Commuting operations factorise

TL;DR

This paper proves that in finite-dimensional quantum systems, commuting operations by two agents can be factorized into separate operations, extending classical results to the fully quantum setting.

Contribution

It demonstrates that commuting quantum operations in finite dimensions can be factorized, generalizing Tsirelson's classical input-output results to the quantum realm.

Findings

Commuting quantum operations factorise in finite dimensions.

Extension of classical factorisation results to quantum systems.

Supports the structure of quantum operations in finite-dimensional settings.

Abstract

Consider two agents, Alice and Bob, each of whom takes a quantum input, operates on a shared quantum system $K$, and produces a quantum output. Alice and Bob's operations may commute, in the sense that the joint input-output behaviour is independent of the order in which they access $K$. Here we ask whether this commutation property implies that $K$ can be split into two factors on which Alice and Bob act separately. The question can be regarded as a "fully quantum" generalisation of a problem posed by Tsirelson, who considered the case where Alice and Bob's inputs and outputs are classical. In this case, the answer is negative in general, but it is known that a factorisation exists in finite dimensions. Here we show the same holds in the fully quantum case, i.e., commuting operations factorise, provided that all input systems are finite-dimensional.