Finite-dimensional Quantum Observables are the Special Symmetric Dagger-Frobenius Algebras of CP Maps

TL;DR

This paper demonstrates that all special symmetric dagger-Frobenius algebras in the CPM category of finite-dimensional Hilbert spaces are canonical, linking quantum observables to classical structures through the principle of purity.

Contribution

It proves that these algebras are derived from classical structures in Hilbert spaces, establishing a canonical form for quantum observables in the CPM framework.

Findings

All special symmetric dagger-Frobenius algebras in CPM(fHilb) are canonical.

Classical structures are included as special cases.

Purity principle is key to the proof.

Abstract

We use purity, a principle borrowed from the foundations of quantum information, to show that all special symmetric dagger-Frobenius algebras in CPM(fHilb) are canonical, i.e. that they arise by doubling of special symmetric dagger-Frobenius algebras in fHilb. In particular, this applies to all classical structures.