text:
\section{Introduction}
\label{sec: introduction}

Formally real Jordan algebras were first proposed as models of
quantum systems by P. Jordan in 1933 \cite{Jordan}. Abstractly, a Jordan algebra is a real vector space $A$ equipped with a commutative bilinear product $\dot$ satisfying the {\em Jordan identity}
$a^2\dot (a \dot b) = a\dot (a^2 \dot b)$ for all $a,b \in A$ (where $a^2 = a \dot a$).
{}%\color{red} We also assume  that $A$ has a unit element, which we denote by $u$.}
$A$ is {\em formally
real} if, for $a_1,...,a_n \in A$,
$\sum_{i=1}^{n} a_{i}^2 = 0$ only when $a_i = 0$ for all $i$. If $A$ is finite-dimensional, this is equivalent to
$A$'s being {\em euclidean}, meaning that
it carries an inner product such that
$\langle a \dot b | c \rangle = \langle a | b \dot c \rangle$ for all $a,b,c \in A$.

 The standard example is the set $\L(\mathbf{H})$ of
self-adjoint operators on a finite-dimensional Hilbert space $\mathbf{H}$, with $a \dot b = (ab + ba)/2$, and
with $\langle a | b \rangle = \mbox{Tr}(ab)$.
In 1934, Jordan, von Neumann and Wigner \cite{JNW} showed that
all finite-dimensional formally real --- equivalently, euclidean ---  Jordan algebras are direct sums
of irreducible, or {\em simple}, such algebras, and that the latter are of just five kinds:
self-adjoint parts of real, complex or quaternionic matrix algebras
(which we can regard as real, complex or quaternionic quantum systems)
{\em spin factors} (which are analogues of qubits in which the ``Bloch
sphere" can have arbitrary finite dimension), and the {\em exceptional
  Jordan algebra} of $3 \times 3$ self-adjoint octonionic matrices.

A reasonable objection is that the physical meaning of the Jordan
product is obscure. (Indeed, it is not obvious why the observables of
a physical system should carry any physically meaningful bilinear
product at all.) Happily, there are alternative characterizations of
euclidean Jordan algebras in terms of ordered vector spaces and
related concepts having readier physical, probabilistic, or
operational interpretations.
The Koecher-Vinberg Theorem
(\cite{Koecher, Vinberg}; see also \cite{KoecherBook},
\cite{FK}, Chapter III or
\cite{Satake}, Chapter I, \S8) identifies
euclidean Jordan algebras with finite-dimensional ordered vector spaces
having homogeneous, self-dual cones; work of Alfsen and Shultz
\cite{ASPaper, AS}
characterizes EJAs in terms of certain projections associated
with closed faces of the positive cone.\footnote{Alfsen and Shultz'
results apply, more generally, to JB-algebras, which in the context of
finite dimension are the formally real Jordan algebras.}
{}%{\color{blue} Exploiting these results},
several recent papers \cite{BMU, MU, Wilce09, Wilce11, Wilce12} have shown that physically reasonable postulates
force a finite-dimensional physical system to have the structure of a euclidean Jordan algebra. To this extent,
euclidean Jordan algebras are a natural class of models for physical systems.

A physical theory, however, is more than a collection of models of
physical systems. It must also describe how systems change
and how they interact. It is natural, therefore, to represent a physical
theory as a {\em category}, in which objects represent physical systems and
morphisms represent physical processes.  To accommodate
composite systems, one wants the category to be monoidal, i.e., to be
equipped with an associative ``tensor product''.   This point of view
has been developed very fruitfully in \cite{Abramsky-Coecke,
  BaezCategories} and elsewhere,  where it is shown that
many features of finite-dimensional quantum mechanics can be recovered
if the category in question is compact closed, or, better still,
dagger-compact (terms we explain in Section \ref{sec: Euclidean JC}).

In this paper, building on work of Hanche-Olsen \cite{HO} and Jamjoom
\cite{JamjoomTensorJW} on tensor products of {}%\color{red} JC-algebras,

  we classify the possible non-signaling composites
  of euclidean Jordan-algebraic systems, subject to a few reasonable
  constraints. } In particular, we show that no such composite exists
if either factor is exceptional, and that a composite is always {}%\color{red}
  a special Jordan algebra, and, indeed, a direct summand of the
  universal tensor product defined by Hanche-Olsen.}

Restricting attention further to Jordan algebras corresponding to
real, complex and quaternionic quantum systems, we then identify two
different monoidal sub-categories extending the
category of finite-dimensional complex matrix algebras and CP maps.
One of these, {}%\color{red} which we call $\mbox{\bf RSE}$}, unifies real, complex and quaternionic quantum mechanics,
but lacks {}%\color{red} certain desirable features. In particular, in
this category, {\em states} are not represented by morphisms;
hence, the category is far from being compact closed.}
The other {}%\color{red} category}, which {\em is} compact closed,
{}%\color{red} also} embraces real, complex and quaternionic quantum systems and processes
(CP maps),
{\em except} that
its rule for composing standard complex quantum systems yields an extra classical
bit.

These results, combined with the those of (any of) the papers cited
above, in which a euclidean Jordan structure emerges from
information-theoretically, physically or operationally natural
assumptions, {}%\color{red} lend support to the idea of}
unified quantum theory that
embraces real, complex and quaternionic quantum systems, and
permits the formation of composites of these.  Consistent with the
results of \cite{BarnumWilceLocalTomography}, the composites that
arise in these constructions do not in general have the property of
``tomographic locality'': states on the composite of two Jordan-algebraic
systems are not, in general, determined by the joint probabilities they
assign to measurement outcomes associated with the two component systems.
Equivalently,
the Jordan algebra $AB$ corresponding to a composite of two formally
real Jordan algebraic systems $A$ and $B$, will generally be
larger than the algebraic (i.e., vector-space) tensor product $A \otimes B$.\\

\noindent{\em Remark:}
A related proposal is advanced by Baez \cite{BaezDivAlg}, who points out that one can regard real and quaternionic quantum systems as pairs $(\mathbf{H},J)$, where $\mathbf{H}$ is a complex Hilbert space and $J$ is an anti-unitary satisfying $J^2 = \mathbf{1}$ (the real case) or $J^2 = -\mathbf{1}$ (the quaternionic case). Such pairs can be organized into a dagger-compact symmetric monoidal category, with morphisms $(\mathbf{H}_1, J_1) \rightarrow (\mathbf{H}_2, J_2)$ linear mappings intertwining $J_1$ and $J_2$, and $(\mathbf{H}_1, J_1) \otimes (\mathbf{H}_2, J_2) = (\mathbf{H}_1 \otimes \mathbf{H}_2, J_1 \otimes J_2)$. This provides a unification of real and quaternionic quantum mechanics at the level of pure states and linear mappings between the relevant Hilbert spaces, where our approach takes in quantum systems over all three of the division rings $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$, at the level of mixed states, observables and completely positive maps. While the precise connection between Baez' approach
and ours is still under study, {}%\color{red} it seems to us likely that an application of Selingers CPM construction \cite{Selinger} to Baez' category will yield a category of the type we consider here.}

Our results rest on a mixture of standard facts about ordered vector spaces, the order structure and the representation theory of euclidean Jordan algebras, and the pioneering work of Hanche-Olsen \cite{HO} on
universal representations and tensor products of JC-algebras.
Since much of this
will be unfamiliar to many readers, we have
included a
good amount of purely expository material.
Section \ref{sec: background probabilistic} provides background on
order unit spaces and
their interpretation as general probabilistic
models, including a fairly general notion of composite for such model. (This material
will be more familiar to many readers, but some of our
notation and terminology, {}%\color{red} and some notions specific to our present purposes,} may not be.)
Section \ref{sec:
  background Jordan} collects background material on euclidean Jordan
algebras, their universal representations, and Hanche-Olsen's universal tensor
product.

The balance of the paper is organized as follows.
In Section \ref{sec: composites Jordan}, we introduce a general
definition for the composite of two euclidean Jordan-algebraic probabilistic
models, and establish some basic properties of any such
composite. Along the way, we see that the composite of two simple EJAs
must be embeddable in a complex matrix algebra, i.e. it must
be \emph{special}
(Theorem \ref{thm: composites special}).
{}%{\color{blue} From this it follows} that no simple
EJA has any composite with an exceptional EJA (Corollary
\ref{cor: no composite with exceptional}), and
that if $A$ and $B$ are simple EJAs, then any composite of $A$ and $B$
must be an ideal  --- that is, a direct summand --- in their universal tensor product (Corollary \ref{thm: simple composites ideals}).

These results warrant our focusing on special EJAs. Section
\ref{sec: Euclidean JC}
develops a canonical, {}%{\color{blue} and naturally associative,} tensor product of embedded EJAs, that is, pairs
$(A,\mathbf{M}_A)$ where $\mathbf{M}_A$ is a {}%{\color{blue} finite-dimensional} complex $\ast$-algebra and $A$ is a Jordan
subalgebra of the self-adjoint part of $\mathbf{M}_A$. (The universal tensor
product is the special case in which $\mathbf{M}_A$ and $\mathbf{M}_B$ are the
universal {}%{\color{blue} complex} enveloping algebras of $A$ and $B$.)
In Section \ref{sec: categories EJC} we introduce a class of mappings we call
{\em completely Jordan-preserving} and use these to construct
symmetric monoidal categories of embedded EJAs, some of which we then show are
compact closed {}%\color{red} or, indeed, dagger-compact}.
Section \ref{sec: conclusion} concludes with further discussion of these
categories and their {}%{\color{blue} physical and information-processing significance}.
To avoid obstructing the flow of the main arguments, we
have removed some technical details to a series of appendices.
\\

\noindent{\bf Acknowledgments} {}%\color{red} Some of our results have previously been announced, without proof, in
\cite{BGraW}.\footnote{There, it is erroneously claimed that $\mbox{\bf RSE}$ is compact closed. This error is corrected
in Section 6 of the present paper; see especially Examples 6.3 and 6.11} } HB and AW wish to thank C. M. Edwards
for introducing them to the paper \cite{HO} of Hanche-Olsen.  HB and MG
thank Cozmin Ududec for valuable discussions. AW is
supported by a grant from the FQXi foundation (FQXi-RFP3-1348). This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

\section{Ordered vector spaces and probabilistic models}
\label{sec: background probabilistic}

In this section and the next, we present enough background material to
make this paper reasonably self-contained. This section summarizes
basic information about ordered vector spaces and the {}%{\color{blue}
  ``convex operational"} (or ``generalized probabilistic theories")
framework for discussing probabilistic physical theories. A good
general reference for ordered vector spaces is Chapter 1 of
\cite{ASbasic} (or the summary
in Appendix A of \cite{AS}).  The use of
ordered vector spaces with
order units as
probabilistic models goes back at least to the work of
Ludwig \cite{Ludwig, LudwigAx};
see also \cite{Davies-Lewis, Edwards, Holevo}.
For a more recent survey, see \cite{BarnumWilceFoils}.

\subsection{Ordered vector spaces}

Let $A$ be a real vector space. A (convex, pointed) {\em
  cone}

  in $A$ is a convex set $K
\subseteq A$ such that $a \in K$ implies $ta \in K$ for all $t \in
\mathbb{R}_+$, and $K \cap -K = \{0\}$. A cone $K$ is \emph{generating}
  iff it spans $A$, i.e., if every $a \in A$ can be expressed as a difference of elements of $K$.

  Any cone (generating or not) induces
  a partial ordering of $A$, given by $a \leq b$ iff $b - a \in K$;
  this is translation-invariant, i.e, $a \leq b$ implies $a + c \leq b
  + c$ for all $a, b, c \in A$, and homogeneous, i.e., $a \leq b$
  implies $ta \leq tb$ for all $t \in \mathbb{R}_+$. Conversely, such an
  ordering determines a cone, namely $K = \{ a | a \geq 0\}$.
  Accordingly, an {\em ordered vector space} is a real vector space
  $A$ equipped with a designated {\em positive cone} cone $A_+$.  It
  is common to assume, and we shall assume here, that $A_+$ is
    closed and generating.

    {\em We also assume,
    without further comment, that all ordered vector spaces are finite
    dimensional}.

If $A$ and $B$ are ordered vector spaces, a linear mapping $f : A
\rightarrow B$ is {\em positive} iff $f(A_+) \subseteq B_+$. If $f$ is
bijective and $f(A_+) = B_+$, then $f^{-1}(B_+) = A_+$, so that
$f^{-1}$ is also positive. In this case, we say that $f$ is an {\em
  order isomorphism}.  An {\em order automorphism} of $A$ is an order
isomorphism from $A$ to itself.

Denoting the dual space of $A$ by $A^{\ast}$, the {\em dual cone},
$A^{\ast}_+$, is the set of positive linear functionals on $A$. Since
we are assuming that $A_+$ is generating, it is easy to see that
$A^{\ast}_+ \cap -A^{\ast}_{+} = \{0\}$. In our finite-dimensional
setting, $A^{\ast}_{+}$ is also generating.  Note that if
$B$ is another ordered vector space and $\phi : A \rightarrow B$ is a
positive linear mapping, then the dual mapping $\phi^{\ast} : B^{\ast}
\rightarrow A^{\ast}$ is also positive.

\subsection{Order units and probabilistic models}  An {\em order unit} in an ordered vector space $A$ is an element $u \in A_+$ such that,
for all $a \in A$, $a \leq tu$ for some $t \in \mathbb{R}_+$. In finite
dimensions, this is equivalent to $u$ belonging to the interior of
$A_+$ (cf. \cite{AT}, Theorem 2.8). An {\em order unit space} is a pair $(A,u)$ where $A$ is an
ordered vector space and $u$ is a designated order unit.

An order unit space provides the machinery to discuss probabilistic
concepts. A {\em state} on $(A,u)$ is a positive linear functional
$\alpha \in A^{\ast}$ with $\alpha(u) = 1$. An {\em effect} is an
element $a \in A_+$ with $a \leq u$.  If $\alpha$ is a state and $a$
is an effect, we have $0 \leq \alpha(a) \leq 1$: we interpret this as
the {\em probability} of the given effect on the given state.  A
{\em discrete  observable} on $A$ with values $\lambda \in \Lambda$ is
represented by an indexed family $\{a_{\lambda} | \lambda \in
\Lambda\}$ of effects summing to $u$, the effect $a_{\lambda}$
representing the event of obtaining value $\lambda$ in a measurement
of the observable. {}%\color{red} Thus, if $\alpha$ is a state, $\lambda \mapsto \alpha(a_{\lambda})$
gives a probability weight on $\Lambda$.}
  (One can extend this discussion to include more
general observables by considering effect-valued
measures (cf. \cite{Edwards}), but we will not need this extra
generality here.)

We denote the set of all states of $A$ by $\Omega$; the set of all
effects {}%{\color{magenta} --- the interval between $0$ and $u$ --- is} denoted
$[0,u]$. In our present finite-dimensional setting, both are compact
convex sets. Extreme points of $\Omega$ are called {\em pure
  states}. \\

\noindent{\bf Examples}

(1) {\em Discrete Classical Probability Theory:} If $S$ is a finite
set, regarded as the {}%\color{red} outcome} space of some classical experiment, let
$A(S) = \mathbb{R}^{S}$, ordered pointwise.  Elements of $A(S)_+$ are then
non-negative random variables, and effects are random variables with
values between $0$ and $1$.
We turn this into an order unit space by taking $u \in A(S)_+$ to be the constant
function with value $1$. It is then easy to show that normalized states on
$A(S)$ correspond exactly to probability weights on $S$; discrete
observables correspond in a natural way to discrete ``fuzzy" random
variables. Extreme effects, i.e, extreme points of $[0,u]$, are easily
seen to be characteristic functions of subsets of $S$; hence, an
observable $\{a_{\lambda}\}$ with $a_{\lambda}$ extreme for each
$\lambda$, corresponds to an ordinary ``sharp" random variable.

(2) {\em Discrete Quantum Probability Theory:} If $\mathbf{H}$ is a
finite-dimensional Hilbert space, let $A(\mathbf{H}) = \L(\mathbf{H})$, the
space of self-adjoint operators on $\mathbf{H}$, ordered by the cone of
positive semidefinite operators; let $u = \mathbf{1}$, the identity operator
on $\mathbf{H}$. Then each normalized state $\alpha$ has the form
$\alpha(a) = \mbox{Tr}(\rho a)$ where $\rho$ is a density operator, and,
conversely, every density operator determines in this way a normalized
state,.  Observables correspond to discrete POVMs; thus, we recover
orthodox finite-dimensional quantum probability theory.

