Mixed states are of interest in quantum mechanics for modelling partial information. More recently categorical approaches to linguistics have also exploited the idea of mixed states to describe ambiguity and hyponym / hypernym relationships. In both these application areas the category Rel of sets and binary relations is often used as an alternative model. Selinger's CPM construction provides the setting for mixed states in Hilbert space based categorical quantum mechanics. By analogy, applying the CPM construction to Rel is seen as introducing mixing into a relational setting. We investigate the category CPM(Rel) of completely positive maps in Rel. We show that the states of an object in CPM(Rel) are in bijective correspondence with certain families of graphs. Via map-state duality this then allows us provide a graph theoretic characterization of the morphisms in CPM(Rel). By identifying an appropriate composition operation on graphs, we then show that CPM(Rel) is isomorphic to a category of sets and graphs between them. This isomorphism then leads to a graph based description of the complete join semilattice enriched dagger compact structure of CPM(Rel). These results allow us to reason about CPM(Rel) entirely in terms of graphs. We exploit these techniques in several examples. We give a closed form expression for the number of states of a finite set in CPM(Rel). The pure states are characterized in graph theoretic terms. We also demonstrate the possibly surprising phenomenon of a pure state that can be given as a mixture of two mixed states.
Mixed states are of interest in quantum mechanics for modelling partial information. More recently categorical approaches to linguistics have also exploited the idea of mixed states to describe ambiguity and hyponym / hypernym relationships. In both these application areas the category Rel of sets and binary relations is often used as an alternative model. Selinger's CPM construction provides the setting for mixed states in Hilbert space based categorical quantum mechanics. By analogy, applying the CPM construction to Rel is seen as introducing mixing into a relational setting. We investigate the category CPM(Rel) of completely positive maps in Rel. We show that the states of an object in CPM(Rel) are in bijective correspondence with certain families of graphs. Via map-state duality this then allows us provide a graph theoretic characterization of the morphisms in CPM(Rel). By identifying an appropriate composition operation on graphs, we then show that CPM(Rel) is isomorphic to a category of sets and graphs between them. This isomorphism then leads to a graph based description of the complete join semilattice enriched dagger compact structure of CPM(Rel). These results allow us to reason about CPM(Rel) entirely in terms of graphs. We exploit these techniques in several examples. We give a closed form expression for the number of states of a finite set in CPM(Rel). The pure states are characterized in graph theoretic terms. We also demonstrate the possibly surprising phenomenon of a pure state that can be given as a mixture of two mixed states.
We recast Selinger's CPM-construction of mixed states completely positive maps [P. Selinger (2007) Dagger compact closed categories and completely positive maps. In: Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), Electronic Notes in Theoretical Computer Science 170, 139–163. Available from Peter Selinger's homepage] as an axiomatization of maximally mixed states. This axiomatization also guarantees categories of completely positive maps to satisfy the preparation-state agreement axiom of [Coecke, B. (2007) De-linearizing linearity: projective quantum axiomatics from strong compact closure. In: Proceedings of the 3rd International Workshop on Quantum Programming Languages (QPL 2005), Electronic Notes in Theoretical Computer Science 170, 49–72. E-print: arXiv:quant-ph/0506134], and admits a physical interpretation in terms of purification of mixed states and CPMs. Internal traces, which are crucial in quantum information theory, are the adjoints to these maximally mixed states.
Monoidal categories have recently been seen to be appropriate structures for establishing abstract axioms for quantum physics. The tensor product in a monoidal category corresponds to the creation of a composite system obtained by joining two independent quantum systems. This correspondence is formalized in the work of Abramsky-Coecke and Selinger. From this point of view, the tensor product in a monoidal category is abstracting the tensor product of Hilbert spaces. The notion of a dagger compact closed category axiomatizes further structure in the Hilbert space category. In particular, the dagger is an abstraction of the familiar Hilbert adjoint. Peter Selinger associates to each dagger compact closed category C its category of completely positive maps, denoted CPM(C). He proves that the resulting category is again a dagger compact closed category. We seek a similar result for tensored †-categories equipped with nuclear ideals. We establish an appropriate notion of completely positive maps in a nuclear ideal. We then define a CPM construction for tensored †-categories equipped with nuclear ideals. Analogous to Selinger's construction, given a nuclear ideal N for a tensored †-category C, our construction yields its category of completely positive maps, CPMN(C). We prove that CPM N(C). is again a tensored †-category. In the process, we also verify that our completely positive maps properly extend Selinger's notion of CPMs. In particular, they preserve nuclear-positive matrices, which are generalizations of von Neumann's positive density operators. As well, we characterize the completely positive maps for the nuclear ideal of Hilbert-Schmidt operators for Hilb and the nuclear ideal of finite relations for LFR. Our construction abstracts the passage from finite-dimensional Hilbert spaces to the category of all Hilbert spaces.
We introduce the CP*-construction on a dagger compact closed category as a generalisation of Selinger's CPM-construction. While the latter takes a dagger compact closed category and forms its category of "abstract matrix algebras" and completely positive maps, the CP*-construction forms its category of "abstract C*-algebras" and completely positive maps. This analogy is justified by the case of finite-dimensional Hilbert spaces, where the CP*-construction yields the category of finite-dimensional C*-algebras and completely positive maps. The CP*-construction fully embeds Selinger's CPM-construction in such a way that the objects in the image of the embedding can be thought of as "purely quantum" state spaces. It also embeds the category of classical stochastic maps, whose image consists of "purely classical" state spaces. By allowing classical and quantum data to coexist, this provides elegant abstract notions of preparation, measurement, and more general quantum channels.
Building on established literature and recent developments in the graph-theoretic characterisation of its CPM category, we provide a treatment of pure state and mixed state quantum mechanics in the category fRel of finite sets and relations. On the way, we highlight the wealth of exotic beasts that hide amongst the extensive operational and structural similarities that the theory shares with more traditional arenas of categorical quantum mechanics, such as the category fdHilb. We conclude our journey by proving that fRel is local, but not without some unexpected twists.
There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an interpretation functor to the dagger-SMC of (finite dimension) Hilbert spaces. In the recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of the extension of these languages beyond pure quantum mechanics, in order to reason on mixed states and general quantum operations, i.e. completely positive maps. Intuitively, such an extension relies on the axiomatisation of a discard map which allows one to get rid of a quantum system, operation which is not allowed in pure quantum mechanics. We introduce a new construction, the discard construction, which transforms any dagger-symmetric monoidal category into a symmetric monoidal category equipped with a discard map. Roughly speaking this construction consists in making any isometry causal. Using this construction we provide an extension for several graphical languages that we prove to be complete for general quantum operations. However this construction fails for some fringe cases like the Clifford+T quantum mechanics, as the category does not have enough isometries.
Dagger compact closed categories were recently introduced by Abramsky and Coecke, under the name “strongly compact closed categories”, as an axiomatic framework for quantum mechanics. We present a graphical language for dagger compact closed categories, and sketch a proof of its completeness for equational reasoning. We give a general construction, the CPM construction, which associates to each dagger compact closed category its “category of completely positive maps”, and we show that the resulting category is again dagger compact closed. We apply these ideas to Abramsky and Coecke's interpretation of quantum protocols, and to D'Hondt and Panangaden's predicate transformer semantics.