A new description of orthogonal bases

TL;DR

This paper characterizes orthogonal bases in finite-dimensional Hilbert spaces using categorical algebra, specifically commutative dagger-Frobenius monoids, providing an axiomatic and operational perspective relevant to quantum mechanics.

Contribution

It introduces a novel categorical axiomatization of orthogonal bases via Frobenius monoids, avoiding explicit vector space references and linking to quantum data operations.

Findings

Orthogonal bases correspond to commutative dagger-Frobenius monoids in FdHilb

Normalised bases are represented by special Frobenius monoids

Operational interpretation involves copying and deleting basis vectors

Abstract

We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative dagger-Frobenius monoid in the category FdHilb, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative dagger-Frobenius monoid is special. Hence orthogonal and orthonormal bases can be axiomatised in terms of composition of operations and tensor product only, without any explicit reference to the underlying vector spaces. This axiomatisation moreover admits an operational interpretation, as the comultiplication copies the basis vectors and the counit uniformly deletes them. That is, we rely on the distinct ability to clone and delete classical data as compared to quantum data to capture basis vectors. For this reason our result has important implications for categorical quantum mechanics.