(3) A class of examples that embraces both (1) and (2) is the
following. Let $\mathbf{M}$ be a unital complex $\ast$-algebra; define $\mathbf{M}_+$ to
consist of all $a \in \mathbf{M}$ with $a = b^{\ast} b$ for some $b \in \mathbf{M}$:
then $(\mathbf{M}_{{sa}}, \mathbf{M}_{+})$ is an ordered vector space, in which the
unit element of $\mathbf{M}$ serves as an order unit.  If $\mathbf{M}$ is
finite-dimensional and commutative, one essentially recovers example
(1); if $\mathbf{M}$ is the algebra $M_n(\mathbb{C})$ of $n \times n$ complex
matrices, one essentially recovers example (2).  More generally, if
$\mathbf{M}$ is finite-dimensional, Wedderburn's theorem tells us that $\mathbf{M}$ is
a direct sum of matrix algebras, so one has finite-dimensional quantum
theory with superselection rules (classical discrete probability
theory being the special case in which all superselection sectors
--- that is, direct summands --- are one-dimensional). \\

One may wish to privilege certain states and/or certain effects of a
probabilistic model as being ``physically possible".  One way of doing
so is to consider ordered subspaces $E$ of $A$, with $u_A \in E$, and
$V$ of $A^{\ast}$: this picks out the set of states $\alpha \in V \cap
A_+$ and the set of effects $a \in E \cap A_+$, $a \leq u$. The pair
$(E,V)$ then serves as a probabilistic model for a system having these
allowed states and effects.
However, in the three
examples above, and in those that concern us in the rest of this
paper, it is always possible to regard {\em all} states in $A^{\ast}$,
and {\em all} effects in $A$, as allowed. Henceforth, then, when we
speak of a {\em probabilistic model} --- or, more briefly, a {\em
  model} --- we simply mean an order unit space $(A,u)$. It will be
convenient to adopt the shorthand $A$ for such a pair, writing $u_A$
for the order unit where necessary. \\

\noindent \textbf{Processes, Symmetries and Dynamics} By a {\em process}
affecting a system represented by a probabilistic model $A$, we mean a
positive linear mapping $\phi : A \rightarrow A$, subject to the
condition that $\phi(u_A) \leq u_A$.
The probability of observing an effect $a$ after the system has been
prepared in a state $\alpha$ and then subjected to a process $\phi$ is
$\alpha(\phi(a))$. One can regard $\alpha(\phi(u))$ as the probability
that the system is not destroyed by the process.  We can, of course,
replace $\phi : A \rightarrow A$ with the adjoint mapping $\phi^{\ast}
: A^{\ast} \rightarrow A^{\ast}$ given by $\phi^{\ast}(\alpha) =
\alpha \circ \phi$, so as to think of a process as a mapping from
states to possibly sub-normalized states. Thus, we can view processes
either as acting on effects (the ``Heisenberg picture"), or on states
(the ``Schr\"{o}dinger picture").

Any non-zero positive linear mapping $\phi : A \rightarrow A$
is a non-negative scalar multiple of a process in the above sense:
since $\Omega(A)$ is compact, $\{\alpha(\phi(u)) | \alpha \in
\Omega(A)\}$ is a compact set of real numbers, not all zero, and so,
has a maximum value $m(\phi) > 0$; $m(\phi)^{-1}\phi$ is then a
process. For this reason, we make little further distinction here
between processes and positive mappings. In particular, if $\phi$ is
an order automorphism of $A$, then both $\phi$ and $\phi^{-1}$ are
scalar multiples of processes in the above sense: each of these
processes ``undoes" the other, {\em up to normalization}, i.e.,
with nonzero probability.  A process that can be reversed with
probability one is represented by an order-automorphism $\phi$ such
that $\phi(u_A) = u_A$, in which case $\phi^{\ast}$ takes normalized
states to normalized states. Such an order-automorphism is called a
{\em symmetry} of $A$.

We denote the group of all order-automorphisms of $A$ by $\mbox{Aut}(A)$.
\footnote{Here our usage diverges from that of \cite{AS} and
  \cite{FK}, who use $\mbox{Aut}(A)$ to denote the group of Jordan
  automorphisms of a Jordan algebra $A$.}  This is a Lie group
{}%{\color{magenta} (see e.g. \cite{HilgertHoffmanLawson}, pp. 182-183);}
  its connected identity component (consisting of
those processes that can be obtained by continuously deforming the
identity map) is denoted $\mbox{Aut}_{0}(A)$.  A possible
(probabilistically) reversible {\em dynamics} for a system modelled by
$A$ is a homomorphism $t \mapsto \phi_t$ from $(\mathbb{R},+)$ to $\mbox{Aut}(A)$,
i.e., a one-parameter subgroup of $\mbox{Aut}(A)$.  The set of symmetries
forms a compact subgroup, $\mbox{Sym}(A)$, of $\mbox{Aut}(A)$.

One might wish to privilege certain processes as reflecting physically
possible motions or evolutions of the system. In that case, one might
add to the basic data $(A,u)$ a preferred
subgroup $G(A)$ of order automorphisms.  We refer to such a structure
as a {\em dynamical} probabilistic model, since the choice of $G(A)$
constrains the permitted probabilistically reversible dynamics
of the model.  \\

\tempout{
{}%{\color{magenta} [HB: with an eye to future publications (and possibly
    already existing ones), I'm not sure I like this terminology; I
    would tend to use this for models with a distinguished
    \emph{semigroup} of positive maps.  (At some point, I was using
    the more general notion in a draft of or predecessor to this paper
    [meaning one that predeceased it ;-)].)  HOWEVER, I can't think of
    any great alternative terminology at the moment so let's not let
    this hold up the paper.  If I think of anything we can discuss it
    then.  On the other hand if it's possible to revert to the
    definition of dynamical composite that uses a (convex) semigroup
    without to much rejiggering, that might be nice.  One would have
    to be explicit about including enough automorphisms in the
    semigroup then.]}
}

\noindent{\bf Self-Duality}
  An inner product $\langle \  | \ \rangle$ on an
ordered vector space $A$ is {\em positive} iff the associated mapping
$A \rightarrow A^{\ast}$, $a \mapsto \langle a|$, is positive,
i.e,. if $\langle a , b \rangle \geq 0$ for all $a, b \in A_+$. We say
that $\langle \  | \ \rangle$ is {\em self-dualizing} if $a \mapsto \langle a |$ maps
$A_+$ {\em onto} $A^{\ast}_+$, so that $a \in A_+$ if and only if
$\langle a , b \rangle \geq 0$ for all $b \in B$.  We say that $A$ (or
its positive cone) is {\em self-dual} if $A$ admits a self-dualizing
inner product.  If $A$ is an order unit space, we ordinarily normalize
such an inner product so that $\langle u_A, u_A \rangle = 1$.  In this
case, we can represent states of $A$ {\em internally}: if $\alpha \in
A^{\ast}_+$ with $\alpha(u) = 1$, there is a unique $a \in A_+$ with
$\langle a , b \rangle = \alpha(b)$ for all $b \in A_+$. Conversely,
if $a \in A_+$ with $\langle a , u \rangle = 1$, then $\langle a |$ is
a state.

The probabilistic models associated with classical and quantum
systems, as discussed above, are self-dual.
Indeed, in non-relativistic quantum theory, where $A = \L(\mathbf{H})$, the standard trace inner product
$\langle a , b \rangle = \mbox{Tr}(ab)$ is self-dualizing. Here it is {\em usual} to identify states internally,
i.e., as density operators.

If $A$ and $B$ are both self-dual and $\phi : A \rightarrow B$ is a positive linear mapping, we can use
self-dualizing inner products on $A$ and $B$ to represent the mapping $\phi^{\ast} : B^{\ast} \rightarrow A^{\ast}$ as a positive linear mapping $\phi^{\dagger} : B \rightarrow A$, setting $\langle a , \phi^{\dagger}(b) \rangle = \langle \phi(a) , b \rangle$ for all $a \in A$ and $b \in B$. If $\phi : A \rightarrow A$ is an order-automorphism, then so is $\phi^{\dagger}$.

\subsection{Composites of probabilistic models} If $A$ and $B$ are probabilistic models of two physical systems, one may want to construct a model of  the pair of systems considered together. In quantum mechanics,
where $A = \L(\mathbf{H}_1)$ and $B = \L(\mathbf{H}_2)$, one would form the
model $AB = \L(\mathbf{H}_1 \otimes \mathbf{H}_2)$ associated with the tensor
product of the two Hilbert spaces. In the framework of general
probabilistic models, there is no such canonical choice for a model of
a composite system. However, one can at least say what one {\em means}
by a composite of two probabilistic models: at a minimum, one should
be able to perform measurements on the two systems separately, and
compare the results. More formally, there should be a mapping $\pi : A
\times B \rightarrow AB$ taking each pair of effects $(a,b) \in A
\times B$ to an effect $\pi(a,b) \in AB$. One would like this to be
       {\em non-signaling}, meaning that the probability of obtaining
       a particular effect on one of the component systems in a state
       $\omega \in \Omega(AB)$ should be independent of what
       observable is measured on the other system. One can show that
       this is equivalent to $\pi$'s being bilinear, {}%\color{red} with $\pi(u_A, u_B) = u_{AB}$.}
       \cite{BarnumWilceFoils}. Finally, one would like to be able to
       prepare $A$ and $B$ separately in arbitrary
       states. Summarizing:

\begin{definition} \label{def: composites}
{\em
A (non-signaling) {\em composite} of probabilistic models $A$ and $B$, is a pair $(AB,\pi)$ where
$AB$ is a probabilistic model, and $\pi : A \times B \rightarrow AB$ is a bilinear mapping such that
\begin{itemize}
\item[(a)] $\pi(a,b) \in (AB)_+$ for all $a \in A_+$ and $b \in B_+$;
\item[(b)] $\pi(u_A, u_B) = u_{AB}$;
\item[(c)] For every pair of states $\alpha \in \Omega(A)$ and $\beta \in \Omega(B)$, there exists a
state $\gamma$ in $\Omega(AB)$ such that $\gamma(\pi(a,b)) = \alpha(a)\beta(b)$.
\end{itemize}
}
\end{definition}

Since $\pi$ is bilinear, it extends uniquely to a linear mapping $A \otimes B \rightarrow AB$, which
we continue to denote by $\pi$ (so that $\pi(a \otimes b) = \pi(a,b)$ for $a \in A, b \in B$).

\begin{lemma} $\pi$ is injective. \end{lemma}

\noindent{\em Proof:} If $\pi(T) = 0$ for some $T \in A \otimes B$,
then for all states $\alpha, \beta$ on $A$ and $B$, we have a state
$\gamma$ on $AB$ with $(\alpha \otimes \beta)(T) = \gamma(\pi(T)) =
0$. But then $T = 0$. $\Box$\\

This warrants our treating $A \otimes B$ as a subspace of $AB$ and writing $a \otimes b$ for $\pi(a,b)$.
Note that if $\omega$ is a state on $AB$, then $\pi^{\ast}(\omega) := \omega \circ \pi$ defines
a joint probability assignment on effects of $A$ and $B$:
\[\pi^{\ast}(\omega)(a,b) = \omega(a \otimes b).\]
This gives us marginal states $\omega_A = \omega(u_A \otimes - )$ and $\omega_B = \omega(- \otimes u_B)$. Where
these are non-zero, we can also define conditional states $\omega_{1|b}(a) := \omega(a \otimes b)/\omega_{B}(b)$
and $\omega_{2|a}(b) = \omega(a \otimes b)/\omega_{A}(a)$.\footnote{In the context of a more general definition
of probabilistic models, in which the cone generated by allowed states might not be the full dual cone
$A^{\ast}_+$, we would need to modify this definition
to enforce that these conditional states belong to the allowed state-space. See \cite{BarnumWilceFoils} for
details.}

When the mapping $\pi : A \otimes B \rightarrow AB$ is surjective, we can identify $AB$ with $A \otimes B$. The joint probability assigment $\pi^{\ast}(\omega)$ then completely determines $\omega$, so that states on $AB$
{\em are} such joint probability assignments. In this case, we say that $AB$ is {\em locally tomographic},
since states of $AB$ can be determined by comparing the results of ``local" measurements, i.e., measurements carried out on $A$ and $B$ alone. In finite dimensions, both classical and {\em complex} quantum-mechanical composites have this feature, while composites of real quantum systems are not locally tomographic \cite{Araki80, Hardy-Wootters}.

When dealing with dynamical probabilistic models, one needs to
supplement conditions (a), (b) and (c) with the further condition that
it should be possible for $A$ and $B$ to evolve independently within
the composite $AB$. That is:

\begin{definition}\label{def: dynamical composites}
{\em A composite of dynamical probabilistic models $A$ and $B$ is a composite $AB$, in the sense of
Definition \ref{def: composites}, plus a mapping $\otimes : G(A) \times G(B) \rightarrow G(AB)$ selecting,
for each $g \in G(A)$ and $h \in G(B)$, {}%{\color{magenta} an element} $g \otimes h \in G(AB)$, such that
\begin{itemize}
\item[(a)] $(g \otimes h)(a \otimes b) = ga \otimes hb$ for all $g \in G(A)$, $h \in G(B)$, $a \in A$ and $b \in B$,
and
\item[(b)] for $g_1, g_2 \in G(A)$ and $h_1, h_2 \in G(B)$,
\[ (g_1 \circ g_2) \otimes (h_1 \circ h_2) = (g_1 \otimes h_1) \circ (g_2 \otimes h_2).\]
\end{itemize}
}
\end{definition}
(Note that since $AB$ may be larger than the algebraic tensor product $A \otimes B$, the symmetry
$(g \otimes h)$ need not be uniquely
{}%{\color{magenta} determined} by condition {}%{\color{magenta} (a)}.)

\subsection{Probabilistic theories as categories} A physical theory is more than a collection of models.
At a minimum, one also needs the means to describe interactions between physical systems.
A natural way of accomplishing this is to treat physical theories as {\em categories}, in which
objects represent physical systems, and morphisms represent processes.
In the setting of this paper, then, it's natural to regard a probabilistic {\em theory} as
a category $\Cat$ in which objects are probabilistic models, i.e., order unit spaces, and in which
morphisms give rise to positive linear mappings between these.

The reason for this phrasing --- morphisms {\em giving rise to}, as opposed to simply {\em being},
positive linear mappings --- is to allow for the possibility that two abstract processes
that behave the same way on effects of their source system, may differ
in other ways---even in detectable ways, such as their effect on
composite systems of which the source and target systems are
components.  If { distinct morphisms between the same two
objects always induce the same positive map}, we say the category, and
the set of morphisms, {\em has} \emph{local process tomography}\footnote{One way to make this more precise is to require that $\Cat$ contain $\mathbb{R}$, ordered as usual and with order unit $1$,
and that $\Cat(I,A)$ {\em be} the cone of positive linear maps $\mathbb{R} \rightarrow A$, so that $\Cat(I,A) \simeq A_+$. Any morphism $\phi \in \Cat(A,B)$ then gives rise to a mapping $\hat{\phi} : \Cat(I,A) \rightarrow \Cat(I,B)$ by $\hat{\phi}(a) = \phi \circ a$ for every $a \in \Cat(I,A)$; this extends to a positive linear mapping $A \rightarrow B$. We shall not pursue this further here; see \cite{BarnumWilce11} for more on these lines. }.
Notice that invertible morphisms $A \rightarrow A$ that preserve the order unit then induce processes in the sense given above, so that every model $A \in \Cat$ carries a distinguished group of reversible processes: models in
$\Cat$, in other words, are automatically {\em dynamical} models.

In order to allow for the formation of composite systems, it is
natural to ask that $\Cat$ be a symmetric monoidal category. That is,
we wish to equip $\Cat$ with a bifunctorial product $\otimes : \Cat
\times \Cat \rightarrow \Cat$ that is naturally associative and
commutative, and for which there is a unit object $I$ with $I \otimes
A \simeq A \simeq A \otimes I$ for objects $A \in \Cat$.  Of course,
we want to take $I = \mathbb{R}$. Moreover, for objects $A, B \in \Cat$, we
want $A \otimes B$ to be a composite in the sense of Definitions
\ref{def: composites} and \ref{def: dynamical composites} above.
In fact, though, every part of those definitions
simply {\em follows from} the monoidality
    of $\Cat$, except for part (b) of \ref{def: composites}; we must
    add ``by hand" the requirement that $u_A \otimes u_B = u_{A
      \otimes B}$.  The category will also pick out, for each object
    $A$, a preferred group $G(A)$, namely, the group of invertible
    morphisms in $\Cat(A,A)$.  The monoidal structure then picks out,
    for $g \in g(A)$ and $h \in G(B)$, a preferred $g \otimes h \in
    G(AB)$.

