Pictures of complete positivity in arbitrary dimension

• Selinger’s CPM-construction, which associates to any dagger compact category of pure states and operations a corresponding dagger compact category of mixed states and operations [27];

• Environment structures, an axiomatic substitute for the CPM-construction which proved to be particularly useful in the derivation of quantum protocols [7,12].

It is well known that assuming compactness imposes finite dimension when exporting these results to the Hilbert space model [16]. This paper introduces variations of each the above two results that rely on dagger structure alone, and in the presence of compactness reduce to the above ones. Hence, these variations accommodate interpretation not just in the dagger compact category of finite dimensional Hilbert spaces and linear maps, but also in the dagger category of Hilbert spaces of arbitrary dimension and continuous linear maps. We show:

• that the generalized CPM-construction indeed corresponds to the usual definitions of infinite-dimensional quantum information theory;

• that the direct correspondence between the CPM-construction and environment structure (up to the so-called doubling axiom) still carries trough.

The next two sections each discuss one of our two variations in turn.

The variation of the CPM-construction relying solely on dagger structure was already publicized by one of the authors as a research report [5].

Here we relate that construction to the usual setting of infinite-dimensional quantum information theory, thereby justifying it in terms of the usual model. An earlier version of this construction appeared in conference proceedings [8]. Whereas composition is not always well-defined there, it did have the advantage that the output category of the construction was automatically small if the input was. The construction here is closer to [5], and has the advantage that it is rigorously well-defined, but the disadvantage that the output category might be large. See the discussion after Proposition 4 and Remark 8. Additionally, we generalise the construction further than [8], to braided monoidal categories that are not necessarily symmetric.

While there are previous results dealing with the transition to a noncompact setting in some way or another, e.g. [1,16,17], what is particularly appealing about the results in this paper is that they still allow the diagrammatic representations of braided monoidal categories [21,28].

Future work Classical information can be modelled in categorical quantum mechanics using so-called classical structures [11,10,3]. It is not clear whether these survive CPM-like constructions; see also [18]. The environment structures of Section 3 could be a useful tool in this investigation.

Compact categories and their graphical calculus originated in [23,24]. For a gentle introduction to dagger (compact) categories [2] and their graphical calculus [27], we refer to [9]. We now recall the CPM-construction [27], that, given a dagger compact category C, produces a new dagger compact category CPM(C) as follows. When wires of both types A and A * arise in one diagram, we will decorate them with arrows in opposite directions. When possible we will suppress coherence isomorphisms in formulae. Finally, recall that ( ) * reverses the order of tensor products, so f * has type

• The objects of CPM(C) are the same as those of C.

• The morphisms A → B of CPM(C) are those morphisms of C that can be written in the form (1

We call X the ancillary system of (1 ⊗ η † ⊗ 1)(f * ⊗ f ), and f its Kraus morphism; these representatives are not unique.

• Identities are inherited from C, and composition is defined as follows.

• The tensor unit I and the tensor product of objects are inherited from C, and the tensor product of morphisms is defined as follows.

e. with ancillary system I and Kraus morphism η A in C).

If C is the dagger compact category FHilb of finite-dimensional Hilbert spaces and linear maps, then CPM(FHilb) is precisely the category of finite-dimensional Hilbert spaces and completely positive maps [27]. To come closer to the traditional setting, we may identify the objects H of CPM(FHilb) with their algebras of operators B(H).

The notion of complete positivity makes perfect sense for normal linear maps between von Neumann algebras B(H) for Hilbert space H of arbitrary dimension. We now present a CP ∞ -construction that works on dagger monoidal categories that are not necessarily compact, and that reduces to the previous construction in the compact case. Subsequently we prove that applying this construction to the category of Hilbert spaces indeed results in the traditional completely positive maps as morphisms.

The idea behind the following construction is to rewrite the morphisms

in a form not using compactness. Composition becomes plugging the hole. Fo