States and effects duality: the “Nijmegen triangle”
transformers
transformers
This view works in many settings, including probabilistic and quantum computation.
Page 6 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
States and effects duality: the “Nijmegen triangle”
transformers
transformers
This view works in many settings, including probabilistic and quantum computation. ◮ Goal: Make this view compositional for quantum computation
Page 6 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Examples of representations (for positive maps)
Page 7 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Examples of representations (for positive maps)
◮ C∗-AlgPU: category of C*-algebras and positive unital maps.
Page 7 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Examples of representations (for positive maps)
◮ C∗-AlgPU: category of C*-algebras and positive unital maps.
modules
sets
Pred
Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Examples of representations (for positive maps)
◮ C∗-AlgPU: category of C*-algebras and positive unital maps.
modules
sets
Pred
Pred and Stat are representations (i.e. full and faithful).
Page 7 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Representation for completely positive maps?
Page 8 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Representation for completely positive maps?
◮ C∗-AlgCPU: category of C*-algebras and completely positive unital maps
modules
sets
Pred
Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Representation for completely positive maps?
◮ C∗-AlgCPU: category of C*-algebras and completely positive unital maps
modules
sets
Pred
Pred and Stat are NOT representations.
Page 8 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Main theorem
Page 9 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Main theorem
◮ NIsom: category of natural numbers and isometries (i.e. matrices F ∈ Mm×n such that F ∗F = I), which induce completely positive unital maps F ∗_ F : Mm → Mn.
Page 9 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Main theorem
◮ NIsom: category of natural numbers and isometries (i.e. matrices F ∈ Mm×n such that F ∗F = I), which induce completely positive unital maps F ∗_ F : Mm → Mn. ◮ Functor M : C∗-AlgCPU → [NIsom, C∗-AlgPU]
M(A) = {Mn(A)}n
M(f : A → B) = {Mn(f ) : Mn(A) → Mn(B)}n
Page 9 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Main theorem
◮ NIsom: category of natural numbers and isometries (i.e. matrices F ∈ Mm×n such that F ∗F = I), which induce completely positive unital maps F ∗_ F : Mm → Mn. ◮ Functor M : C∗-AlgCPU → [NIsom, C∗-AlgPU]
M(A) = {Mn(A)}n
M(f : A → B) = {Mn(f ) : Mn(A) → Mn(B)}n
Theorem
The functor M : C∗-AlgCPU → [NIsom, C∗-AlgPU] yields a representation
Page 9 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Variation on the index category
Page 10 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Variation on the index category
◮ NCPU: category whose objects are natural numbers and where a morphism m → n is a completely positive unital map Mm → Mn.
Page 10 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Variation on the index category
◮ NCPU: category whose objects are natural numbers and where a morphism m → n is a completely positive unital map Mm → Mn. ◮ Functor M : C∗-AlgCPU → [NCPU, C∗-AlgPU]
M(A) = {Mn(A)}n
M(f : A → B) = {Mn(f ) : Mn(A) → Mn(B)}n
Page 10 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Variation on the index category
◮ NCPU: category whose objects are natural numbers and where a morphism m → n is a completely positive unital map Mm → Mn. ◮ Functor M : C∗-AlgCPU → [NCPU, C∗-AlgPU]
M(A) = {Mn(A)}n
M(f : A → B) = {Mn(f ) : Mn(A) → Mn(B)}n
Theorem
The functor M : C∗-AlgCPU → [NCPU, C∗-AlgPU] yields a representation
Page 10 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
A recipe for representations of completely positive unital maps
Page 11 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
A recipe for representations of completely positive unital maps
◮ Take a representation F of C∗-AlgPU in a category R.
Page 11 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
A recipe for representations of completely positive unital maps
◮ Take a representation F of C∗-AlgPU in a category R.
◮
Pred : C∗-AlgPU → (effect modules)
◮
Stat : C∗-Algop
PU → (convex sets)
Page 11 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
A recipe for representations of completely positive unital maps
◮ Take a representation F of C∗-AlgPU in a category R.
◮
Pred : C∗-AlgPU → (effect modules)
◮
Stat : C∗-Algop
PU → (convex sets)
◮ Mix it with our representation M of C∗-AlgCPU in [NIsom, C∗-AlgPU]
faithful.
Page 11 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
A recipe for representations of completely positive unital maps
◮ Take a representation F of C∗-AlgPU in a category R.
◮
Pred : C∗-AlgPU → (effect modules)
◮
Stat : C∗-Algop
PU → (convex sets)
◮ Mix it with our representation M of C∗-AlgCPU in [NIsom, C∗-AlgPU]
faithful. ◮ You get a representation of C∗-AlgCPU in [NIsom, R] !
