Complete positivity and natural representation of quantum computations 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 � . ◮ • C*-algebra = algebra of physical observables (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 � . ◮ • C*-algebra = algebra of physical observables (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 � . ◮ • C*-algebra = algebra of physical observables (measurable quantities of a physical system). Bool: � bool � = C ⊕ C ◮ Qubit: � qubit � = M 2 = B ( C 2 ) ◮ 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 � . ◮ • C*-algebra = algebra of physical observables (measurable quantities of a physical system). Bool: � bool � = C ⊕ C ◮ Qubit: � qubit � = M 2 = B ( C 2 ) ◮ 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 � . ◮ • C*-algebra = algebra of physical observables (measurable quantities of a physical system). Bool: � bool � = C ⊕ C ◮ Qubit: � qubit � = M 2 = B ( C 2 ) ◮ 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 � . ◮ • C*-algebra = algebra of physical observables (measurable quantities of a physical system). Bool: � bool � = C ⊕ C ◮ Qubit: � qubit � = M 2 = B ( C 2 ) ◮ 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) ◮ • unital: preserves the unit. 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) ◮ • unital: preserves the unit. • positive: preserves observables 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) ◮ • unital: preserves the unit. • positive: preserves observables positive element: a = x ∗ x for some x . ◮ observables are determined by positive elements. ◮ 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) ◮ • completely positive : allows to run the computation on a 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) ◮ • completely positive : allows to run the computation on a subsystem of a bigger system. M 2 n ( f ) : M 2 n ( B ) → M 2 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) ◮ • completely positive : allows to run the computation on a subsystem of a bigger system. id � qubit � ⊗ 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) ◮ • completely positive : allows to run the computation on a subsystem of a bigger system. id � qubit � ⊗ 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) ◮ • completely positive : allows to run the computation on a subsystem of a bigger system. id � qubit � ⊗ 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 ◮ • full and faithful functor 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 ◮ • Natural isomorphism 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 ◮ • Natural isomorphism 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” � � � � predicate state ⊤ transformers transformers Pred Stat � � computations 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” � � � � predicate state ⊤ transformers transformers Pred Stat � � computations 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” � � � � predicate state ⊤ transformers transformers Pred Stat � � computations 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
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