\section{Background on Euclidean Jordan algebras}
\label{sec: background Jordan}

In this section, we summarize the essential background information
on euclidean Jordan algebras and their universal tensor products that
will be used in the sequel. General references for this material are
the monographs \cite{AS} of Alfsen and Shultz and \cite{FK}
of Faraut and Koranyi and \cite{HO-Stormer} of Hanche-Olsen and
St{\o}rmer, plus the paper \cite{HO} of Hanche-Olsen.

\subsection{Euclidean Jordan algebras}

As already mentioned, a {\em euclidean Jordan algebra} (hereafter:
EJA) is a finite-dimensional commutative (but not necessarily
associative) real algebra $(A,\dot)$ with a multiplicative unit
element $u$, satisfying the {\em Jordan identity} $a^2 \dot (a \dot
b) = a \dot (a^2 \dot b)$ for all $a,b \in A$, and equipped with an
inner product satisfying $\langle a \dot b , c \rangle = \langle b ,
a \dot c \rangle$ for all $a,b,c \in A$.  Obviously, any commutative,
associative real algebra provides an example, but not an interesting
one. The basic interesting example is the self-adjoint part of a
complex matrix algebra $M$, with $a \dot b = (ab + ba)/2$ and with
$\langle a, b \rangle = \mbox{Tr}(ab)$.  Any Jordan subalgebra of an EJA is
also an EJA.  Since real and quaternionic matrix algebras have
representations as subalgebras of complex matrix algebras, their
self-adjoint parts are again EJAs. So, too, is the {\em spin factor}
$V_n = \mathbb{R} \times \mathbb{R}^n$, with the obvious inner product and with
\[(t,x) \dot (s,y) = (ts + \langle x, y \rangle, ty + sx);\]
this can be embedded in $M_{2^{k}}(\mathbb{C})$ if $n = 2k$ or $2k+1$, as discussed in more detail in Appendix \ref{appendix: spin factors} \\

\noindent{\bf Classification} Direct sums of EJAs are also EJAs, so we
can obtain more examples by forming direct sums of the EJAs of the
types mentioned above. The {\em Jordan-von Neumann-Wigner
  Classification Theorem} \cite{JNW} provides a near converse: every euclidean
Jordan algebra is a direct sum of {\em simple} EJAs, each of which is
isomorphic to a spin factor $V_n$, or to the self-adjoint part of a
matrix algebra $M_n(\mathbb{F})$ where $\Bbb{F}$ is one of the classical division
rings $\mathbb{R}, \mathbb{C}$ or $\mathbb{H}$, {\em or}, if $n = 3$,  the {\em octonions},
$\O$.  This last example, which is not embeddable into a complex matrix
algebra, is called the {\em exceptional Jordan algebra}, or the {\em
  Albert algebra}. \\

A Jordan algebra that {\em is} embeddable in the self-adjoint part
of a complex matrix algebra is said to be {\em special}.   In addition to
$M_{n}(\mathbb{C})_{{sa}}$, the simple EJAs $M_n(\mathbb{R})_{{sa}}$, $M_n(\mathbb{H})_{{sa}}$ and $V_n$ are
all special. It follows
from the classification theorem that any EJA decomposes as a direct
sum $A_{\sp} \oplus A_{\mbox{\em ex}}$ where $A_{\sp}$ is special and $A_{\mbox{\em ex}}$
is a direct sum of copies of the exceptional Jordan algebra.
\\

\noindent{\bf Operator commutation} For each $a \in A$, define $L_a : A \rightarrow A$ to be
the operation of Jordan-multiplication by $a$: $L_a(x) = a \dot x$ for all $x \in A$. Elements
$a, b \in A$ are said to {\em operator commute} iff $L_a \circ L_b = L_b \circ L_a$. If
$A$ is a Jordan subalgebra of $M_{{sa}}$, where $M$ is a complex $\ast$-algebra, then
for all $x \in A$,
\[4 L_a (L_b x)  = a(bx + xb) + (bx + xb)a = abx + axb + bxa + xba\]
and similarly
\[4 L_b(L_a x) = bax + bxa + axb + xab.\]
If $a$ and $b$ operator-commute, the left-hand sides are equal. Subtracting, we have
\[abx + xba - bax - xab = 0\]
or
\[[a,b]x + x[b,a] = 0\]
which is to say, $[a,b]x - x[a,b] = 0$. If $M$ is unital and $A$ is a unital subalgebra, so that $u_A = 1_{\mathbf{M}}$,
then setting $x = 1_{M} \in A$ gives us $[a,b] = -[a,b]$, i.e., $a$ and $b$ commute in $M$.
\\

\noindent
{\bf Projections and the Spectral Theorem}
A {\em projection}
in an EJA $A$ is an element $a \in A$ with $a^2 = a$. If $p, q$ are
projections with $p \dot q = 0$, we say that $p$ and $q$ are
orthogonal.  In this case, $p + q$ is another projection. A projection
not representable as a sum of other projections is said to be {\em
  minimal} or {\em primitive}. A {\em Jordan frame} is a set $E
\subseteq A$ of pairwise orthogonal minimal projections that sum to
the Jordan unit.  The {\em Spectral Theorem} for EJAs (see e.g.
\cite{FK}, Theorem III.1.1, or \cite{AS}, Theorem 2.20 {}%{\color{blue} for}
an infinite-dimensional version) asserts that every element $a \in A$
can be expanded as a linear combination $a = \sum_{x \in E} t_x x$
where $E$ is some Jordan frame {}%\color{red} and $t_x$ is a coefficient in $\mathbb{R}$ for each $x \in E$.}

The group of Jordan automorphisms acts transitively on the set of
Jordan frames (\cite{FK}, Theorem IV.2.5).  Hence all
Jordan frames for a given euclidean Jordan algebra $A$ have the same
number of elements.  This number is called the {\em rank} of $A$. By
the Classification Theorem, all simple Jordan algebras having rank $4$
or higher are special. \\

\subsection{Euclidean Jordan algebras as ordered vector spaces}

Any euclidean Jordan algebra $A$ can be regarded as an ordered real
vector space, with positive cone $A_+ = \{ a^2 | a \in A\}$. (That this
{\em is} a cone is a non-trivial fact (see \cite{FK}, Theorem III.2.1, or \cite{ASbasic}, pp. 36-28). By the
spectral theorem, $a = b^2$ for some $b \in A$ iff $a$ has a spectral
decomposition $a = \sum_{i} \lambda_i x_i$ in which all the
coefficients $\lambda_i$ are non-negative.

 The Jordan unit $u$ is also an order unit; thus, any EJA can serve as a probabilistic
model, as defined in Section 2:  physical states correspond to states {\em qua} normalized positive linear
functionals on $A$, while
measurement outcomes are represented by \emph{effects}, i.e,
elements $a \in A_+$ with $0 \leq a \leq u$, and (discrete) observables,
by sets $\{e_i\}$ of events with $\sum_i e_i = u$.

 As discussed earlier, the inner product on $A$ allows us to represent states internally, i.e.,
for every state $\alpha$ there exists a unique $a \in A_+$ with $\alpha(x) = \langle a | x \rangle$
for all $x \in A$; conversely, every vector $a \in A_{+}$ with $\langle a | u \rangle = 1$ defines
a state in this way. Now, if $a$ is a projection, i.e., $a^2 = a$, let $\hat{a} = \|a\|^{-1} a$:
then
\[\langle \hat{a} | u \rangle = \frac{1}{\|a\|^2} \langle a | u \rangle =
\frac{1}{\|a\|^2} \langle a^2 | u \rangle =
\frac{1}{\|a\|^2} \langle a | a \rangle = 1.\]
Thus, $\hat{a}$ represents a state. A similar computation shows that $\langle \hat{a} | a \rangle = 1$.
Thus, every projection, regarded as an effect, has probability $1$ in some state. The following is the
converse:

\begin{lemma} \label{lemma: projections}
Let $A$ be an EJA, and let $a \in A_+$ with $a \leq u$. If there exists a state $\alpha$ on $A$
with $\alpha(a) = 1$, then $a$ is a projection. \end{lemma}

\noindent{\em Proof:} By the Spectral Theorem and the fact that $0 \le a$, $a = \sum_{x \in
  E} t_x x$ for some Jordan frame $E$ and coefficients $t_x \geq 0$,
$x \in E$.  Since $a \le u$, for any fixed $x_o \in E$ we have
$t_{x_o} = \langle x_o, a \rangle \leq \langle x_o, u \rangle =
1$. Now suppose there is a state $\alpha$ with $\alpha(a) = 1$. Then
we have $\sum_{x \in E} t_x \alpha(x) = 1$. Since $\sum_{x \in E}
\alpha(x) = \alpha(u) = 1$, it follows that $1$ is a convex
combination of the coefficients $\{t_x\}$, each of which lies between
$0$ and $1$.  Hence, $t_x = 1$ or $t_x = 0$ for every $x \in E$,
whence, $a = \sum_{x \in A} x$ for a set $A \subseteq E$.  Since the
elements of $A$ are pairwise orthogonal projections, $a$ is also a
projection. $\Box$ \\

A {\em sharp} observable is one in which every outcome has probability one in some state.
The preceding Lemma tells us that these are the same as projection-valued observables. A maximally
fine-grained sharp observable is therefore a collection $\{p_i\}$ of  {\em minimal} non-zero projections
summing to the unit $u$, i.e., a Jordan frame.\\

{}%{\color{blue} \noindent{\bf  Order-automorphisms} The order structure of an EJA $A$, together with its inner product an order unit, entirely determines its Jordan structure. One manifestation of this is that
a symmetry of $A$ --- that is, an order-automorphism preserving the unit $u_A$ --- is the same thing as a
Jordan automorphism (\cite{AS}, Theorem 2.80).}

Another class of order automorphisms is given by the {\em quadratic representations} of certain elements of $A$.
The quadratic representation of $a \in A$ is
the mapping $U_a : A \rightarrow A$ given by
\[U_a = 2L_{a}^{2} - L_{a^2}\]
i.e., $U_a(x) = 2a(ax) - a^2(x)$.
These mappings have direct physical interpretations as {\em filters} in the sense of \cite{Wilce12}.
The following non-trivial facts will be used repeatedly in what follows:

\begin{proposition}\label{prop: quadratic representation} Let $a \in A$. Then
\begin{itemize}
\item[(a)] $U_a$ is a positive mapping;
\item[(b)] If $a$ lies in the interior of $A_+$, $U_a$ is invertible,  with
inverse given by $U_{a^{-1}}$;
\item[(c)] $e^{L_{a}} = U_{e^{a/2}}$
\end{itemize}
\end{proposition}

\noindent{\em Proof:} For (a),
see  Theorem 1.25 of \cite{AS}; for (b), \cite{AS} Lemma 1.23 or \cite{FK}, Proposition II.3.1.
Part (c) is Proposition II.3.4 in \cite{FK}.  $\Box$\\

Combining (a) and (b), $U_a$ is an order automorphism for every $a$ in the interior of $A_+$.
Regarding (c), note that $e^{L_a}$ is the ordinary operator exponential; in other words,
$\phi_{t} := e^{tL_{a}} = U_{\frac{t}{2}a}$ is a one-parameter group of order-automorphisms in $G(A)$ with $\phi'(0) = L_{a}$.

{}%{\color{blue} Since $U_a(u_A) = 2a^2 - a^2 = a^2$, it follows that the group of order-automorphisms of $A$ act transitively on
 the interior of $A_+$. Abstractly, an ordered vector space having this property is said to be {\em homogeneous}.
 It follows that if $\phi$ is any order-automorphism with $\phi(u_A) = a^2 \in A_+$, then $U_{a}^{-1} \circ \phi$ is a symmetry of $A$. Hence, every order-automorphism of $A$ decomposes as $\phi = U_{a} \circ g$ where $g$ is a symmetry.
In fact, one can show that $a$ can be chosen to belong to the interior of $A_+$, and that, with this
 choice, the decomposition is unique (\cite{FK}, III.5.1)\\

\noindent{\bf Self-Duality and the KV Theorem} The inner product on an EJA $A$ is self-dualizing. Recall that this
means that an element $a \in A$ belongs to the cone $A_+$ iff $\langle a | b \rangle \geq 0$ for all elements $b$ of $A_+$ --- equivalently, iff $\langle a |$ belongs to the dual cone $A_{+}^{\ast}$.
Thus, as an ordered vector space, $A$ is both self-dual and homogeneous. Conversely, if $(A,u)$ is an homogenous  finite-dimensional order unit space with a self-dualizing inner product, there exists a {\em unique} bilinear
product on $A$ making $A$, with the given inner product, a euclidean Jordan algebra, and the order unit, the Jordan unit. This is the content of the {\em Koecher-Vinberg Theorem}. {}%\color{red} While we make no use of this result here, it is at the
center of efforts to provide an operational motivation for euclidean Jordan algebras as models of physical systems,
e.g., in \cite{Wilce12, BMU}. }

\subsection{Representations of EJAs}
\label{subsec: reps of EJAs}

A {\em representation} of a Jordan algebra $A$ is a Jordan
homomorphism $\pi : A \rightarrow M_n(\mathbb{C})_{{sa}}$ for some $n$.\footnote{This is a finite dimensional, concrete representation.  For the finite-dimensional algebras we are concerned with,
this definition suffices.}
\footnote{}%{\color{blue} AW: Do we want to allow range to be an arbitrary f.d. complex matrix algebra?}
So a Jordan algebra is special iff it has an injective, {}%{\color{blue} or {\em faithful}}, representation.
Recall that every EJA decomposes
as a direct sum $A = A_{\sp}
\oplus A_{\mbox{\em ex}}$, where $A_{\sp}$ is special and $A_{\mbox{\em ex}}$ has no
non-trivial
representations.
The latter, in turn, is a direct sum of copies of the exceptional EJA $M_3(\O)_{{sa}}$. {}%\color{red} See \cite{AS}, Theorem 4.3 for details.}\\

\noindent {\bf Standard Representations} For the non-exceptional
simple EJAs, it will be useful to record what we will call their {\em
  standard} representations.
It will also prove helpful to adopt the following abbreviations:
\[R_n = M_n(\mathbb{R})_{{sa}}; \ C_n = M_n(\mathbb{C})_{{sa}}, \ Q_n = M_n(\H)_{{sa}}.\]
 As above, we write $V_n$ for the spin factor $\mathbb{R} \times \mathbb{R}^{n}$.  With this notation
we have obvious embeddings $R_n \leq C_n \leq M_n(\mathbb{C})_{{sa}}$. For $Q_n$, note that a quaternion
$a + bi + cj + dk$ can be written as $(a + bi) + (c + di)j$, and so, can be represented by
the pair of complex numbers $(a + bi, c + di)$.
Thus,  any $n \times n$ matrix of quaternions can be represented
as a $2n \times 2n$ complex matrix having the form $\left [ \begin{array}{rc} \Gamma_1 & \Gamma_2 \\ -\bar{\Gamma}_2 & \bar{\Gamma}_1 \end{array} \right ]$ where the blocks $\Gamma_1$ and $\Gamma_2$ are $n \times n$ complex matrices.
This gives us a faithful representation of $Q_n$ in $M_{2n}(\mathbb{C})$, {}%\color{red} known as the {\em symplectic representation}
\cite{Graydon11}.}
There is also
what we will call a {\em standard representation}
 of $V_n$ in $M_{2^k}(\mathbb{C})$, where $n = 2k$ or $2k + 1$.  This is less obvious; the
details are given in Appendix \ref{appendix: spin factors}.\\

\noindent{\bf Involutions and Reversibility}
By an {\em involution} on a complex $\ast$-algebra $M$, we mean
a  $\ast$-anti-automorphism of $M$ of period $2$ --- in more detail, a linear mapping
$\Phi : M \rightarrow M$ such that $\Phi(a^{\ast}) = \Phi(a)^{\ast}$, $\Phi(ab) = \Phi(b)\Phi(a)$, and
$\Phi(\Phi(a)) = a$ for all $a \in M$. A {\em real} involution is defined similarly, except that the
mapping is only required to be real-linear.\footnote{Our usage is slightly nonstandard here: an involution
on a complex $\ast$-algebra is more frequently defined to be a conjugate-linear. The only involution in this sense
that will concern us is $a \mapsto a^{\ast}$.} It is straightforward that the set
$M^{\Phi}_{{sa}} = \{ a \in M | a = a^{\ast} = \Phi(a)\}$ of all self-adjoint fixed-points of $M$ under $\Phi$ is a Jordan subalgebra of $M_{{sa}}$.
Indeed, if $a, b \in M^{\phi}_{{sa}}$, then $a \dot b = \tfrac{1}{2}(ab + ba) \in M_{{sa}}$, and
\[ \Phi(ab + ba) = \Phi(b)\Phi(a) + \Phi(a)\Phi(b) = ba + ab = ab + ba\]
so that $a \dot b \in M^{\Phi}_{{sa}}$ as well. In fact, more is true: if $a_1,...,a_n \in M^{\Phi}_{{sa}}$, then
\[\Phi(a_1 \cdots a_n + a_n \cdots a_1) = a_n \cdots a_1 + a_1 \cdots a_n \]
so that $a_1 \cdots a_n + a_n \cdots a_1 \in M^{\Phi}_{{sa}}$. }
More generally, a Jordan subalgebra of $M_{{sa}}$ having this property is said to be {\em reversible}. An
abstract EJA $A$ is reversible iff it has a faithful (that is, injective) representation as a reversible Jordan subalgebra
of some $\ast$-algebra $M$.  If {\em all} of $A$'s faithful representations are reversible, then $A$ is said to
be {\em universally reversible} (hereafter: UR). \\

