A Coalgebraic Model of Quantum Bisimulation

Recent works have shown that defining a behavioural equivalence that matches the observational properties of a quantum-capable, concurrent, non-deterministic system is a surprisingly difficult task. We explore coalgebras over distributions taking weights from a generic effect algebra, which subsumes probabilities and quantum effects, a physical formalism that represents the probabilistic behaviour of an open quantum system. To abide by the properties of quantum theory, we introduce monads graded on a partial commutative monoid, intuitively allowing composition of two processes only if they use different quantum resources, as prescribed by the no-cloning theorem. We investigate the relation between an open quantum system and its probabilistic counterparts obtained when instantiating the input with a specific quantum state. We consider Aczel-Mendler and kernel bisimilarities, advocating for the latter as it characterizes quantum systems that exhibit the same probabilistic behaviour for all input states. Finally, we propose operators on quantum effect labelled transition systems, paving the way for a process calculi semantics that is parametric over the quantum input.