Classical and quantum structuralism

TL;DR

This paper explores the categorical framework of symmetric dagger-monoidal categories to formalize quantum and classical data, highlighting how classical data can be represented, distinguished, and interact with quantum systems through a graphical calculus.

Contribution

It introduces a unified categorical approach to classical and quantum data, distinguishing classical operations and modeling their interactions with quantum systems diagrammatically.

Findings

Classical data correspond to Frobenius algebras with copying and deleting capabilities.

The framework distinguishes deterministic, nondeterministic, and probabilistic classical operations.

Graphical calculus enables diagrammatic representation of classical-quantum interactions.

Abstract

In recent work, symmetric dagger-monoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to Frobenius algebras with some additional properties. They express the distinguishing capabilities of classical data: in contrast with quantum data, classical data can be copied and deleted. The algebraic approach thus shifts the paradigm of "quantization" of a classical theory to "classicization" of a quantum theory. Remarkably, the simple SDM framework suffices not only for this conceptual shift, but even allows us to distinguish the deterministic classical operations (i.e. functions) from the nondeterministic classical operations (i.e. relations), and the probabilistic classical operations (stochastic maps). Moreover, a combination of some basic categorical constructions (due to Kleisli, resp. Grothendieck) with the categorical presentations of quantum states, provides a resource sensitive account of various quantum-classical interactions: of classical control of quantum data, of classical data arising from quantum measurements, as well as of the classical data processing in-between controls and measurements. A salient feature here is the graphical calculus for categorical quantum mechanics, which allows a purely diagrammatic representation of classical-quantum interaction.