  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.
