Fibred Coalgebraic Logic and Quantum Protocols

TL;DR

This paper introduces a fibred coalgebraic logic tailored for modeling quantum systems, enabling complex modality construction and uniform descriptions of quantum protocols, with proven semantic preservation and local reasoning capabilities.

Contribution

It extends coalgebraic logic with fibred semantics and modality calculus, specifically designed for quantum systems, allowing compositional reasoning about quantum protocols.

Findings

Modalities can describe standard quantum protocols.

Semantic properties are preserved under composition.

Supports local reasoning about quantum states.

Abstract

Motivated by applications in modelling quantum systems using coalgebraic techniques, we introduce a fibred coalgebraic logic. Our approach extends the conventional predicate lifting semantics with additional modalities relating conditions on different fibres. As this fibred setting will typically involve multiple signature functors, the logic incorporates a calculus of modalities enabling the construction of new modalities using various composition operations. We extend the semantics of coalgebraic logic to this setting, and prove that this extension respects behavioural equivalence. We show how properties of the semantics of modalities are preserved under composition operations, and then apply the calculational aspect of our logic to produce an expressive set of modalities for reasoning about quantum systems, building these modalities up from simpler components. We then demonstrate how these modalities can describe some standard quantum protocols. The novel features of our logic are shown to allow for a uniform description of unitary evolution, and support local reasoning such as "Alice's qubit satisfies condition" as is common when discussing quantum protocols.