Causality in Higher Order Process Theories

TL;DR

This paper develops a unified categorical framework for higher order quantum processes, called HOPT, linking causality, determinism, and signalling within a process-theoretic and mathematical structure.

Contribution

It provides an equivalent axiomatic construction of the HOPT framework, connecting foundational quantum features with categorical structures.

Findings

HOPT models lower and higher order transformations in a unified way

Established links between causality, determinism, and signalling in quantum processes

Connected process-theoretic features with *-autonomy in the categorical framework

Abstract

Quantum supermaps provide a framework in which higher order quantum processes can act on lower order quantum processes. In doing so, they enable the definition and analysis of new quantum protocols and causal structures. Recently, key features of quantum supermaps were captured through a general categorical framework, which led to a framework of higher order process theories (HOPT). The HOPT framework models lower and higher order transformations in a single unified theory, with its mathematical structure shown to coincide with the notion of a closed symmetric monoidal category. Here we provide an equivalent construction of the HOPT framework from four simple axioms of process-theoretic nature. We then use the HOPT framework to establish connections between foundational features such as causality, determinism and signalling, alongside exploring their interaction with the mathematical structure of *-autonomy.