Page 11 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
A recipe for representations of completely positive unital maps
◮ Take a representation F of C∗-AlgPU in a category R.
◮
Pred : C∗-AlgPU → (effect modules)
◮
Stat : C∗-Algop
PU → (convex sets)
◮ Mix it with our representation M of C∗-AlgCPU in [NIsom, C∗-AlgPU]
faithful. ◮ You get a representation of C∗-AlgCPU in [NIsom, R] ! ◮ Crucial point: Representing a completely positive map as a natural family of maps rather than as a unique map.
Page 11 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality How to build representations of completely positive maps
Where we are, sofar
Types for quantum computation How to build representations of completely positive maps Application: Quantum domain theory Concluding remarks
Convex dcpos
Page 12 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Convex dcpos
◮ Convex set (X, ⊕r : X 2 → X) x ⊕r y = r · x + (1 − r) · y (r ∈ [0, 1]) (+ extra conditions which describe the convex structure of X).
Page 12 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Convex dcpos
◮ Convex set (X, ⊕r : X 2 → X) x ⊕r y = r · x + (1 − r) · y (r ∈ [0, 1]) (+ extra conditions which describe the convex structure of X). ◮ A convex dcpo is a convex set equipped with a dcpo structure such that the functions that constitute its convex structure are Scott-continuous.
Page 12 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Convex dcpos
◮ Convex set (X, ⊕r : X 2 → X) x ⊕r y = r · x + (1 − r) · y (r ∈ [0, 1]) (+ extra conditions which describe the convex structure of X). ◮ A convex dcpo is a convex set equipped with a dcpo structure such that the functions that constitute its convex structure are Scott-continuous. ◮ dConv: category of convex dcpos and affine Scott-continuous maps between them.
Page 12 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Convex dcpos
◮ Convex set (X, ⊕r : X 2 → X) x ⊕r y = r · x + (1 − r) · y (r ∈ [0, 1]) (+ extra conditions which describe the convex structure of X). ◮ A convex dcpo is a convex set equipped with a dcpo structure such that the functions that constitute its convex structure are Scott-continuous. ◮ dConv: category of convex dcpos and affine Scott-continuous maps between them. ◮ Example: unit interval of the reals.
Page 12 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Representing state spaces as convex dcpos
Page 13 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Representing state spaces as convex dcpos
◮ W*-algebras are C*-algebras with nice domain-theoretic properties.
Page 13 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Representing state spaces as convex dcpos
◮ W*-algebras are C*-algebras with nice domain-theoretic properties.
◮ W∗-AlgPU: category of W*-algebras and (normal) positive unital maps ◮ W∗-AlgCPU: category of W*-algebras and (normal) completely positive unital maps
Page 13 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Representing state spaces as convex dcpos
◮ W*-algebras are C*-algebras with nice domain-theoretic properties.
◮ W∗-AlgPU: category of W*-algebras and (normal) positive unital maps ◮ W∗-AlgCPU: category of W*-algebras and (normal) completely positive unital maps ◮ NS(A) = W∗-AlgPU(A, C) for a W*-algebra A
Page 13 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Representing state spaces as convex dcpos
◮ W*-algebras are C*-algebras with nice domain-theoretic properties.
◮ W∗-AlgPU: category of W*-algebras and (normal) positive unital maps ◮ W∗-AlgCPU: category of W*-algebras and (normal) completely positive unital maps ◮ The functor NS : W∗-Algop
PU → dConv
taking A to NS(A) is full and faithful.
Page 13 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Representing state spaces as convex dcpos
◮ W*-algebras are C*-algebras with nice domain-theoretic properties.
◮ W∗-AlgPU: category of W*-algebras and (normal) positive unital maps ◮ W∗-AlgCPU: category of W*-algebras and (normal) completely positive unital maps ◮ The functor NS : W∗-Algop
CPU → [Nop CPU, dConv]
taking A to {NS(Mn(A))}n is full and faithful.
Page 13 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Quantum (pre)domains
Page 14 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Quantum (pre)domains
◮ NCPU: category whose objects are natural numbers and where a morphism m → n is a completely positive unital map Mm → Mn.
