Constructor Theory as Process Theory

Abstract

Constructor theory is a meta-theoretic approach that seeks to characterise concrete theories of physics in terms of the (im)possibility to implement certain abstract "tasks" by means of physical processes. Process theory, on the other hand, pursues analogous characterisation goals in terms of the compositional structure of said processes, concretely presented through the lens of (symmetric monoidal) category theory. In this work, we show how to formulate fundamental notions of constructor theory within the canvas of process theory. Specifically, we exploit the functorial interplay between the symmetric monoidal structure of the category of sets and relations, where the abstract tasks live, and that of symmetric monoidal categories from physics, where concrete processes can be found to implement said tasks. Through this, we answer the question of how constructor theory relates to the broader body of process-theoretic literature, and provide the impetus for future collaborative work between the fields. Conceivable TasksConstructor theory is concerned with the study of physical theories in terms of the question "which tasks are performable within this physical theory?": there is an abstract notion of conceivable tasks and a concrete notion of possible tasks. In seminal work by Deutsch [17], it was remarked that, in full generality, the only real requirement on conceivable tasks is arbitrary composability in sequence and in parallel, i.e. that they form a symmetric monoidal category (SMC). 1 Back then, however, the same author made a specific choice to model tasks as relations between sets: constructor theory literature has stuck by this choice ever since, and so will we.Remark 2.1. In this work, we take all monoidal categories to be strict, and in particular we assume that objects obj(D) in a monoidal category D form a strict monoid. In the case of the SMCs Rel and Set, considered in Definition 2.2 below, this implies a choice of singleton set 1 := { * } to act as a strict unit for the Cartesian product:This also affords us the freedom to write triples (and other tuples) without having to care about nesting:1 It is possible that Deutsch meant for substrates to have an individual identity as physical systems, rather than just a "type": that is, it is possible that Deutsch would prefer for "this qubit" and "that qubit" to be modelled by different-albeit isomorphicobjects in a process theory. In this case, it would make no sense to consider parallel compositions of tasks involving the "same" physical system, and partially-monoidal categories as defined in [22] would be preferable as a process-theoretical universe.