Toy Quantum Categories

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. Several authors have developed the idea that quantum mechanics (QM) can be expressed using categories rather than the traditional apparatus of Hilbert space [1,2]. Specifically, dagger symmetric monoidal categories (†-SMCs) with ‘enough’ basis structures are a suitable arena for describing many features of quantum mechanics [3,4,5]. This is unsurprising since FdHilb, the category of finite dimensional Hilbert spaces, linear maps, which ‘hosts’ standard (finite-dimensional) QM machinery, is such a category. Here, the basis structures correspond with orthonormal bases and ‘enough’ means ‘there exist incompatible observables’. However, many features of quantum mechanics can be modelled in any category of this sort. This paper explores some concrete examples of ‘discrete’ †-SMCs with ‘enough’ basis structures to model important QM features. We demonstrate two important facts: • Spekkens’s toy model [9] is an (interesting) instance of categorical quantum axiomatics; • Within the category FRel of finite sets, relations with the cartesian product as a tensor, the two element set {0, 1} comes equipped with two complementary observables 5 . 1 B. C. is supported by EPSRC Advanced Research Fellowship EP/D072786/1 entitled The Structure of Quantum Information and its Ramifications for IT. 2 Email: [email protected] 3 B. E. is supported by an EPSRC DTA Scholarship. 4 Email: [email protected] 5 We became aware of this pair of complementary observables when discussing our axiomatisation of Spekkens’s toy model with Jamie Vicary. This paper is electronically published in Electronic Notes in Theoretical Computer Science URL: www.elsevier.nl/locate/entcs