Complete Positivity and Natural Representation of Quantum Computations

The referenced article is about semantic models of quantum computation. In common with other approaches to programming language semantics [Win93], the general idea is to interpret a type A as a space JAK of observations about A. One interprets a computation x : A ` t : B, that produces a term t of type B but depends on a term x of type A, as a predicate transformer JBK→ JAK which maps a predicate on B to its weakest precondition (See e.g. [DP06, Ren13, Cho14]). In more detail, one interprets a type A as a C*-algebra of operators JAK, and the computations describe maps between C*-algebras that are in particular positive: it is actually only the positive elements of the algebra that describe the observables, and these must be preserved by predicate transformers. In fact, they should be completely positive. Informally this means that one can run the computation on a subsystem of a bigger system; for example, we could adjoin an extra qubit to the system and still run the computation. More formally it means that not only does the map JtK : JBK→ JAK preserve positive elements, but also idJqubitK⊗JtK : JqubitK⊗ JBK→ JqubitK⊗ JAK preserves positive elements. The first contribution of this paper is a technique for building representations of quantum computation in terms of completely positive maps. In the second half of the paper we demonstrate our technique by making some first steps in the development of a ‘quantum domain theory’.