In this language, then, all Jordan algebras of the form $M^{\Phi}_{{sa}}$ are reversible.
All self-adjoint parts of real and complex matrix algebras are universally reversible, as are
the quaternionic ones of rank 3 and higher {}%\color{red} (whence, all EJAs of rank $\geq 4$ are UR)}.  The quaternionic bit
$M_2(\mathbb{H})_{{sa}}$, which is isomorphic to $V_5$, is not universally reversible, but
it is reversible, since its
standard embedding into $M_4(\mathbb{C})$ is reversible: indeed, it's the set fixed points
of the involution $\Phi(x) = -J x J^{t}$ where $J = \left [ \begin{array}{rc} 0 & \mathbf{1} \\ -\mathbf{1} & 0 \end{array} \right ]$. Spin factors $V_n$ with $n=4$ or $n
\geq 6$ are not reversible at all. For details, see \cite{HO}. Thus,
the reversible {\em simple} EJAs are precisely those of the form $M_n(\mathbb{R})_{{sa}}$, $M_{n}(\mathbb{C})_{{sa}}$ and $M_{n}(\mathbb{H})_{{sa}}$.  \\

\noindent{\bf The universal representation}  In addition to the standard representations discussed above,
every special EJA has a {\em universal} representation.

\begin{definition}
{\em
A \emph{universal $\Cu$ algebra} for a euclidean Jordan algebra $A$
\footnote{}%{\color{blue} Should
we just say this is for JC-algebras?} is a
{}%{\color{blue} complex $\ast$-algebra}
$\Cu(A)$, plus an embedding $\psi_A : A \rightarrow \Cu(A)_{{sa}}$,
such that for $C^{\ast}$-algebra $\mathbf{M}$ and any Jordan
homomorphism $\phi : A \rightarrow \mathbf{M}_{{sa}}$, there exists a unique
$\ast$-homomorphism $\hat{\phi} : \Cu(A) \rightarrow \mathbf{M}$ with
$\phi = \hat{\phi} \circ \psi_A$.
\footnote{}%{\color{blue} We really need to sort out whether we need exactly this construction, or just the universal enveloping algebra from Jacobson, in our EJA setting.}
}
\end{definition}

For the existence of $\Cu(A)$, see \cite{AS} or \cite{HO-Stormer}.
Universal $C^*$-algebras for a given EJA $A$ are
unique up to {}%\color{red} canonical $\ast$}-isomorphism, warranting our speaking of
``the'' universal $C^*$-algebra of $A$.\footnote{}%{\color{blue} In fact, by privileging
any particular construction of $\Cu(A)$, we can take this literally.}
It is easy to see that any Jordan homorphism $A \rightarrow B$ lifts, via
  $\psi_A$ and $\psi_B$, to a $C^*$-homomorphism $\Cu(A) \rightarrow
  \Cu(B)$, and that this lifting respects composition of Jordan homomorphisms,
  so that $\Cu(\ \cdot \ )$ defines a functor from the category of EJAs to the
  category of {}%{\color{blue} complex} $\ast$-algebras.  It can be shown that this
  functor is exact (\cite{HO}, Theorem 4.1).
 It is an important fact that $A$ is exceptional iff $\Cu(A) = \{0\}$.
  Otherwise, the embedding $\psi_A$ of $A$ into $\Cu(A)$ is faithful,
  and takes the unit of $A$ to the unit of $\Cu(A)$.  In
  this case, we will often identify $A$ with its embedded image in
  $C^*(A)$.

  If $A$ is special, $\Cu(A)$ comes equipped with a unique
    involution that fixes every point of $A$.  To see
    this, note that that the opposite algebra $\Cu(A)^{\mbox{op}}$ (the same
    vector space, equipped with reversed multiplication) is equally a
    universal $C^{\ast}$-algebra for $A$. Hence, there is a unique
    $\ast$-isomorphism $\Phi_{A} : \Cu(A)^{\mbox{op}} \rightarrow \Cu(A)$
    fixing all points of $A$. We can equally well regard $\Phi$ as a
    $\ast$-antiautomorphism of $\mbox{Aut}(A)$; so regarded, $\Phi$ is
    self-inverse, {}%{\color{magenta} hence} an involution.

{}%{\color{magenta}

\begin{definition}\label{def: canonical involution}
We call the involution $\Phi: C^*(A) \rightarrow C^*(A)$ just described, the
\emph{canonical involution}, $\Phi_A$, on $C^*(A)$.
\end{definition}
}

As discussed above, the self-adjoint fixed-points of an involution $\Phi$ on a complex $\ast$-algebra $M$
  constitute a
  Jordan subalgebra, $M^{\Phi}_{{sa}}$, of $M_{{sa}}$. With $\Phi_{A}$ the canonical
  involution on $\Cu(A)_{{sa}}$, we have a Jordan embedding $A \leq \Cu(A)^{\Phi}_{{sa}}$.

\begin{proposition}[\cite{HO}, Lemma 4.2 and Theorem 4.4]
\label{prop: UR fixed points}
With notation as above, $A$ is UR iff $A = \Cu(A)^{\Phi}_{{sa}}$.
  More generally, {}%\color{red} if $A$ is UR and} there exists an embedding $A \leq
  M_{{sa}}$ for a complex $\ast$-algebra $M$, and an involution $\Phi$
  on $M$ fixing points of $A$, then the $\ast$-subalgebra of
  $M$ generated by $A$ is isomorphic to $\Cu(A)$ and $\Phi$,
  restricted to this subalgebra, is the canonical involution.
\end{proposition}

This characterization allows the explicit computation of
  universal $C^{\ast}$-algebras of UR EJAs \cite{HO}. For
    instance, $R_n = M_n(\mathbb{R})_{{sa}}$ generates $M_n(\mathbb{C})_{{sa}}$ as a
    (complex) $\ast$-algebra, in which it sits as the set of symmetric
    self-adjoint matrices: in other words, the set of fixed points of
    the involution $\Phi(a) = a^{T}$. Thus, $\Cu(R_n) =
    M_n(\mathbb{C})$. Similarly, the image of $M_n(\mathbb{H})_{{sa}}$ under the
    standard (symplectic) embedding into $M_{2n}(\mathbb{C}) =
    M_2(M_n(\mathbb{C}))$ is fixed by the involution $\Phi(a) = -J a^T J$
    where $J$ is the block matrix $\left [ \begin{array}{cc} 0 & \mathbf{1}
        \\ -\mathbf{1} & 0 \end{array} \right ]$, with $\mathbf{1}$ the $n \times n$
    identity matrix. For a third example, consider the
    embedding $M_n(\mathbb{C})_{{sa}} \rightarrow (M_n(\mathbb{C}) \oplus
    M_n(\mathbb{C}))_{{sa}}$ given by $\psi(a) = (a,a^T)$: the image of this
    embedding generates $M_n(\mathbb{C}) \oplus M_n(\mathbb{C})$ as a $\ast$-algebra,
    and is exactly the set of fixed-points of the involution
    $\Phi(a,b) = (b^T, a^T)$. Thus, $\Cu(C_n) = M_n(\mathbb{C}) \oplus
    M_n(\mathbb{C})$.  Note, in passing, that this also
    shows that there is no involution on $M_n(\mathbb{C})$ fixing points of
    $C_n = M_n(\mathbb{C})_{{sa}}$. {}%\color{red} (Otherwise, $M_n(\mathbb{C})$ would be the universal
    $\ast$-algebra for $C_n$.)}

The universal $C^{\ast}$-algebras for all simple, non-exceptional EJAs
are are summarized in Table 1 below, {}%\color{red} along with the canonical involutions in the UR cases.}
{}%\color{red} For a spin factor $V_n$, the universal $C^{\ast}$-algebra is the complex Clifford
algebra $\mbox{Cliff}_{\mathbb{C}}(n)$ on $n$ generators \cite{HO}; these are tabulated separately in Table 1(b).
 }

For contrast, Table
1(c) lists the complex matrix algebras into which $R_n, C_n$ and $Q_n$ are
{\em standardly} embedded.  We also list in Fig. 1 (c) algebras
supporting what we are calling standard embeddings of
the spin factors $V_n$; these agree with the
universal ones for $n = 2k$, but for $n=2k+1$, embed $V_n$ in
$M_{2^k}(\mathbb{C})$ rather than the $M_{2^k}(\mathbb{C}) \oplus M_{2^k}(\mathbb{C})$ of the
universal embedding.  Thus, all the targets of embeddings in Fig.
1(c) are \emph{simple} complex matrix algebras.
\[
\begin{array}{ccc}
\begin{array}{l|l|l}
A  & \Cu(A) & \Phi \\
\hline
R_n & M_n(\mathbb{C}) & a \mapsto a^T\\
C_n & M_n(\mathbb{C}) \oplus M_n(\mathbb{C}) & (a,b)  \mapsto (b^T,a^T) \\
Q_n & M_{2n}(\mathbb{C}) \ \mbox{if} \ n > 2 & a \mapsto -J a^T J  \\
V_n & \mbox{Cliff}_{\mathbb{C}}(n) &
 \\
\vspace{.05in} &
\end{array}
& &
\begin{array}{l|l}
n ~(k \in \mathbb{N}_+ )  & { \mbox{Cliff}_{\mathbb{C}}(n)}  \\
\hline
2k & M_{2^k}(\mathbb{C}) \\
2k+1 & M_{2^k}(\mathbb{C}) \oplus M_{2^k}(\mathbb{C})\\
\vspace{.4in}
\end{array}\\
\mbox{(a)} & & \mbox{(b)} \\
 \mbox{ Universal embeddings } & & \mbox{ Clifford algebras}
\end{array}
\]
\[
\begin{array}{c}
\begin{array}{l|l}
A  & \mathbf{M} \\
\hline
R_n & M_n(\mathbb{C}) \\
C_n & M_n(\mathbb{C}) \\
Q_n & M_{2n}(\mathbb{C})   \\
V_{2k} & M_{2^k}(\mathbb{C}) \\
V_{2k+1} & M_{2^k}(\mathbb{C})\\
\end{array} \\
\mbox{(c)}\\
\mbox{ Standard embeddings}\\
\\
\mbox{{\bf Table 1:} Unversal and standard embeddings}
\end{array}\]
\vspace{.3in}

Note that the spin factors $V_2, V_3, V_5$ correspond to the three
types of quantum bits: $V_2 \simeq R_2$; $V_2 \simeq C_2$ and $V_5 \simeq Q_2$.
This last, together with line 2 of Fig. 1 table (b), gives us the missing item in table (a):
\[\Cu(Q_2) \simeq M_4(\mathbb{C}) \oplus M_4(\mathbb{C}).\]

It will be helpful to record here two further facts about
  universal $C^{\ast}$ algebras.  First, $\Cu(A \oplus B) = \Cu(A)
  \oplus \Cu(B)$. This follows from the exactness of the $\Cu(
  \ )$-functor.  Combining this with Proposition
\ref{prop: UR fixed points}, it follows that
  if $A$ and $B$ are both UR, so is $A \oplus B$. Details can be found
  in Appendix A.

\subsection{The universal tensor product}

The universal representation allows one to define a natural tensor product of special EJAs, {}%{\color{blue} first studied
by H. Hanche-Olsen \cite{HO}:}

\begin{definition}
{\em
The {\em universal} tensor
product of two special EJAs $A$ and $B$, denoted $A \widetilde{\otimes} B$
is the Jordan subalgebra of
$\Cu(A) \otimes \Cu(B)$ generated by $A \otimes B$.
}
\end{definition}

Some important facts about the universal tensor product are collected
in the following:

\begin{proposition} Let $A$, $B$ and $C$ denote EJAs.
\label{prop: universal tensor product properties}
\begin{itemize}
\item[(a)]  If $\phi : A \rightarrow C$, $\psi : B \rightarrow C$ are unital Jordan homomorphisms with operator-commuting ranges, then
there exists a unique Jordan homomorphism $A \widetilde{\otimes} B \rightarrow C$ taking $a
 \otimes b$ to $\phi(a) \dot \psi(b)$ for all $a \in A$, $b \in B$.
\item[(b)] $\Cu(A \widetilde{\otimes} B) = \Cu(A) \otimes \Cu(B) \ \ \mbox{and} \ \ \Phi_{A \widetilde{\otimes} B} = \Phi_{A} \otimes \Phi_{B}$.
\item[(c)] $A \widetilde{\otimes} B$ is universally reversible unless one of the
  factors has a one-dimensional summand and the other has a
  representation onto a spin factor $V_n$ with $n=4$ or $n \ge 6$.
\item[(d)] If $A$ is universally reversible, then $A \widetilde{\otimes}
  M_n(\mathbb{C})_{{sa}} = (\Cu(A) \otimes M_{n}(\mathbb{C}))_{{sa}}$.
\item[(e)] $u_{A \widetilde{\otimes} B} = u_A \otimes u_B = u_{C^*(A \widetilde{\otimes} B)}$.
\end{itemize}
\end{proposition}

\noindent{\em Proof:} (a), (c), and (d) are Propositions 5.2, 5.3 and
5.4, respectively, in \cite{HO}; (b) is observed in the
vicinity of these propositions in \cite{HO}, while (e) follows easily from
the fact that $\psi^A(u_A) = u_{C^*(A)}$.  $\Box$\\

Table 1 (c) shows that if $A$ and $B$ are simple and
non-trivial, $A \widetilde{\otimes} B$ will always be UR, and hence, by
{}%{\color{magenta} Proposition \ref{prop: UR fixed points} and Proposition
\ref{prop: universal tensor product properties} (b)},
the fixed-point set of $\Phi_A \otimes \Phi_B$.  Using this, one can compute $A
\widetilde{\otimes} B$ for simple $A$ and $B$ (with $n, k > 2$ in the case of
$Q_n$ and $Q_k$) { \cite{HO}}:
\[
\begin{array}{c}
\begin{array}{l|ccc}
\widetilde{\otimes}    & R_k & C_k & Q_k \\
 \hline
R_n & R_{nk} & C_{nk} & Q_{nk}    \\
C_n & C_{nk} & C_{nk} \oplus C_{nk} & C_{2nk}  \\
Q_n & Q_{nk} & C_{2nk} & R_{4nk}
\end{array} \\
\\
\mbox{{\bf Table 2:} Universal tensor products of simple UR EJAs }
\end{array}
\]
For $Q_2 \widetilde{\otimes} Q_2$, we obtain the direct sum of four copies of $R_{16} = M_{16}(\mathbb{R})_{{sa}}$. The details
{}%\color{red} can be found in} \cite{Graydon16}.
\footnote{What about $Q_2 \widetilde{\otimes} B$ for other $B$?? For example, what's $Q_2 \widetilde{\otimes} C_2$?}

\section{Composites of Jordan-Algebraic Systems}
\label{sec: composites Jordan}

If $A$ is a euclidean Jordan algebra, recall that we {}%\color{red} take} $G(A)$
{}%\color{red} to be} the connected
component of the identity in the group $\mbox{Aut}(A)$ of
order-automorphisms of $A$. Note that if $\phi \in G(A)$, then
$\phi^{\dagger} \in G(A)$ as well.
{}%{\color{blue} Henceforth, we shall treat Jordan models as dynamical models, using $G(A)$ as the dynamical group. This is
a reasonable choice. First, elements of $G(A)$ are exactly those automorphisms of $A_+$ that figure
in the system's possible dynamics, as elements of
one-parameter groups of automorphisms.
This suggests that the ``physical" dynamical group of a dynamical
model based on $A$ should
at least be a subgroup of $G(A)$, so that the latter is the least constrained choice.
\footnote{}%{\color{blue} Though we might want to think about whether we actually {\em need} all of $G(A)$...}
Moreover, $G(A)$, like the full group of order-automorphisms, acts transitively on the interior of the
cone $A_+$, and its unit-preserving subgroup acts transitively on the set of {\em Jordan frames}, i.e., maximally informative sharp observables --- or, equivalently, on maximal lists $(\alpha_1,...,\alpha_n)$ of sharply-distinguishable  states. These transitivity properties, abstracted from the
Jordan-algebraic setting, were among the postulates used (in somewhat different ways) by
\cite{Wilce09, Wilce11} and \cite{BMU} to derive the Jordan structure of probabilistic models, so it
is not unreasonable to require that the dynamical group enjoy them. We conjecture that the only such
subgroup of $G(A)$ is $G(A)$ itself. }\footnote{}%{\color{blue} AW: I'm not so confident that this is correct. I believe
this is the weakest point of this paper.}

