We provide axioms that guarantee a category is equivalent to that of continuous linear functions between Hilbert spaces. The axioms are purely categorical and do not presuppose any analytical structure. This addresses a question about the mathematica
l foundations of quantum theory raised in reconstruction programmes such as those of von Neumann, Mackey, Jauch, Piron, Abramsky, and Coecke.
We present a universal construction that relates reversible dynamics on open systems to arbitrary dynamics on closed systems: the restriction affine completion of a monoidal restriction category quotiented by well-pointedness. This categorical comple
tion encompasses both quantum channels, via Stinespring dilation, and classical computing, via Bennetts method. Moreover, in these two cases, we show how our construction can be essentially undone by a further universal construction. This shows how both mixed quantum theory and classical computation rest on entirely reversible foundations.
The Ehresmann-Schein-Nambooripad theorem gives a structure theorem for inverse monoids: they are inductive groupoids. A particularly nice case due to Jarek is that commutative inverse monoids become semilattices of abelian groups. It has also been ca
tegorified by DeWolf-Pronk to a structure theorem for inverse categories as locally complete inductive groupoids. We show that in the case of compact inverse categories, this takes the particularly nice form of a semilattice of compact groupoids. Moreover, one-object compact inverse categories are exactly commutative inverse monoids. Compact groupoids, in turn, are determined in particularly simple terms of 3-cocycles by Baez-Lauda.
We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equi
valently be described as Frobenius structures with a ternary multiplication. We study such ternary Frobenius structures and the relationship to binary ones, generalising that between connectors and groupoids.
We interpret ontological models for finite-dimensional quantum theory as functors from the category of finite-dimensional Hilbert spaces and bounded linear maps to the category of measurable spaces and Markov kernels. This uniformises several earlier
results, that we analyse more closely: Pusey, Barrett, and Rudolphs result rules out monoidal functors; Leifer and Maroneys result rules out functors that preserve a duality between states and measurement; Aaronson et als result rules out functors that adhere to the Schrodinger equation. We also prove that it is possible to have epistemic functors that take values in signed Markov kernels.
Topos quantum mechanics, developed by Isham et. al., creates a topos of presheaves over the poset V(N) of abelian von Neumann subalgebras of the von Neumann algebra N of bounded operators associated to a physical system, and established several resul
ts, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global section of the spectral presheaf; (b) a version of the spectral theorem for self-adjoint operators; (c) a connection between states of N and measures on the spectral presheaf; and (d) a model of dynamics in terms of V(N). We consider a modification to this approach using not the whole of the poset V(N), but only its elements of height at most two. This produces a different topos with different internal logic. However, the core results (a)--(d) established using the full poset V(N) are also established for the topos over the smaller poset, and some aspects simplify considerably. Additionally, this smaller poset has appealing aspects reminiscent of projective geometry.
Quantum algorithms are usually described as monolithic circuits, becoming large at modest input size. Near-term quantum architectures can only manage a small number of qubits. We develop an automated method to distribute quantum circuits over multipl
e agents, minimising quantum communication between them. We reduce the problem to hypergraph partitioning and then solve it with state-of-the-art optimisers. This makes our approach useful in practice, unlike previous methods. Our implementation is evaluated on five quantum circuits of practical relevance.
A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We show that und
er mild conditions subunits endow any monoidal category with a kind of topological intuition: there are well-behaved notions of restriction, localisation, and support, even though the subunits in general only form a semilattice. We develop universal constructions completing any monoidal category to one whose subunits universally form a lattice, preframe, or frame.
Reversible computing models settings in which all processes can be reversed. Applications include low-power computing, quantum computing, and robotics. It is unclear how to represent side-effects in this setting, because conventional methods need not
respect reversibility. We model reversible effects by adapting Hughes arrows to dagger arrows and inverse arrows. This captures several fundamental reversible effects, including serialization and mutable store computations. Whereas arrows are monoids in the category of profunctors, dagger arrows are involutive monoids in the category of profunctors, and inverse arrows satisfy certain additional properties. These semantics inform the design of functional reversible programs supporting side-effects.
We develop a notion of limit for dagger categories, that we show is suitable in the following ways: it subsumes special cases known from the literature; dagger limits are unique up to unitary isomorphism; a wide class of dagger limits can be built fr
om a small selection of them; dagger limits of a fixed shape can be phrased as dagger adjoints to a diagonal functor; dagger limits can be built from ordinary limits in the presence of polar decomposition; dagger limits commute with dagger colimits in many cases.
