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Abstract: We show that an orthogonal basis for a finite-dimensional Hilbert space canbe equivalently characterised as a commutative dagger-Frobenius monoid in thecategory FdHilb, which has finite-dimensional Hilbert spaces as objects andcontinuous linear maps as morphisms, and tensor product for the monoidalstructure. The basis is normalised exactly when the corresponding commutativedagger-Frobenius monoid is special. Hence orthogonal and orthonormal bases canbe axiomatised in terms of composition of operations and tensor product only,without any explicit reference to the underlying vector spaces. Thisaxiomatisation moreover admits an operational interpretation, as thecomultiplication copies the basis vectors and the counit uniformly deletesthem. That is, we rely on the distinct ability to clone and delete classicaldata as compared to quantum data to capture basis vectors. For this reason ourresult has important implications for categorical quantum mechanics.

Author: Bob Coecke, Dusko Pavlovic, Jamie Vicary

Source: https://arxiv.org/

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