\tempout{HB's comments --- meld with above:
{}%{\color{green}
Besides being mathematically natural, requiring that the entire
connected identity component of each ``local'' automorphism group be
able to act independently in this manner is also operationally
appealing.
 The unit-preserving subgroup (which is a maximal compact
subgroup) of $G(A)$ corresponds to reversible dynamics in which total
probability is preserved, and accomodating it in a
theory-\emph{cum}-category has important consequences for the
information-theoretic and physical processes that are possible in such
a theory.  For example, the fact that this subgroup acts transitively
on the set of Jordan frames, abstracted beyond the Jordan setting as
transitive action of the normalization-preserving subgroup of $G(A)$
on the set of maximal lists of mutually distinguishable pure states,
is one of the postulates used in \cite{BMU}'s characterization of
simple Jordan-algebraic (plus classical) systems, and a similar
postulate, involving transitivity on ordered Jordan frames, is used in \cite{Wilce09, Wilce11}.  Similarly, homogeneity is
linked in \cite{BGW} to important operational properties such as
steering and purification, and further used in \cite{Wilce09, Wilce11} in a
characterization of Jordan systems.  So it is, again, desirable to
allow for the action of a subgroup of $G(A)$ that implements this
homogeneity, i.e. acts transitively on the interior of $A_+$.  The
reversible unit-preserving subgroup by itself does not thus act, but
the entire group $G(A)$ does, and also includes the
unit-preserving subgroup.  We conjecture that for
Jordan-algebraic systems $G(A)$ is the smallest subgroup with these
two operationally significant, properties.}}

This suggests, then, the following

\begin{definition}\label{def: new Jordan composite}
{\em A {\em composite} of EJAs $A$ and $B$ is an EJA $AB$,
plus a bilinear mapping $\pi : A \otimes B \rightarrow AB$, such that
\begin{itemize}
\item[(a)] $\pi$ makes $(AB,G(AB))$ a dynamical composite of $(A,G(A))$ and $(B,G(B))$, in the sense of Definition \ref{def: dynamical composites}
\item[(b)] $(\phi \otimes \psi)^{\dagger} = \phi^{\dagger} \otimes \psi^{\dagger}$ for all $\phi, \psi \in G(A)$.
\item[(c)] $AB$ is generated, as a Jordan algebra, by (images of) pure tensors.
\end{itemize}
{}%\color{red} By Lemma 2.2}, the
mapping $\pi : A \otimes B \rightarrow AB$ is injective;
hence, we can, and shall, identify $A \otimes B$ with its image in $AB$, writing
$\pi(a,b)$ as $a \otimes b$.}
\end{definition}

  Condition (b) is rather strong, but natural if we keep in mind that
  our ultimate aim is to construct dagger-compact categories of EJAs.
  Regarding condition (c), suppose $\pi : A \times B \rightarrow AB$
  satisfied only (a) and (b): letting $A \odot B$ denote the Jordan
  subalgebra of $AB$ generated by $\pi(A \otimes B)$, one can show
  (Appendix \ref{appendix: weak composites}) that the co-restriction of
  $\pi$ to $A \odot B$ also satisfies (a) and (b); thus, any composite
  in the weaker sense defined by (a) and (b) contains a composite
  satisfying all three conditions.

In Section \ref{sec: Euclidean JC}, Proposition \ref{prop: canonical
  product composite}, we will show that $A \widetilde{\otimes} B$ is a composite
in the sense of Definition \ref{def: new Jordan composite}.  The main
result of the present section is to show that
any such composite $AB$
is a direct summand of $A \widetilde{\otimes} B$.  In view of the table in Table 2, this severely
limits the possibilities for $AB$.

\begin{lemma}\label{lemma: product projections}
Let $p \in A$ and $q \in B$ be projections. Then $p \otimes q$ is a
projection in $AB$, for any composite $AB$ of $A$ and $B$.
\end{lemma}

\noindent{\em Proof:} By Lemma \ref{lemma: projections}, there exist
states $\alpha$ and $\beta$ on $A$ and $B$, respectively, with
$\alpha(p) = \beta(q) = 1$. Hence, by condition (c) of Definition 2.1,
there exists a state $\gamma$ on $AB$ with $\gamma(p \otimes q) =
\alpha(p) \beta(q) = 1$. Invoking Lemma \ref{lemma: projections}
again, $p \otimes q$ must be a projection in $AB$. $\Box$\\

This has an important consequence regarding the inner product on $AB$:\footnote{Does this really belong here?}

\begin{proposition}\label{prop: inner} Let $AB$ be a composite of EJAs $A$ and $B$. Then
for all $a, x \in A$ and all $b, y \in B$, $\langle a \odot x | b \odot y \rangle = \langle a | x \rangle \langle b | y \rangle$.
\end{proposition}

The proof
is given in Appendix \ref{appendix: Inner}.
{}%\color{red} It follows that, for $a, a' \in A$ and $b, b' \in B$, $\langle a \otimes b, a' \otimes b' \rangle = 0$ iff
either $\langle a, a' \rangle = 0$ or $\langle b, b' \rangle = 0$. }

\begin{proposition}\label{prop: operator commutation}
For all $a \in A$, $b \in B$, $a \otimes u_B$ and $u_A \otimes b$
operator-commute in $AB$.
\end{proposition}

\noindent{\em Proof:}  Suppose $p \in A$ and $q \in B$ are projections, and let $p' = u_A - p$ and $q' = u_B - q$.
Then we have
\[u_{AB} = u_{A} \otimes u_{B} = (p + p') \otimes (q + q') = p \otimes q + p' \otimes q + p \otimes q' + p' \otimes q'.\]
The four projections appearing on the right are
mutually orthogonal, by \ref{prop: inner}, and sum to the unit in $AB$.  Hence, $p \otimes
u_B = p \otimes q + p \otimes q'$ and $u_A \otimes q = p \otimes q +
p' \otimes q$ operator commute by \cite{AS}, Lemma 1.48.  Now let $a \in A$
and $b \in B$ be arbitrary: by spectral theory, we have $a = \sum_i
t_i p_i$ and $b = \sum_j s_j q_j$ for pairwise orthogonal projections
$p_i$ and $q_j$. Since $p_i \otimes u_B$ and $u_A \otimes q_j$
operator commute for all $i, j$, it follows that
\[{}%\color{red} a \otimes u_B} = \sum_i t_i p_i \otimes u_B \ \ \mbox{and} \ \ {}%\color{red} u_A \otimes b} = \sum_j s_j u_A \otimes q_j\]
also operator commute. $\Box$\\

\subsection{The identity $\boldsymbol{(a \otimes u)} \bullet \boldsymbol{(x \otimes y)
= \boldsymbol (a} \bullet \boldsymbol{ x ) \otimes y }$.}

To this point, we have limited information---mainly Proposition
\ref{prop: operator commutation}---about how the Jordan
structure of a composite $AB$ interacts with the Jordan structures of
$A$ and $B$. Our goal in this subsection is to establish, for
any $a \in A$ and any $x, y \in B$, the identity $(a \otimes u) \dot
(x \otimes y) = (a \dot x) \otimes y$ --- in other words, that $L_{a
  \otimes u_B}$ acts on $A \otimes B \leq AB$ as $L_{a} \otimes \mathbf{1}_B$,
where $\mathbf{1}_B$ is the identity operator on $B$. This is non-trivial,
since $AB$ need not be spanned by $A \otimes B$.\\

\noindent{\bf One-parameter groups and exponentials} It will be
helpful first to recall some basic facts about operator exponentials,
or, equivalently, one-parameter groups of linear operators on
finite-dimensional spaces (see, e.g., \cite{Curtis}). Let $V$ be a finite-dimensional real vector
space, and $X$, a linear operator on $V$. Recall that
$\phi(t) := e^{tX}$ is the unique function $\mathbb{R} \rightarrow {\cal
  L}(\mathbf{V})$ satisfying the initial-value problem
\[\phi'(t) = X \phi(t) ; \ \ \phi(0) = \mathbf{1}\]
(where $\mathbf{1}$ is the identity operator on $V$). In particular, $\phi'(0) =
X$. The function $\phi$ satisfies $\phi(t + s) = \phi(t)\phi(s)$
{}%\color{red} and hence, $\phi(t)\phi(-t) = \phi(0) = \mathbf{1}$,}
 {}%\color{red} hence, as $\phi(0) = \mathbf{1}$,
$\phi(t)$ is invertible, with $\phi(t)^{-1} = \phi(-t)$. In other
words,} $\phi$ is a one-parameter group of linear operators on $V$.
Conversely, if $\phi : \mathbb{R} \rightarrow {\cal L}(V)$ is any continuous
one-parameter group of linear operators on $V$, then $\phi$ is
differentiable, and $\phi(t) = e^{tX}$ where $X = \phi'(0)$. Notice,
also, that in such a case we have
\[Xa = \frac{d}{dt} \phi(t)a|_{t = 0}\]
for any vector $a \in V$.

For later reference, the following lemma collects some standard facts:

\begin{lemma} \label{lemma: operator commutation and one parameter groups}
Let $X, Y$ be linear operators on a finite-dimensional inner product space $V$. Then
\begin{itemize}
\item[(a)] $X$ commutes with $e^{tX}$ for all $t$;
\item[(b)] If
$e^{tX}$ commutes with $e^{sY}$ for all $t, s$, then $X$ commutes with $Y$;
\item[(c)] $(e^{tX})^{\dagger} = e^{tX^{\dagger}}$.
\end{itemize}
\end{lemma}

\tempout{
\noindent{\em Proof:} (a) is standard. For (b), define $F(s,t) =
e^{tX} e^{sY} = e^{sY} e^{tX}$. By the equality of mixed partials, we
have
\[XY F(s,t) = \frac{\partial^2}{\partial s \partial t} F(s,t) = \frac{\partial^2}{\partial s \partial t} F(s,t)
= YX F(s,t)\] for all $s, t$. In particular, setting $s, t = 0$, so
that $F(s,t) = \mathbf{1}$ (the identity operator), we have $XY = YX$. For
(c), use the observation that if $\phi(t)$ is a one-parameter group,
then $\langle \phi'(0)x, y \rangle = \frac{d}{dt} \langle \phi(t) x, y
\rangle|_{t = 0}$. $\Box$\\}

Note that, by (c), if $\phi(t)$ is a one-parameter group with $\phi'(0) = X$ hermitian, then $\phi(t)$ is hermitian for all $t$.

Now let $A$ be an EJA. For $a \in A$,
define
\[\phi_{a}(t) := e^{t L_{a}} = e^{L_{ta}},\]
i.e., $\phi_{a}$ is the solution to the initial-value problem
$\frac{d}{dt}\phi_{a} = L_{a} \phi_{a}, \ \ \phi_{a}(0) = \mathbf{1}$.
By part (c) of Proposition \ref{prop: quadratic representation}, part (c), $\phi_{a}(t) = U_{e^{ta/2}}$; by part
(b) of the same Proposition, this last is a positive mapping. Since
$e^{tL_{a}}$ is invertible with inverse $e^{-tL_{a}} = e^{L_{-ta}}$, $\phi_{a}(t)$ is an order-automorphism
belonging to $G(A)$.
\tempout{By changing the parametrization, one gets $\phi_{a}(t)$ an
automorphism for all $t$. (In more detail: set $\psi_{a}(s) :=
\phi_{a}(ts)$, with $t$ fixed; then $\psi_{a}$ is still a one-parameter
group, with $\psi_{a}'(0) = t \phi_{a}'(0) = t L_{a} = L_{ta}$, and
$\psi_{a}(1) = \phi_{a}(t)$ an automorphism.)[Didn't we just establish this directly?]}
{}%\color{red}
It follows that $L_a \in {\frak g}_{A}$, the Lie group of the identity component $G(A)$ of $A$.
Note that $\langle L_{a} x, y \rangle = \langle a x, y \rangle = \langle x, ay \rangle = \langle x, L_{a} y\rangle$
for all $x, y \in A$; that is, $L_a$ is self-adjoint. One can show that, conversely, a self-adjoint
element of ${\frak g}_{A}$ has the form $L_{a}$ for a unique $a \in A$. (See \cite{FK}, pp. 6 and 49, for the details.)}

\tempout{
\begin{corollary}{}%\color{red} Do we use this?}
Let $\phi_{a}$ and $\phi_{b}$ be the one-parameter groups
associated with $L_a$ and $L_b$, as discussed above. If $\phi_{a}(t)$ and
$\phi_{b}(s)$ commute for all $t, s$, then so do $L_a$ and $L_b$,
i.e., $a$ and $b$ operator commute.\footnote{}%\color{red} [AW: Do we actually use this anywhere?]}
\end{corollary}

\noindent{\em Proof:}
Immediate from Lemma \ref{lemma: operator commutation and one parameter groups}.
$\hfill \Box$\\
}

We are now ready for the main result of this section.

\begin{proposition}\label{prop: main equation}
Let $AB$ be a composite (in the sense of Definition
\ref{def: new Jordan composite}) of Jordan algebras $A$ and $B$. For
all $a, x \in A$ and $b, y \in B$,
\[(a \otimes u_B)\dot (x \otimes y) = a \dot x \otimes y \ \ \mbox{and} \ \ (u_A \otimes b)\dot (x \otimes y) = x \otimes b \dot y\]
\end{proposition}

\noindent{\em Proof:} We prove the first identity; the second is
handled similarly. Let $\phi(t)$ be a
one-parameter group on $A$ with $\phi'(0) = L_a$.  Then $\psi(t) :=
\phi(t) \otimes \mathbf{1}$ is a one-parameter group of automorphisms, by
condition (c) of \ref{def: dynamical composites}.
 Let $Y = \psi'(0) {}%\color{red} \in {\frak g}_{AB}}$; then, for all $x \in A$ and $y \in B$,
\begin{eqnarray*}
Y(x \otimes y)
& = & \left[ \frac{d}{dt} \psi(t)\right]_{t=0}  (x \otimes y) \\
& = & \left[ \frac{d}{dt} (\psi(t)(x \otimes y))\right]_{t = 0}\\
& = & \left[ \frac{d}{dt} \left( \phi(t)x \otimes y \right) \right]_{t = 0} \\
& = & \left( \left[ \frac{d}{dt} \phi(t)\right ]_{t = 0} x \right) \otimes y
\ = \ L_{a} x \otimes y \ = \ ax \otimes y.
\end{eqnarray*}
Subject to condition (c) of Definition
\ref{def: new Jordan composite},
we have
\[(\phi_{a}(t) \otimes \mathbf{1})^{\dagger} = \phi_{a}(t)^{\dagger} \otimes \mathbf{1} = \phi_{a}(t) \otimes \mathbf{1}.\]
Hence, $Y$ is self-adjoint.
{}%\color{red} As discussed above,} it follows
that
there exists some $v \in AB$ with $Y = L_{v}$ on $A \otimes B$.
\footnote{{}%{\color{green} [HB: I still need check over the remainder of this proof once more.]}{}%{\color{blue} [AW: proof now much
shortened!]}}
Thus,
\[v (x \otimes y) = L_{v} (x \otimes y) = Y(x \otimes y) = ax \otimes y.\]
Setting $x = u_A$ and $y = u_B$, we have
 $v = v u_{AB} = v(u_{A} \otimes u_{B}) = au_A \otimes u_B = a \otimes u_B$, which gives the
 advertised result. $\Box$ \\

\tempout{
\[\langle v | x \otimes y \rangle = \mbox{Tr}(ax \otimes y) = \mbox{Tr}(ax) \mbox{Tr}(y) = \mbox{Tr}(ax)\mbox{Tr}(uy) = \langle a  \otimes u | x \otimes y \rangle\]
(where the last identity is part of the definition of the composite $AB$)
Hence, the projection of $v$ onto $A \otimes B \leq AB$ is precisely $a \otimes u$. It follows that
\begin{equation} \label{eq: another equation}
v = (a \otimes u) + z
\end{equation}
where $z$ is orthogonal to $A \otimes B$, i.e., $\langle z | T \rangle = 0$ for every $T \in A \otimes B$.
We want to show that $z = 0$.

