A categorical semantics of quantum protocols

TL;DR

This paper introduces a new abstract categorical framework for quantum protocols, capturing essential quantum information processes and their underlying structures, and analyzing the axiomatic conditions needed for protocols like teleportation.

Contribution

It develops a novel categorical semantics for quantum mechanics, unifying quantum protocols within a purely structural and abstract setting, revealing axiomatic freedoms and foundational insights.

Findings

Categorical structures can model quantum protocols like teleportation and entanglement-swapping.

Scalars and the Born rule emerge naturally from the categorical framework.

Axiomatic conditions on the scalar ring determine the feasibility of quantum protocols.

Abstract

We study quantum information and computation from a novel point of view. Our approach is based on recasting the standard axiomatic presentation of quantum mechanics, due to von Neumann, at a more abstract level, of compact closed categories with biproducts. We show how the essential structures found in key quantum information protocols such as teleportation, logic-gate teleportation, and entanglement-swapping can be captured at this abstract level. Moreover, from the combination of the --apparently purely qualitative-- structures of compact closure and biproducts there emerge `scalars` and a `Born rule'. This abstract and structural point of view opens up new possibilities for describing and reasoning about quantum systems. It also shows the degrees of axiomatic freedom: we can show what requirements are placed on the (semi)ring of scalars C(I,I), where C is the category and I is the tensor unit, in order to perform various protocols such as teleportation. Our formalism captures both the information-flow aspect of the protocols (see quant-ph/0402014), and the branching due to quantum indeterminism. This contrasts with the standard accounts, in which the classical information flows are `outside' the usual quantum-mechanical formalism.