Quantum sets

Abstract

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.

Quantum CPOs

“…The remainder of the paper lays out the theory of quantum cpos, their relation to classical cpos, and the basic categorical results that are needed to use quantum cpos as semantic models. We begin with quantum sets [9], which describe combined classical/quantum systems that are discrete in the sense that every complete Boolean algebra of propositions is isomorphic to a power set. The results in [9] draw heavily on the work in [10] and [25].…”

“…We begin with quantum sets [9], which describe combined classical/quantum systems that are discrete in the sense that every complete Boolean algebra of propositions is isomorphic to a power set. The results in [9] draw heavily on the work in [10] and [25].…”

“…We briefly review the essential notions of [9]. A quantum set is a collection of finite-dimensional Hilbert spaces, e.g., X = {C 2 , C 5 }.…”

“…) that corresponds to this function, in the sense of the duality between quantum sets and hereditarily atomic von Neumann algebras [9], is simply conjugation by the Hadamard matrix above, appropriately normalized. Example 3.2.…”

“…qCPOs in the sense of the duality between quantum sets and hereditarily atomic von Neumann algebras [9], includes C 2 into M 2 (C) diagonally. Example 3.3.…”