We begin with the case in which $a = p$, a projection. By assumption
(see Definition \ref{def: new Jordan composite}), $p \otimes u$ is also a projection in $AB$.  Then
by (\ref{eq: an equation}) we have
\[v (p \otimes u) = p^2 \otimes u = p \otimes u\]
while by Eq. \ref{eq: another equation} also
\[v (p \otimes u) = ((p \otimes u) + z)(p \otimes u) = (p \otimes u)^2 + z (p \otimes u) = (p \otimes u) + z(p \otimes u).\]
Subtracting, we have $z (p \otimes u) = 0$. Now consider $p' = u - p$: again, $p' \otimes u$ is a projection, and $p' \otimes u = (u - p) \otimes
u = (u \otimes u) - (p \otimes u) = (p \otimes u)'$:
By (1),
\[v(p' \otimes u) = pp' \otimes u = 0;\]
but also $v(p' \otimes u) = (p \otimes u)(p' \otimes u) + z(p' \otimes u) = z(p' \otimes u)$.
{}%{\color{magenta} It follows that} $z(p' \otimes u) = 0$, whence, $z = z(p \otimes u + p' \otimes u) = 0$.
Hence, $z (p' \otimes u) = 0$. We now have
\[ \ \ z = z (u \otimes u) = z ((p \otimes u) + (p' \otimes u)) = z(p \otimes u) + z(p' \otimes u) = 0 + 0 = 0.\]

The general case now follows from the spectral theorem, plus
linearity: if $a = \sum_i t_i p_i$, where the $p_i$ are projections,
then by the foregoing, $(p_i \otimes u)(x \otimes y) = p_i x \otimes y$ for $i = 1,...,n$, whence
\[\ \ \ \ (a \otimes u)(x \otimes y) = \sum_i t_i (p_i \otimes u)(x \otimes y) = \sum_i t_i (p_i x \otimes y) =
ax \otimes y.\]
The identity $(u_A \otimes b)(x \otimes y) = x \otimes by$ follows by symmetry. $\Box$\\}

Recall that, for $a \in A$, the mapping $U_a : A \rightarrow A$ is defined by
$U_a = 2L_{a}^2 - L_{a^2}$.

\begin{corollary}\label{cor: cor to main equation} In any composite $AB$ {}%\color{red} of EJAs $A$ and $B$}, and
for any $a \in A$, $b \in B$, $U_{a \otimes u_B}$ and $U_{u_A \otimes A}$ act on $A \otimes B$ as
$U_a \otimes \mbox{id}_B$ and $\mbox{id}_A \otimes U_b$, respectively. \end{corollary}

\subsection{Composites of Simple EJAs}

We now show that if $A$ and $B$ are non-trivial {\em simple} EJAs --- that is, if neither has any non-trivial direct summands --- then any composite $AB$ must be special, universally reversible, and an ideal (a direct summand) of the maximal tensor product $A \widetilde{\otimes} B$. The rough idea is that, since $A$ and $B$ have rank at least two, the fact that
products of distinguishable effects are distinguishable will yield at least four distinguishable effects in $AB$.
If the latter were simple, this would be the end of the story; but we know from the case of universal tensor products
(which we will ultimately show are dynamical composites in our sense) that composites can have non-trivial direct
summands. Thus, need to work a bit harder, and show that every irreducible direct summand of $AB$ has rank at least 4.

{}%\color{red} An element} $s \in A$ is {}%\color{red} called} a {\em symmetry} iff $s^2 = u$.\footnote{}%\color{red} Not to be confused with a symmetry {\em qua} order-automorphism.} In this case
$U_s$ is a Jordan automorphism of $A$, with $U_{s}^{2} = \mbox{id}$
(\cite{AS}, Prop. 2.34). Also
note that $p := \frac{1}{2}(s + u)$ is a projection, and, conversely,
if $p$ is a projection, then $s := 2p - u$ is a symmetry.

Two projections $p, q \in A$ are {\em exchanged by a symmetry} $s \in
A$ iff $U_s(p) = q$ (in which case, $p = U_{s}(q)$).  More
generally, $p$ and $q$ are {\em equivalent} iff there exists a finite
sequence of symmetries $s_1,...s_{\ell}$ with $q = (U_{s_{\ell}} \circ
\cdots \circ U_{s_1})(p)$.

Now let $AB$ be a composite in the sense of Definition \ref{def: new Jordan composite}.

\begin{lemma}\label{lemma: exchange} Let $s \in A$ be a symmetry exchanging projections
$p_1, p_2 \in A$, and let $t \in B$ be a symmetry exchanging projections $q_1, q_2 \in B$.
Then $s \otimes u_B$ and $u_A \otimes t$
  are symmetries in $AB$, and $U_{u_A \otimes t} U_{s \otimes u_B}
  (p_1 \otimes q_1) = p_2 \otimes q_2$. In particular, the projections
  $p_1 \otimes q_1$ and $p_2 \otimes q_2$ are equivalent.\end{lemma}

\noindent{\em Proof:} $(s \otimes u_B)^2 = s^2 \otimes u_B = u_A
\otimes u_B = u_{AB}$ by Proposition \ref{prop: main equation}. Similarly for $u_A \otimes
t$. Now by Corollary \ref{cor: cor to main equation}, we have
\[\ \ U_{u_A \otimes t} U_{s \otimes u_B} (p_1 \otimes q_1) = U_{u_A \otimes t}(U_s (p_1) \otimes q_1) = U_{s}(p_1) \otimes U_{t}(q_1) = p_2 \otimes q_2. \ \Box\]

\begin{theorem}\label{thm: composites special}{ Let
$AB$ be a composite of simple, {}%\color{red} non-trivial} Jordan algebras $A$ and $B$.
Then $AB$ is a special, {}%\color{red} universally reversible} EJA.}
\end{theorem}

\noindent{\em Proof:} {}%\color{red} We shall show that every irreducible direct
summand of $AB$ has rank $\geq 4$, from which the result follows.}
Decompose $AB$ as a direct sum of simple ideals,
say $AB = \bigoplus_{\alpha} M_{\alpha}$. Let $\pi_{\alpha} : AB
\rightarrow M_{\alpha}$ be the corresponding projections, and let
$u_{\alpha} := \pi_{\alpha}(u_{AB})$ be the unit in $M_{\alpha}$.
Suppose now that $\{p_1,...,p_n\}$ is a Jordan frame in $A$ and
$\{q_1,...,q_m\}$ is a Jordan frame in $B$.  By \cite{AS} Lemma
3.19\footnote{which is a pointer to \cite{HO-Stormer}, Lemma 5.3.2},
there are symmetries in $AB$ exchanging the $p_i$, and there are
symmetries in $B$ exchanging the $q_j$. By Lemma \ref{lemma: exchange}, therefore, the
projections $p_i \otimes q_j$ are pairwise equivalent. By \cite{AS} Lemma
3.9, therefore, these projections have the same central cover
$c$. This means that for each $\alpha$, the projection $\pi_{\alpha} :
AB \rightarrow M_{\alpha}$ takes {}%{\color{blue} none} of the projections $p_i \otimes
q_j$ to the zero projection in $M_{\alpha}$, or it takes {}%{\color{blue} all} of them
to zero --- the former case arising exactly when $u_{\alpha}
\leq c$, and the latter, when $u_{\alpha}c = c u_{\alpha} = 0$. If
$M_{\alpha}$ is of the first type, $\{\pi_{\alpha}(p_i \otimes q_j) |
i = 1,...,n, j = 1,...,m\}$ consists of $nm$ distinct orthogonal
projections in $M_{\alpha}$, summing to the unit $\pi_{\alpha}(u) =:
u_{\alpha}$. Hence, the rank of $M_{\alpha}$ is at least
$nm$. In particular, since $A$ and $B$
are non-trivial, $n, m \geq 2$, whence, $M_{\alpha}$ has rank at least
4, and hence, is special.

Now let $p, q$ be arbitrary projections in $A$ and $B$, respectively:
extending each to a Jordan frame, as above, we see that for all
$\alpha$, if $\pi_{\alpha}(p \otimes q) \not = 0$, then $M_{\alpha}$
is special. Hence, $p \otimes q$ belongs to the direct sum of the
special summands of $AB$, i.e., to $M_{\mbox{sp}}$.
Since projections $p \otimes q$ generate $A \odot B$, the latter is
special.

{}%\color{red} The argument also shows that each simple direct summand $M_{\alpha}$, in addition to being
special, is not a spin factor,
and hence, is UR. It follows from this, plus the fact that direct sums of universally reversible EJAs are
again UR, that $AB$ must be UR. $\Box$ }

\begin{corollary}\label{cor: no composite with exceptional}
If $A$ is simple and $B$ is exceptional, there exists
no composite $AB$ satisfying the
conditions of Definition 1.\end{corollary}

\noindent{\em Proof:}  The mapping $B \rightarrow AB$ given by $b \mapsto u_{A} \otimes b$ is a faithful Jordan homomorphism, by Proposition \ref{prop: main equation}. But there exists no faithful Jordan homomorphism from $B$ into a special euclidean Jordan algebra. $\Box$\\

\begin{theorem}\label{thm: simple composites ideals} Let $A$ and $B$ be simple, non-trivial JC-algebras. Then $AB$ is an ideal in $A \widetilde{\otimes} B$.\end{theorem}

\noindent{\em Proof:}  By Proposition \ref{prop: main equation}, we have Jordan homomorphisms $A, B \rightarrow AB$ with operator-commuting ranges. Since $AB$ is
special, i.e., a JC-algebra, elements of $AB$ operator commute iff their images in $\Cu(AB)$ operator commute
 (\cite{HO}, Lemma 5.1). Thus, we have Jordan homomorphisms $A, B \rightarrow \Cu(AB)$ with operator-commuting ranges. The universal
property of $A \widetilde{\otimes} B$ yields a Jordan homomorphism $\phi : A \widetilde{\otimes} B \rightarrow \Cu(AB)$ taking
(the image of) $a \otimes b$ in $A \widetilde{\otimes} B$ to (the image of) $a \otimes b$ in $\Cu(AB)$. Since both
$A \widetilde{\otimes} B$ and $AB$ are generated by pure tensors, $\phi$ takes $A \widetilde{\otimes} B$ onto $AB$. Letting $K$ denote the kernel of $\phi$, an ideal of $A \widetilde{\otimes} B$, we have $A \widetilde{\otimes} B = K' \oplus K$, where $K'$ is the complementary ideal;
the mapping $\phi$ factors through the projection $A \widetilde{\otimes} B \rightarrow K'$ to give an isomorphism
$K' \simeq AB$\footnote{{}%\color{red} [AW: I've now checked, and this does work. But I must {}%\color{red} write down the details!}]}}. $\Box$ \\

Combined with table 2, Corollary \ref{thm: simple composites ideals} sharply restricts the possibilities
for composites of simple EJAS. In particular, it follows that {\em
if $A \widetilde{\otimes} B$ is itself simple}, then $AB
\simeq A \widetilde{\otimes} B$. In other words, in this case the {}%\color{red} universal tensor product}
is the only ``reasonable" tensor product (to the extent that we
think the
conditions of Definition \ref{def: new Jordan composite}
constitute reasonableness, in this context).
If $A = B = C_n$,
so that $A \widetilde{\otimes} B = M_{n^2}({\Bbb C})_{sa} \oplus M_{n^2}({\Bbb
  C})_{sa}$, we have another candidate, i.e., the usual
quantum-mechanical composite $M_{n^2}({\Bbb C})_{sa}$. If $A = B = M_2(\H)$ (that is, if $A$ and $B$
are two quaternionic bits), we have $A \widetilde{\otimes} B = M_{16}(\mathbb{R})_{{sa}} \oplus M_{16}(\mathbb{R})_{{sa}} \oplus M_{16}(\mathbb{R})_{{sa}} \oplus M_{16}(\mathbb{R})_{{sa}}$, giving us four possibilities for $AB$. {\em These exhaust the possibilities for real, complex
and quaternionic quantum composites!}
\footnote{}%{\color{blue} [AW: What about spin factors?]}
\footnote{\hbcomment{}%{\color{green} I suggest displaying the claims that are in this
paragraph, as they're among the most important conclusions of the paper.}}\\

\tempout{
{}%\color{red} A final consequence of Theorem\ref{thm: composites special}, which will be very useful in computing particular
composites, is the following. Let $A$ and $B$ be EJAs with composite $AB$ contained $M_{{sa}}$ for a $\ast$-algebra $M$.
We may suppose $AB$ generates $M$ as a $\ast$-algebra (if not, replace it by the $\ast$-subalgebra so generated).
Now suppose $M$ carries an involution fixing points of $A \otimes B$: then this also fixes points of the
Jordan subalgebra of $M$ generated by $A \otimes B$, which is to say, it fixes points of $AB$. Since the latter
is UR, it follows from (\cite{HO}, Theorem ...) that $M \simeq \Cu(AB)$, and that $AB = \mathbf{M}^{\Phi}_{{sa}}$.
}
}

\section{EJC-algebras}
\label{sec: Euclidean JC}

In view of Theorem \ref{thm: composites special} and Corollary
\ref{cor: no composite with exceptional}, we now restrict our
attention to special EJAs.  A {\em JC-algebra} is variously defined as
a norm-closed Jordan subalgebra of $\L(\mathbf{H})$ for a real or complex
Hilbert space $\mathbf{H}$, or as a Jordan algebra that is
Jordan-isomorphic to such an algebra. In finite dimensions, any JC
algebra is euclidean, and any special euclidean Jordan algebra is JC
(on the second definition).  Here, we consider EJAs
that are embedded, not necessarily in $\L(\mathbf{H})$ for a specific
Hilbert space, but in some definite complex $\ast$-algebra.

\begin{definition}{\em An {\em embedded euclidean JC-algebra} is a pair $(A,\mathbf{M}_A)$ where $\mathbf{M}_A$ is a {}%\color{red} finite-dimensional} complex $\ast$-algebra and $A$ is a unital Jordan subalgebra of $(\mathbf{M}_{A})_{{sa}}$.} \end{definition}

For aesthetic reasons, we abbreviate the phrase ``embedded euclidean JC" by EJC (rather than EEJC), letting the initial E stand simultaneously for {\em embedded} and {\em euclidean}.
In this section, we develop a canonical tensor product for EJC
algebras, {}%\color{red} and use this to construct several symmetric monoidal
categories of EJC-algebras.}
In one case, we obtain a category of reversible EJAs, with
a monoidal product extending that of ordinary complex matrix algebras;
{}%\color{red} in another, which we call $\mbox{}%\color{red} \bf InvQM}$}, we
restrict attention to EJAs that arise as fixed-point algebras of
involutions on {}%\color{red} complex} $\ast$-algebras (a class that includes all
universally reversible EJCs, but also includes the quaternionic bit,
$M_{2}(\mathbb{H})_{{sa}}$, {}%\color{red} as symplectically embedded in $M_{4}(\mathbb{C})$}. Here, the monoidal structure agrees with the Hanche-Olsen tensor product
{}%\color{red} in the cases in which the factors are UR.}

For any $\ast$-algebra $\mathbf{M}$ and any set $S \subseteq \mathbf{M}_{{sa}}$, let
$J(S)$ denote the Jordan subalgebra of
$\mathbf{M}_{{sa}}$  generated by $S$. (We
should probably write this as $J_{\mathbf{M}}(S)$, but context will generally
make the usage unambiguous. See below for an example!)

\begin{definition} \label{def: canonical tensor product}
{\em The {\em canonical tensor product} of $(A,\mathbf{M}_A)$ and
  $(B,\mathbf{M}_B)$ is $(A \odot B, \mathbf{M}_{A} \otimes \mathbf{M}_{B})$, where
$A \odot B := J(A \otimes B) \subseteq \mathbf{M}_{A}
  \otimes \mathbf{M}_{B}$, the Jordan subalgebra of $(\mathbf{M}_A \otimes \mathbf{M}_B)_{{sa}}$ generated
  by $A \otimes B$.
\footnote{When $\mathbf{M}_A = M_n(\mathbb{C})$ for some $n$ and similarly for $B$,
  this agrees with a construction by Jamjoom, Def. 2.1 in
  \cite{JamjoomTensorJW}.}}\end{definition}

Note that this makes it a matter of definition that $\mathbf{M}_{A \odot B} = \mathbf{M}_{A} \otimes \mathbf{M}_{B}$.
In particular, if $A$ and $B$ are universally embedded, so that $\mathbf{M}_{A} = \Cu(A)$ and $\mathbf{M}_B = \Cu(B)$, then
$A \odot B = A \widetilde{\otimes} B$, so that $\mathbf{M}_{A \widetilde{\otimes} B} = \Cu(A) \otimes \Cu(B)$ by definition;
but the fact that this last is $\Cu(A \widetilde{\otimes} B)$ {}%\color{red} (whence, $A \odot B$ is UR)} is a theorem.

