Composites and Categories of Euclidean Jordan Algebras

TL;DR

This paper explores the structure of composites of Euclidean Jordan algebras, showing limitations on exceptional algebras and constructing categories that unify various quantum systems, including classical and quaternionic.

Contribution

It introduces a symmetric monoidal category of Euclidean Jordan algebras and a dagger-compact category of embedded EJAs, unifying finite-dimensional quantum systems with new compositional insights.

Findings

No composite has the exceptional Jordan algebra as a summand.

Composite of simple, non-exceptional EJAs is a summand of their universal tensor product.

The constructed categories include real, complex, and quaternionic quantum mechanics, with some composites having capacities exceeding the product of parts.

Abstract

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras (EJAs), satisfying some reasonable additional constraints motivated by the desire to construct dagger-compact categories of such models. We show that no such composite has the exceptional Jordan algebra as a direct summand, nor does any such composite exist if one factor has an exceptional summand, unless the other factor is a direct sum of one-dimensional Jordan algebras (representing essentially a classical system). Moreover, we show that any composite of simple, non-exceptional EJAs is a direct summand of their universal tensor product, sharply limiting the possibilities. These results warrant our focussing on concrete Jordan algebras of hermitian matrices, i.e., euclidean Jordan algebras with a preferred embedding in a complex matrix algebra}. We show that these can be organized in a natural way as a symmetric monoidal category, albeit one that is not compact closed. We then construct a related category InvQM of embedded euclidean Jordan algebras, having fewer objects but more morphisms, that is not only compact closed but dagger-compact. This category unifies finite-dimensional real, complex and quaternionic mixed-state quantum mechanics, except that the composite of two complex quantum systems comes with an extra classical bit. Our notion of composite requires neither tomographic locality, nor preservation of purity under tensor product. The categories we construct include examples in which both of these conditions fail. In such cases, the information capacity (the maximum number of mutually distinguishable states) of a composite is greater than the product of the capacities of its constituents.