A Categorical Reconstruction of Quantum Theory

TL;DR

This paper reconstructs finite-dimensional quantum theory using purely categorical principles, avoiding tomography assumptions, and characterizes quantum and real Hilbert space theories over suitable rings.

Contribution

It provides a fully categorical reconstruction of quantum theory, generalizing previous approaches and including real Hilbert space quantum theories.

Findings

Reconstruction of quantum theory from categorical principles.

Identification of the ring $S$ as complex numbers or reals for different theories.

Axioms do not require tomography assumptions.

Abstract

We reconstruct finite-dimensional quantum theory from categorical principles. That is, we provide properties ensuring that a given physical theory described by a dagger compact category in which one may `discard' objects is equivalent to a generalised finite-dimensional quantum theory over a suitable ring $S$. The principles used resemble those due to Chiribella, D'Ariano and Perinotti. Unlike previous reconstructions, our axioms and proof are fully categorical in nature, in particular not requiring tomography assumptions. Specialising the result to probabilistic theories we obtain either traditional quantum theory with $S$ being the complex numbers, or that over real Hilbert spaces with $S$ being the reals.