Page 14 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Quantum (pre)domains
◮ NCPU: category whose objects are natural numbers and where a morphism m → n is a completely positive unital map Mm → Mn. ◮ Quantum predomain: functor D : Nop
CPU → dConv such that
D(f ⊕r g) = D(f ) ⊕r D(g) r ∈ [0, 1]
Page 14 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Quantum (pre)domains
◮ NCPU: category whose objects are natural numbers and where a morphism m → n is a completely positive unital map Mm → Mn. ◮ Quantum predomain: functor D : Nop
CPU → dConv such that
D(f ⊕r g) = D(f ) ⊕r D(g) r ∈ [0, 1] ◮ Quantum domain: quantum predomain D such that D(1) has a least element.
Page 14 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Motivation: introducing lifting in (classical) domain theory
Page 15 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Motivation: introducing lifting in (classical) domain theory
Set
identity on objects
Pfn
(_)+1
(_)⊥
Dom!
forgetful
Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Motivation: introducing lifting in (classical) domain theory
Set
identity on objects
Pfn
(_)+1
(_)⊥
Dom!
forgetful
Predomain: set + partial order + least upper bounds of ω-chains.
Page 15 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Motivation: introducing lifting in (classical) domain theory
Set
identity on objects
Pfn
(_)+1
(_)⊥
Dom!
forgetful
Predomain: set + partial order + least upper bounds of ω-chains. ◮ Domain: predomain with a least element.
Page 15 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Motivation: introducing lifting in quantum domain theory
Page 16 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Motivation: introducing lifting in quantum domain theory
W∗-AlgCPU
identity on objects
W∗-AlgCPSU
(_)⊕C
(_)⊥
QDom!
forgetful
Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Motivation: introducing lifting in quantum domain theory
W∗-AlgCPU
identity on objects
W∗-AlgCPSU
(_)⊕C
(_)⊥
QDom!
forgetful
Quantum predomain: functor D : Nop
CPU → dConv such that
D(f ⊕r g) = D(f ) ⊕r D(g) r ∈ [0, 1] ◮ Quantum domain: quantum predomain D such that D(1) has a least element.
Page 16 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Application: Quantum domain theory
Where we are, sofar
Types for quantum computation How to build representations of completely positive maps Application: Quantum domain theory Concluding remarks
Next step: Improving quantum domains
Page 17 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Concluding remarks
Next step: Improving quantum domains
◮ Axiomatization of the algebraic structure of quantum domains
Page 17 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Concluding remarks
Next step: Improving quantum domains
◮ Axiomatization of the algebraic structure of quantum domains A ⊕ A → M2(A) in W∗-AlgCPU = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = NS(M2(A)) ⇒ NS(A) ⊕ NS(A) in QDom
Page 17 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Concluding remarks
Next step: Improving quantum domains
◮ Axiomatization of the algebraic structure of quantum domains A ⊕ A → M2(A) in W∗-AlgCPU = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = NS(M2(A)) ⇒ NS(A) ⊕ NS(A) in QDom
Page 17 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Concluding remarks
Next step: Improving quantum domains
◮ Axiomatization of the algebraic structure of quantum domains A ⊕ A → M2(A) in W∗-AlgCPU = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = NS(M2(A)) ⇒ NS(A) ⊕ NS(A) in QDom
◮ Algebraic compactness and quantum (pre)domains (to appear).
Page 17 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Concluding remarks
Main points
Page 18 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Concluding remarks
Main points
◮ C*-algebras with completely positive maps are a widely accepted model of first-order quantum computation.
Page 18 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Concluding remarks
Main points
◮ C*-algebras with completely positive maps are a widely accepted model of first-order quantum computation. ◮ There are representations of various categories of C*-algebras with positive maps.
Page 18 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Concluding remarks
Main points
◮ C*-algebras with completely positive maps are a widely accepted model of first-order quantum computation. ◮ There are representations of various categories of C*-algebras with positive maps. ◮ Our contribution is a general method for extending these representations to completely positive maps.
Page 18 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Concluding remarks
Main points
◮ C*-algebras with completely positive maps are a widely accepted model of first-order quantum computation. ◮ There are representations of various categories of C*-algebras with positive maps. ◮ Our contribution is a general method for extending these representations to completely positive maps.
Theorem
The functor M : C∗-AlgCPU → [NCPU, C∗-AlgPU] yields a representation
Page 18 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Concluding remarks
Main points
◮ C*-algebras with completely positive maps are a widely accepted model of first-order quantum computation. ◮ There are representations of various categories of C*-algebras with positive maps. ◮ Our contribution is a general method for extending these representations to completely positive maps.
Theorem
The functor M : C∗-AlgCPU → [NCPU, C∗-AlgPU] yields a representation
◮ Trick for quantum domain theory: replacing Scott-continuous maps by natural families of Scott-continuous maps.
Page 18 of 18 Rennela, Staton 15th July 2015 Complete positivity as naturality Concluding remarks