Let us call an EJC $(A,\mathbf{M}_A)$ {\em reversible} iff $A$ is reversible in $\mathbf{M}_A$. Note that if $A$ is a
simple EJC {\em standardly}
embedded in $\mathbf{M}_A$, then $A$ is reversible iff $A = R_n, C_n$ or $Q_n$ for some $n$.

\begin{proposition} \label{prop: canonical product composite}
If $(A,M_A)$ and $(B,M_B)$ are {\em }%\color{red} reversible} EJC-algebras
then their canonical tensor product $A \odot B$ is a
{}%\color{red} composite in the sense of \ref{def: new Jordan composite}.}
\end{proposition}

\noindent{\em Proof:}  We must show that $A \odot B$ is a dynamical composite, in the
sense of Definition \ref{def: dynamical composites}. In particular, it must
be a composite of probabilistic models in the sense of Definition \ref{def: composites}.
Conditions (a)-(c) of that definition are easily verified:
That $u_{AB} = u_{A} \otimes u_{B}$ follows from the unitality of
the embeddings $A \mapsto \mathbf{M}_A$, $B \mapsto \mathbf{M}_B$ and $AB \mapsto \mathbf{M}_{AB}$.
For all $a, x \in (\mathbf{M}_A)_{{sa}}$ and $b, y \in (\mathbf{M}_B)_{{sa}}$, we have
\[\langle a \otimes b | x \otimes y \rangle = \mbox{Tr}(ax \otimes by) = \mbox{Tr}(ax)\mbox{Tr}(by) = \langle a |x \rangle \langle b | y \rangle\]
Thus,  for any states $\alpha = \langle a |$ and $\beta = \langle b |$, where $a \in A_+$ and $b \in B_+$, we have
a state $\gamma = \langle a \otimes b |$ on $AB$ with $\gamma(x \otimes y) = \alpha(x) \beta(y)$ for
all $x \in A_+$, $y \in B_+$.

 That $A \odot B$ satisfies the additional conditions required
to be a \emph{dynamical} composite in the sense of Definition \ref{def: dynamical composites} is not trivial.
 However, using a result of Upmeier on
 the extension of derivations on reversible JC-algebras
 \cite{Upmeier}, one can show that any $\phi \in G(A)$ extends to an
 element $\hat{\phi} \in G((\mathbf{M}_A)_{{sa}})$ that preserves $A$, and that
 $\hat{\phi} \otimes \mathbf{1}_{M_B}$ is an order-automorphism of $(\mathbf{M}_A
 \otimes \mathbf{M}_B)_{{sa}}$ preserving $A \odot B$.  It follows that
 $\hat{\phi} \otimes \hat{\psi} = (\hat{\phi} \otimes \mathbf{1}_{\mathbf{M}_B}) \circ (\mathbf{1}_{\mathbf{M}_A} \otimes \hat{\psi})$
 preserves $A \odot B$ as well. It is not difficult to verify that
 $(\hat{\phi} \otimes \hat{\psi})^{\dagger} = \hat{\phi}^{\dagger} \otimes \hat{\psi}^{\dagger}$.
 The details are presented in Appendix \ref{appendix: extending}.
Thus $A \odot B$ satisfies
 conditions (a) and (b) of Definition \ref{def: new Jordan composite}.  Condition
(c) is immediate from the definition of $A \odot B$.
$\Box$\\

Combining this with Theorem \ref{thm: composites special}, we see that if $A$ and $B$ are simple reversible EJCs
for which $A \widetilde{\otimes} B$ is also simple, then the {}%\color{red} canonical} and universal tensor products of $A$ and $B$ coincide.
This covers all cases except those involving two factors of the form $C_n$ and $C_k$, and those involving
$Q_2$ as a factor. In fact, many of the latter are covered by the following.

\begin{corollary}\label{cor: canonical composites with involutions}
Let $(A,\mathbf{M}_A)$ and $(B,\mathbf{M}_B)$ be reversible EJCs with $A$ generating
$M_A$ and $B$ generating $M_B$ as $\ast$-algebras.  Suppose $M_A$ and
$M_B$ carry involutions $\Phi$ and $\Psi$, respectively, with $\Phi$
fixing points of $A$ and $\Psi$ fixing points of $B$ (i.e.
  $A \subseteq \mathbf{M}_A^\Phi$ and $B \subseteq \mathbf{M}_B^\Phi$). Then $A \odot
B = (\mathbf{M}_{A} \otimes \mathbf{M}_{B})^{\Phi \otimes \Psi}_{{sa}}$, the set of
self-adjoint fixed points of $\Phi \otimes \Psi$.
\end{corollary}

\noindent{\em Proof:} By the preceding proposition, $A \odot B$ is a dynamical
composite of $A$ and $B$; hence, by Theorem \ref{thm: composites
  special}, $A \odot B$ is universally reversible. Since $\Phi \otimes
\Psi$ fixes points of $A \otimes B$, it also fixes points of the
Jordan algebra this generates in $M_A \otimes M_B$, i.e., of $A \odot
B$. But then, by Proposition \ref{prop: UR fixed points}, $A \odot B$
is exactly the set of fixed points of $\Phi \otimes \Psi$. $\Box$\\

As an example, the quaternionic bit $Q_2$, standardly embedded in $M_4(\mathbb{C}) = M_2(M_2(\mathbb{C}))$ as the set of
block matrices of the form $\left [ \begin{array}{rc} a & b \\ -\bar{b} & \bar{a} \end{array} \right ]$
with $a$ self-adjoint, is exactly the fixed-point set of the involution $\Phi(x) = -J x^T J$, where
$J = \left [ \begin{array}{rc} 0 & \mathbf{1} \\ -\mathbf{1} & 0 \end{array} \right ]$. Thus, $Q_2 \odot Q_2$ is
the set of self-adjoint fixed points of $\Phi \otimes \Phi$ acting on $M_4(\mathbb{C}) \otimes M_4(\mathbb{C}) = M_{16}(\mathbb{C})$.
From this, it follows that $Q_2 \odot Q_2 \simeq R_{16} = M_{16}(\mathbb{R})_{{sa}}$. For details, see
Appendix \ref{appendix: the quabit}. Notice that Corollary \ref{cor: canonical composites with involutions}
applies equally to all simple, universally embedded EJCs, {\em and} to all {\em standardly} embedded
EJCs {\em except} for $C_n$ (recalling
here that no (complex-linear) involution on $M_n(\mathbb{C})$ fixes all
points of $M_n(\mathbb{C})_{{sa}}$).  \\

\tempout{\hbcomment{ I think the more serious worry about the claim that $A
  \odot B$ is a composite may concern conditioning.  As mentioned
  above, I think we need to either show, from Definition
\ref{def: composite probabilistic} or
\ref{def: new Jordan composite}, or stipulate in a
modified definition, that conditional
  states and effects are in the correct state space.  If we must
  stipulate it in the definition, then we need to show it in the above
proof.  This means that for \emph{any} states $y \in A \odot B$ and states
$a \in A$ respectively, we need to show
that the functional $b \mapsto \mbox{tr} (y (a \otimes b))$ from $B_+$ to $\mathbb{R}$ is
in $B_+^*$. {}%{\color{blue} [AW: I've added a note after the def. of composites explaining why (for
models with full-dual state spaces) we always do have conditional states. Regarding the functional above:
$a \otimes b$ is positive in $AB$ for all fixed positive $a$ and positive $b$; since $y$ is positive
in $AB$ and $T(x,y) := \mbox{Tr}(xy)$ is positive when both factors are positive (i.e., is a positive inner product),
we do have positivity here.]} }}

\noindent{\bf Canonical composites of standardly embedded, reversible EJAs}
It follows from Proposition
\ref{prop: canonical product composite}, together with
Corollary \ref{thm: simple composites ideals} that
the canonical and universal tensor products coincide for simple, reversible EJCs whose universal
tensor products are simple. This covers all cases except for those in which one factor is
$M_2(\mathbb{H})_{{sa}}$ (the quaternionic bit), and those in which both factors have the form $M_n(\mathbb{C})_{{sa}}$ for some $n$.
Restricting attention to standardly embedded EJCs, these missing cases can be computed directly: see appendix \ref{appendix: the quabit} and \cite{Graydon16}. The results are summarized in the following table. Recall here the abbreviations $R_n = M_{n}(\mathbb{R})_{{sa}}$, $C_n = M_n(\mathbb{C})_{{sa}}$ and $Q_n = M_{n}(\mathbb{H})_{{sa}}$.
\[\begin{array}{c}
\begin{array}{c|ccc}
\odot & R_n & C_n & Q_n \\
\hline
R_m   & R_{mn} & C_{mn} & Q_{mn} \\
C_m   & C_{mn} & C_{mn} & C_{2mn} \\
Q_m   & Q_{mn} & C_{2mn} & R_{4mn}
\end{array}\\
\\
\mbox{Table 3: $\odot$ for standardly embedded, reversible EJCs}
\end{array} \]

\noindent{\bf Associativity} We now establish that the canonical
tensor product is associative, setting the stage for the construction
of symmetric monoidal categories of EJCs in Section \ref{sec: categories EJC}.

\begin{proposition}  \label{prop: associativity}
$\odot$ is associative. More precisely,  the associator mapping $\alpha : \mathbf{M}_{A} \otimes (\mathbf{M}_{B} \otimes \mathbf{M}_{C}) \rightarrow (\mathbf{M}_{A} \otimes \mathbf{M}_{B}) \otimes \mathbf{M}_{C}$ carries $A \odot (B \odot C)$ isomorphically onto
$(A \odot B) \odot C$. \end{proposition}

The essence of the proof is the following Lemma.

\begin{lemma} \label{lemma: associativity}
Let $\mathbf{M}$ and $\mathbf{N}$ be $\ast$-algebras, and let $A$ and $B$ be Jordan subalgebras of $\mathbf{M}_{{sa}}$ and $\mathbf{N}_{{sa}}$, respectively. Then $J(A \otimes J(B)) = J(A \otimes B)$\footnote{Note that on the left, $J(A \otimes J(B))$ refers to the Jordan subalgebra of $(A \otimes B)_{{sa}}$ generated by $A \otimes J(B)$, where $J(B)$ refers to the Jordan subalgebra of $B_{{sa}}$ generated by $F$.}.\end{lemma}

\noindent{\em Proof of Lemma \ref{lemma: associativity}}:
We make use of the following notation. If $X$
and $Y$ are subsets (note: not necessarily subspaces) of $\mathbf{M}$ and
$\mathbf{N}$, respectively, let
\[X \bullet Y \ := \ \{\, a \bullet b \ | \ a \in X, b \in Y \, \};\]
similarly, $X + Y$ is the set of sums $a + b$
with $a \in X$, $b \in Y$, and, for $t \in \mathbb{R}$, $tX$ consists of multiples $ta$ of elements $a \in X$.
Finally, we write $X \circledcirc Y$ for the set $\{ a \otimes b | a \in X, b \in Y\}$ of pure tensors
of elements in $X$ and $Y$. Note that if $X$ and $Y$ happen to be subspaces of $\mathbf{M}$ and $\mathbf{N}$, respectively,
then $J(X \circledcirc Y) = J(X \otimes Y)$.

Now let $\cal N$ denote the set of all subsets $Y$ of $\mathbf{N}$ such that
\begin{itemize}
\item[(i)] $B \subseteq Y \subseteq J(B)$, and
\item[(ii)]$A \circledcirc Y \subseteq J(A \circledcirc B)$.
\end{itemize} Note that $B \in {\cal N}$; in particular, then, ${\cal N}$ is non-empty.  We are going to
show first that $J(B) = \bigcup {\cal N}$.

{\em Claim 1:} If $Y_1, Y_2 \in {\cal N}$, then the sets $Y_1 \bullet Y_2$, $Y_1 + Y_2$ and $tY_1$ ($t$ any real
scalar) belong to ${\cal N}$.

To see this, suppose that $y_1 \in Y_1 \in {\cal N}$ and $y_2 \in Y_2 \in {\cal N}$.
Then for any $a \in A$, we have
\begin{eqnarray*}
a \otimes (y_1 \bullet y_2 ) & = & a \otimes \frac{1}{2} (y_1y_2 + y_2 y_1) \\
& = & \frac{1}{2} [(a \otimes y_1 y_2) + (a \otimes y_2 y_1)]\\
& = & \frac{1}{2} [(a \otimes y_1)(1 \otimes y_2) + (1 \otimes y_2)(a \otimes y_1) ]\\
& = & (a \otimes y_1) \bullet (1 \otimes y_2).
\end{eqnarray*}
Since $A \circledcirc Y_1$ and $A \circledcirc Y_2$ are both, by assumption,
contained in $J(A \circledcirc B)$, the last expression defines an element
of $J(A \otimes B)$. Since $a$ is arbitrary here, all pure tensors
from $A \circledcirc (Y_1 \bullet Y_2)$ belong to $J(A \circledcirc B)$.
{}%\color{red} Since
the latter is a subspace of $(\mathbf{M} \otimes \mathbf{N})_{{sa}}$, it follows that
linear combinations of pure tensors from $A \circledcirc (N_1 \bullet N_2)$
are contained in $J(A \circledcirc B)$ as well. [Needed?]} Finally, notice that $B
\subseteq Y_1$ and $B \subseteq Y_2$ implies that $B \subseteq Y_1
\bullet Y_2$ (for instance, express $b \in B$ as $b \bullet 1$). The
corresponding claims for $Y_1 + Y_2$ and $t Y_1$ are straightforward,
since $a \otimes (y_1 + y_2) = (a \otimes y_1) + (a \otimes y_2)$ and
$a \otimes ty_1 = t(a \otimes y_1)$.

Now let $J = \bigcup {\cal N}$. {\em Claim 2:} $J = J(B)$. Since $B
\subseteq J \subseteq J(B)$, it's enough to show that $J$ is a Jordan
subalgebra of $\mathbf{N}_{{sa}}$. Let $y_1, y_2 \in J$: then for some $Y_1, Y_2
\in {\cal M}$, we have $y_1 \in Y_1$ and $y_2 \in Y_2$.  By Claim 1,
$y_1 \bullet y_2 \in Y_1 \bullet Y_2 \in {\cal N}$, $y_1 + y_2 \in N_1
+ Y_2 \in {\cal N}$, and $t y_1 \in tY_1 \in {\cal N}$; hence, $y_1
\bullet y_2$, $y_1 + y_2$ and $t y_1$ belong to $J$. In other words,
$J$ is closed under scalar multiplication, addition and the Jordan
product. This proves Claim 2.

We now use the fact that $J(B) = \bigcup {\cal N}$ to show that $J(A
\otimes J(B)) = J(A \otimes B)$. Let $\tau = \sum_i a_i \otimes y_i$
be an arbitrary element of $A \otimes J(B)$. Since $J(B) = \bigcup {\cal
  N}$, for each $i$, $y_i \in Y_i$ for some $Y_i \in {\cal N}$. But
then, by definition of ${\cal N}$, $a_i \otimes y_i \in A \otimes Y_i
\subseteq J(A \otimes B)$; since the latter is a subspace of $\mathbf{M}
\otimes \mathbf{N}$, $\tau \in J(A \otimes B)$ as well. Thus, $A \otimes J(B)
\subseteq J(A \otimes B)$.

