Finite dimensional Hilbert spaces are complete for dagger compact closed categories

TL;DR

This paper establishes that any equation valid in finite dimensional Hilbert spaces is derivable from the axioms of dagger compact closed categories, linking categorical axioms with Hilbert space properties.

Contribution

It proves an equivalence between equations in dagger compact closed categories and those valid in finite dimensional Hilbert spaces.

Findings

Equations valid in finite dimensional Hilbert spaces follow from dagger compact closed category axioms.

The categorical framework precisely captures properties of finite dimensional Hilbert spaces.

The result bridges categorical quantum mechanics and Hilbert space formalism.

Abstract

We show that an equation follows from the axioms of dagger compact closed categories if and only if it holds in finite dimensional Hilbert spaces.