Higher-order causal theories are models of BV-logic

TL;DR

This paper extends the Caus[-] construction to produce models of BV logic, integrating additive and sequential products, and applies it to probabilistic and quantum higher-order processes, revealing new logical and causal structures.

Contribution

It introduces a richer internal logic for the Caus[-] construction, modeling BV logic and higher-order quantum channels with causal constraints.

Findings

Models of BV logic including additives and sequential products

Probabilistic coherence spaces as BV models with normalized maps

Higher-order quantum channels as BV models with indefinite causal structures

Abstract

The Caus[-] construction takes a compact closed category of basic processes and yields a *-autonomous category of higher-order processes obeying certain signalling/causality constraints, as dictated by the type system in the resulting category. This paper looks at instances where the base category C satisfies additional properties yielding an affine-linear structure on Caus[C] and a substantially richer internal logic. While the original construction only gave multiplicative linear logic, here we additionally obtain additives and a non-commutative, self-dual sequential product yielding a model of Guglielmi's BV logic. Furthermore, we obtain a natural interpretation for the sequential product as "A can signal to B, but not vice-versa", which sits as expected between the non-signalling tensor and the fully-signalling (i.e. unconstrained) par. Fixing matrices of positive numbers for C recovers the BV category structure of probabilistic coherence spaces identified by Blute, Panangaden, and Slavnov, restricted to normalised maps. On the other hand, fixing the category of completely positive maps gives an entirely new model of BV consisting of higher order quantum channels, encompassing recent work in the study of quantum and indefinite causal structures.