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Electronic Notes in Theoretical Computer Science

Toy Quantum Categories (Extended Abstract)

Bob Coecke, Bill Edwards

Published Feb 1, 2011• 25 citations
Publisher

Abstract

We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact very general categorytheoretic properties. We also show the remarkable fact that we can already interpret complementary quantum observables on the two-element set in FRel. Keywords: Quantum category, dagger symmetric monoidal category, finite dimensional Hilbert spaces, FdHilb, FRel

DOI
10.1016/j.entcs.2011.01.004

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We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact very general categorytheoretic properties. We also show the remarkable fact that we can already interpret complementary quantum observables on the two-element set in FRel. Keywords: Quantum category, dagger symmetric monoidal category, finite dimensional Hilbert spaces, FdHilb, FRel
We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact very general category-theoretic properties. We also show the remarkable fact that we can already interpret complementary quantum observables on the two-element set FRel.
We provide two mathematical descriptions of Spekkens's toy qubit theory, an inductively one in terms of a small set of generators, as well as an explicit closed form description. It is a subcategory MSpek of the category of finite sets, relations and the cartesian product. States of maximal knowledge form a subcategory Spek. This establishes the consistency of the toy theory, which has previously only been constructed for at most four systems. Our model also shows that the theory is closed under both parallel and sequential composition of operations (= symmetric monoidal structure), that it obeys map-state duality (= compact closure), and that states and effects are in bijective correspondence (= dagger structure). From the perspective of categorical quantum mechanics, this provides an interesting alternative model which enables us to describe many quantum phenomena in a discrete manner, and to which mathematical concepts such as basis structures, and complementarity thereof, still apply. Hence, the framework of categorical quantum mechanics has delivered on its promise to encompass theories other than quantum theory.
A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within categorical axiomatizations of quantum mechanics. We develop a definition of H*-algebra that can be interpreted in any symmetric monoidal dagger category, reduces to the classical notion from functional analysis in the category of (possibly infinite-dimensional) Hilbert spaces, and hence provides a categorical way to speak about orthonormal bases and quantum observables in arbitrary dimension. Moreover, these algebras reduce to the usual notion of Frobenius algebra in compact categories. We then investigate the relations between nonunital Frobenius algebras and H*-algebras. We give a number of equivalent conditions to characterize when they coincide in the category of Hilbert spaces. We also show that they always coincide in categories of generalized relations and positive matrices.
Toy models have been used to separate important features of quantum computation from the rich background of the standard Hilbert space model. Category theory, on the other hand, is a general tool to separate components of mathematical structures, and analyze one layer at a time. It seems natural to combine the two approaches, and several authors have already pursued this idea. We explore categorical comprehension construction as a tool for adding features to toy models. We use it to comprehend quantum propositions and probabilities within the basic model of finite-dimensional Hilbert spaces. We also analyze complementary quantum observables over the category of sets and relations. This leads into the realm of test spaces, a well-studied model. We present one of many possible extensions of this model, enabled by the comprehension construction. Conspicuously, all models obtained in this way carry the same categorical structure, extending the familiar dagger compact framework with the complementation operations. We call the obtained structure dagger mix autonomous, because it extends star autonomous categories, popular in computer science, in a similar way like dagger compact structure extends compact categories. Dagger mix autonomous categories seem to arise quite naturally in quantum computation, as soon as complementarity is viewed as a part of the global structure.
A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within categorical axiomatizations of quantum mechanics. We develop a definition of H*-algebra that can be interpreted in any symmetric monoidal dagger category, reduces to the classical notion from functional analysis in the category of (possibly infinite-dimensional) Hilbert spaces, and hence provides a categorical way to speak about orthonormal bases and quantum observables in arbitrary dimension. Moreover, these algebras reduce to the usual notion of Frobenius algebra in compact categories. We then investigate the relations between nonunital Frobenius algebras and H*algebras. We give a number of equivalent conditions to characterize when they coincide in the category of Hilbert spaces. We also show that they always coincide in categories of generalized relations and positive matrices.
A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum sets are certain binary relations that can be characterized in terms of this dagger compact structure, and the resulting category of quantum sets and functions generalizes the category of ordinary sets and functions in the manner of noncommutative mathematics. In particular, this category is dual to a subcategory of von Neumann algebras. The basic properties of quantum sets are presented thoroughly, with the noncommutative dictionary in mind, and with an eye to convenient application. As a motivating example, a notion of quantum graph coloring is derived within this framework, and it is shown to be equivalent to the notion that appears in the quantum information theory literature.