Programming Research Group CLASSICAL AND QUANTUM STRUCTURES

Dagger-compact categories have been proposed as a categorical framework suitable for quantum reasoning [1, LiCS’04]. Modelling classical operations in this framework seemed to require additional completeness assumptions, most notably biproducts. In the present paper, we show that classical operations naturally arise from the quantum structures, with no additional assumptions. Formally, the distinct capabilities of classical data – that they can be copied and deleted – are captured by means of special coalgebra structures of classical objects. Conceptually, this suggests that the connection of the classical and the quantum reasoning extends the connection of the classical and the resource sensitive logics. Technically, the connection of the classical and the quantum structures thus echoes the connection of the additive and the multiplicative connectives in Linear Logic. In this familiar conceptual framework, we propose a comprehensive formulation of axioms of quantum informatics, which essentially improves on previous work. The underlying graphic calculus allows a very succinct presentation of several quantum informatic protocols. The underlying structural analysis also provides the elements of an abstract stochastic calculus, and points towards possible refinements of resource sensitive logics, which arise from the quantitative content of quantum mechanics and the limited observability of quantum data.