Some Nearly Quantum Theories

TL;DR

This paper explores the structure of non-signaling composites of probabilistic models based on Euclidean Jordan algebras, revealing limitations and constructing categories that unify various quantum theories with some exceptions.

Contribution

It introduces new categorical frameworks for Jordan-algebraic models that unify real, complex, and quaternionic quantum mechanics, and establishes no-go results for extending these models.

Findings

No composite with the exceptional Jordan algebra as a summand.

Constructed categories unify real, complex, and quaternionic quantum mechanics.

Identified phenomena like failure of local tomography and supermultiplicativity.

Abstract

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras. Subject to some reasonable constraints, we show that no such composite exists having the exceptional Jordan algebra as a direct summand. We then construct several dagger compact categories of such Jordan-algebraic models. One of these neatly unifies real, complex and quaternionic mixed-state quantum mechanics, with the exception of the quaternionic "bit". Another is similar, except in that (i) it excludes the quaternionic bit, and (ii) the composite of two complex quantum systems comes with an extra classical bit. In both of these categories, states are morphisms from systems to the tensor unit, which helps give the categorical structure a clear operational interpretation. A no-go result shows that the first of these categories, at least, cannot be extended to include spin factors other than the (real, complex, and quaternionic) quantum bits, while preserving the representation of states as morphisms. The same is true for attempts to extend the second category to even-dimensional spin-factors. Interesting phenomena exhibited by some composites in these categories include failure of local tomography, supermultiplicativity of the maximal number of mutually distinguishable states, and mixed states whose marginals are pure.