# Composites and Categories of Euclidean Jordan Algebras

Howard Barnum1,2,3, Matthew A. Graydon4,5, and Alexander Wilce6

1Riemann Center for Geometry and Physics, Institute for Theoretical Physics, Leibniz Universität Hannover
2University of New Mexico
3Currently unaffiliated hnbarnum@aol.com
4Department of Applied Mathematics, University of Waterloo
5Institute for Quantum Computing, University of Waterloo m3graydo@uwaterloo.ca
6Department of Mathematical Sciences, Susquehanna University wilce@susqu.edu

### 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 $\mbox{InvQM}$ 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.}

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