It now follows that $A \otimes B \subseteq A \otimes J = A \otimes
J(B) \subseteq J(A \otimes B)$, hence, that $J(A \otimes B) \subseteq
J(A \otimes J(B)) \subseteq J(A \otimes B)$. $\Box$\\

\noindent{\em Proof of Proposition \ref{prop: associativity}:} Let $(A,\mathbf{M}_A)$,
$(B,\mathbf{M}_B)$ and $(C,\mathbf{M}_C)$ be EJC-algebras. We
need to show that the associator mapping $\alpha : \mathbf{M}_A \otimes (\mathbf{M}_B
\otimes \mathbf{M}_C) \rightarrow (\mathbf{M}_A \otimes \mathbf{M}_B) \otimes \mathbf{M}_C$ carries $A
\odot (B \odot C)$ onto $(A \odot B) \odot C$.  Applying Lemma 2, we
have
\[ A \odot (B \odot C) = J(A \otimes J(B \otimes C)) = J(A \otimes (B \otimes C))\]
and
\[ (A \odot B) \odot C = J(J(A \otimes B) \otimes C) = J((A \otimes B) \otimes C).\]
The associator mapping is a $\ast$-isomorphism, and carries $A \otimes (B \otimes C)$ to
$(A \otimes B) \otimes C$. Hence, it also carries $J(A \otimes (B \otimes C))$ onto
$J((A \otimes B) \otimes C)$. $\Box$\\ \

\noindent{\bf Direct sums} If $(A,\mathbf{M}_A)$ and $(B,\mathbf{M}_B)$ are two embedded EJAs, define
$(A,\mathbf{M}_A) \oplus (B,\mathbf{M}_B) = (A \oplus B, \mathbf{M}_A \oplus \mathbf{M}_B)$, where the embedding of $A \oplus B$ in $\mathbf{M}_A \oplus \mathbf{M}_B$ is the obvious one. One can easily check that
for sets $X \subseteq \mathbf{M}_A$ and $Y \subseteq \mathbf{M}_B$, $J(X \oplus Y) = J(X) \oplus J(Y)$. Using this, and
the distributivity of tensor products over direct sums in the contexts of vector spaces and $\ast$-algebras, we have
\begin{eqnarray*}
A \odot (B \oplus C) = J(A \otimes (B \oplus C)) & = & J((A \otimes B) \oplus (A \otimes C)) \\
& = & J(A \otimes B) \oplus J(A \otimes C) \\
& = & (A \odot B) \oplus (A \odot C)
\end{eqnarray*}
(where $J$ refers variously to generated Jordan subalgebras of $\mathbf{M}_{A}
\otimes (\mathbf{M}_{B} \oplus \mathbf{M}_{C}))$, $(\mathbf{M}_{A} \otimes \mathbf{M}_{B}) \oplus
\mathbf{M}_{A} \otimes \mathbf{M}_{C})$, $\mathbf{M}_{A} \otimes \mathbf{M}_{B}$, and $\mathbf{M}_{A} \otimes
\mathbf{M}_{C}$).\\

\tempout{
\noindent{\bf Standard composites of simple, standardly embedded, reversible EJAs} In appendix \ref{appendix: standard tps}, we compute
the canonical composites of all simple, reversible EJAs in their standard embeddings. For reference, the table is given below. Recall here
that $R_n = M_n(\mathbb{R})_{{sa}}$, $C_n = M_n(\mathbb{C})_{{sa}}$ and $Q_n = M_n(\H)_{{sa}}$. Note that this
agrees with the correponding table for the universal tensor product, except in that (i) $C_m \odot C_n = C_{mn}$, where
$C_m \widetilde{\otimes} C_n = C_{mn} \oplus C_{mn}$, and (ii) $Q_2 \odot Q_2 = R_{16}$, where $Q_2 \widetilde{\otimes} Q_2$ is a direct sum of four copies of $R_{16}$.
\[\begin{array}{c}
\begin{array}{c|ccc}
\odot & R_n & C_n & Q_n \\
\hline
R_m   & R_{mn} & C_{mn} & Q_{mn} \\
C_m   & C_{mn} & C_{mn} & C_{2mn} \\
Q_m   & Q_{mn} & C_{2mn} & R_{4mn}
\end{array}\\
\\
\mbox{Table 3}
\end{array} \]
}

\section{Monoidal Categories of EJC-algebras}
\label{sec: categories EJC}

As discussed in Section 2, a physical theory ought to be represented
by a category, in which objects correspond to systems and
morphisms, to physical processes. An obvious candidate for a category
in which objects are embedded JC-algebras is the following.

{}%\color{red}
\begin{definition}{\em Let $(A,\mathbf{M}_A)$ and $(B,\mathbf{M}_B)$ be EJC-algebras. A linear mapping
 $\phi : \mathbf{M}_A \rightarrow \mathbf{M}_B$
is {\em Jordan preserving} iff  $\phi(A) \subseteq B$. The category $\mbox{\bf EJC}$ has, as objects,
EJC-algebras, and, as morphisms, {\em completely positive} Jordan-preserving mappings. }
\end{definition}
}

In view of Proposition \ref{prop: associativity}, one might guess that
$\mbox{\bf EJC}$ is symmetric-monoidal under $\odot$. There is certainly a
natural choice for the monoidal unit, namely $I = (\mathbb{R}, \mathbb{C})$. However,
the following examples show that tensor products of $\mbox{\bf EJC}$ morphisms
are generally not morphisms.

\tempout{
{}%{\color{magenta} [HB: shall we promote the following two examples to
    theorems?  Or at least propositions?  I realize maybe you don't
    think these are technically hard enough to prove to want to call
    them theorems, but I worry a little that calling them examples
    reduces the saliency of what are fairly important points in this
    paper.  Not a big deal though...] }

{}%{\color{magenta} [HB: ... wondering, do these examples work even if
$A$ is the tensor unit?] }
}

\begin{example}\label{ex: no states} {\em Let $(A,\Cu(A))$ and
$(B,\Cu(B))$ be simple, non-trivial, universally embedded EJCs, and suppose
that $B$ is not UR (e.g., the spin factor $V_4$).
{}%\color{red} Suppose, further, that $A \widetilde{\otimes} B$ is irreducible ---
for instance, let $A = R_n$ for any $n$.}
 Let
$\hat{B}$ be the set of fixed points of the canonical involution
$\Phi_B$ (cf. Definition \ref{def: canonical involution}). Then by Corollary \ref{thm: simple composites ideals}, $A \odot B = A \widetilde{\otimes} B$, the set of
fixed points of $\Phi_A \otimes \Phi_B$. In particular, $u_A \otimes
\hat{B}$ is contained in $A \odot B$. Now let $\alpha$ be a state on
$\Cu(A)$: this is CP, and trivially Jordan-preserving, and so, a
morphism in $\mbox{\bf EJC}$. But
\[(\alpha \otimes \mbox{id}_{B})(u_A \otimes \hat{B}) = \alpha(u_A)\hat{B} = \hat{B},\]
which {}%{\color{magenta} by Proposition \ref{prop: UR fixed points}
is larger than $B$ because $B$ is not UR}. So $f \otimes \mbox{id}_{B}$ isn't Jordan-preserving.}
\end{example}

The next example is similar:

\begin{example}\label{ex: no states 2} {\em Let $C_n = M_n(\mathbb{C})_{{sa}}$ and $R_k = M_k(\mathbb{R})_{{sa}}$, as usual.
Consider these as standardly embedded, i.e., consider the EJCs $(C_n, M_n(\mathbb{C}))$ and $(R_k, M_k(\mathbb{C}))$, where
in the latter, $R_k$ is embedded as the set $k \times k$ complex matrices with real entries,
i.e., the set of self-adjoint symmetric $k \times k$ complex matrices. Then we have
\[(C_n, M_n(\mathbb{C})) \odot (R_k, M_k(\mathbb{C})) = (C_{nk}, M_{nk}(\mathbb{C}))\]
where the embedding of $C_{nk}$ in $M_{nk}(\mathbb{C})$ is the {}%\color{red} standard} one. Now let $\alpha$ be a state on
$M_n(\mathbb{C})$, and let $b$ be any self-adjoint matrix in $M_k(\mathbb{C})$. Then $\mathbf{1}_n \otimes b$ is self-adjoint
in $M_{n}(\mathbb{C}) \otimes M_{k}(\mathbb{C}) = M_{nk}(\mathbb{C})$, and $(\alpha \otimes \mbox{id}_{M_{k}}(\mathbb{C}))(\mathbf{1}_n \otimes b) = b$. Since
$b$ needn't belong to $R_k$, the mapping $\alpha \otimes \mbox{id}_{M_k}(\mathbb{C})$ isn't Jordan-preserving. }
\end{example}

\tempout{
{}%{\color{blue} [Should we bring all categorical considerations regarding EJCs into this section?]}
Proposition ... may encourage us to think of $\mbox{\bf EJC}$ as a symmetric monoidal category under $\odot$. However,
Example... shows that this is not the case, as the tensor product of two Jordan-preserving CP mappings
need not be Jordan preserving. In this section, we construct several subcategories of $\mbox{\bf EJC}$ that {\em are}
symmetric monoidal under $\odot$. }

\tempout{
The category $\mbox{\bf EJC}$ has several interesting sub-categories. One of these is the
category $\UEJC$ of universally embedded EJCs, still with CJP mappings.
As noted above the Hanche-Olsen tensor product of $A$ and $B$ can be understood as the
canonical tensor product of the pairs
$(A,\Cu(A))$ and $(B,\Cu(B))$. As Example ... illustrates, there is generally
no CJP tensor product of CJP mappings.
and therefore $\UEJC$ is a monoidal sub-category of $\mbox{\bf CPEJC}$. As
 Since $\mbox{\bf CPUREJC}$ isn't strictly monoidal under $\widetilde{\otimes}$, neither is
 $\mbox{\bf CPEJC}$ monoidal under $\odot$.

Here is another example. Let $\Cat_{o}$ be the subcategory of $\Cat$
consisting of simple universally reversible euclidean JC
algebras, each represented in the ``obvious" way.  More precisely,
recalling the shorthand
\[R_n = M_{n}({\Bbb R})_{h}, \ \ C_n = M_{n}({\Bbb C})_{{sa}}, \  \mbox{and} \ M_n = H_n({\Bbb H})_{{sa}},\]
there are canonical representations
\[R_n \subseteq M_{n}({\Bbb C}), \ C_n \subseteq M_n({\Bbb C}), \ \mbox{and} \
 H_n \subseteq M_{2n}({\Bbb C}).\footnote{I am being a trifle sloppy
   here --- this is a case where talking about representations of,
   rather than represented, JC-algebras would make sense.} \] With
 these representations understood, the canonical tensor products of
 these are given by Table 3 above. As all entries are again simple and UR,
 $\mathbb{C}_o$ is closed under $\odot$.}
  .
\tempout{
\[
\begin{array}{c}
\begin{array}{l|ccc}
\odot    & R_k & C_k & H_k \\
 \hline
R_n & R_{nk} & C_{nk} & H_{nk}    \\
C_n & C_{nk} & C_{nk} & C_{2nk}  \\
H_n & H_{nk} & C_{2nk} & R_{4nk}
\end{array} \\
\\
\mbox{Figure 3}
\end{array}\]
As all entries are simple, $\Cat_o$ is closed under $\odot$. \\
}

\subsection{Completely Jordan Preserving Maps}

Examples \ref{ex: no states} and \ref{ex: no states 2} suggest the following adaptation
of the notion of complete positivity to our setting.

\begin{definition}
{\em
A linear mapping $\phi : (A,\mathbf{M}_A) \rightarrow (B,\mathbf{M}_B)$ is
\emph{completely Jordan preserving} (CJP)
iff, for any embedded JC-algebra $(C,\mathbf{M}_C)$, the mapping $\phi \otimes
\mbox{id}_{\mathbf{M}_C}$ is positive, and takes $A \odot C$ into $B \odot C$.
}
\end{definition}

{}%\color{red} Note that this implies that $\phi$ is both Jordan-preserving (take $C = \mathbb{R}$) and completely positive (take
$C = C_n$ for any $n$). }

\begin{lemma}\label{lemma: CJP1}  If $\phi : \mathbf{M}_A \rightarrow \mathbf{M}_B$ is CJP, then for any $(C,\mathbf{M}_C)$, $\phi \otimes \mbox{id}_{\mathbf{M}_{C}}$ is again CJP.\end{lemma}

\noindent{\em Proof:}  If $(D,\mathbf{M}_D)$ is another embedded JC-algebra,
consider $(C \odot D, \mathbf{M}_{C} \otimes \mathbf{M}_{D})$. The associativity
of $\odot$ gives us
\[(\phi \otimes \mbox{id}_{\mathbf{M}_{C}}) \otimes \mbox{id}_{\mathbf{M}_D} = \phi \otimes (\mbox{id}_{\mathbf{M}_C \otimes \mathbf{M}_D}).\]
Since $\phi$ is CJP, the latter {}%\color{red} caries} $A \odot (C \odot D)$ into $B \odot (C \odot D)$.
Since $\odot$ is associative, this tells us that
{}%\color{red} $(\phi \otimes \mbox{id}_{C}) \otimes \mbox{id}_{D}$ carries} $(A \odot C) \odot D$  into $(B \odot C) \odot D$.
$\Box$\\

While all CJP morphisms are CP,
Example \ref{ex: no
  states} shows that the converse is false. On the other hand, the
class of CJP morphisms is still quite large.

\begin{example} {\em Let $\phi : \mathbf{M}_A \rightarrow \mathbf{M}_B$ be a {}%\color{red} $\ast$-homomorphism} taking $A$ to $B$. Then
$\phi \otimes \mbox{id}_{\mathbf{M}_C}$ is again a $\ast$-homomorphism, and hence,
    takes the Jordan subalgebra generated by $A \otimes C$ to that
    generated by $B \otimes C$, i.e., sends $A \odot C$ into $B \odot
    C$. So $\phi$ is CJP. }\end{example}

\begin{example}
{\em Let $(A,\mathbf{M}_A)$ be an embedded JC-algebra, and let $a \in A$. The
  mapping $U_a : \mathbf{M}_A \rightarrow \mathbf{M}_A$ given by $U_a(b) = aba$ is
  {}%\color{red} completely} positive, and can be expressed in terms of the Jordan product on
  $\mathbf{M}_A$ as
\[U_a(b) = 2 a \bullet (a \bullet b) - (a^2)\bullet b\]
i.e., $U_a = 2 L_{a}^{2} - L_{a^2}$, where $L_a$ is the operator of
left Jordan multiplication. Since $A$ is a Jordan subagebra of
$(\mathbf{M}_A)_{{sa}}$, $U_a(b) \in A$ for all $a, b \in A$. Thus, $U_a$ is a
morphism.  Now if $(C,\mathbf{M}_C)$ is another embedded JC-algebra, we have
\[U_{a} \otimes U_{c} = U_{a \otimes c}\]
for all $c \in \mathbf{M}_C$; in particular,
\[U_a \otimes \mbox{id}_{\mathbf{M}_C} = U_a \otimes U_{\mathbf{1}} = U_{a \otimes \mathbf{1}}.\]
Since $a \otimes \mathbf{1} \in A \odot C$, it follows that $U_{a}$ is
CJP. }\end{example}

\begin{proposition}
\label{prop: CJP composes and tensors}
Let $(A,\mathbf{M}_A)$, {}%\color{red} $(A', \mathbf{M}_{A'})$, $(B,\mathbf{M}_B)$, $(B', \mathbf{M}_{B'})$ and and $(C,\mathbf{M}_{C})$}  be EJC-algebras.
\begin{itemize}
\item[(i)] If $\phi : \mathbf{M}_{A} \rightarrow \mathbf{M}_{B}$ and $\psi : \mathbf{M}_{B} \rightarrow \mathbf{M}_{C}$
are CJP, then so is $\psi \circ \phi$, and
\item[(ii)] if $\phi : \mathbf{M}_{A} \rightarrow \mathbf{M}_{B}$ and $\psi : \mathbf{M}_{A'} \rightarrow \mathbf{M}_{B'}$ are CJP,
then so is $\phi \otimes \psi : \mathbf{M}_{A \odot A'} = \mathbf{M}_{A} \otimes \mathbf{M}_{A'} \rightarrow \mathbf{M}_{B} \otimes \mathbf{M}_{B'} = \mathbf{M}_{B \odot B'}$.
\[\phi \otimes \psi = (\phi \otimes \mbox{id}_{\mathbf{M}_{B'}}) \circ (\mbox{id}_{\mathbf{M}_{A}} \otimes \psi)\]
is again CJP.
\end{itemize}
\end{proposition}

\noindent{\em Proof:}  (i) is straightforward, and (ii) follows from (i) and Lemma \ref{lemma: CJP1}.  $\Box$\\

Thus, {}%\color{red} EJC-algebras and CJP maps form}
a symmetric monoidal
category, which we will call $\mbox{\bf CJP}$, with tensor unit $I =
(\mathbb{R},\mathbb{C})$. \\

\noindent{\bf Relatively CJP Mappings} The category $\mbox{\bf CJP}$ has an undesirable feature:
Any positive mapping $f : \mathbf{M}_A \rightarrow \mathbb{R}$ is automatically
Jordan preserving, yet, as Example \ref{ex: no states} illustrates, for a non-UR $B$, $f \otimes \mbox{id}_B$ {}%\color{red} need
not be}. In other words, there {}%\color{red} need be} no CJP morphisms $A \rightarrow I$. In particular,
states of $A$ {}%\color{red} need not be} CJP morphisms.

{}%\color{red} In some cases,} we can remedy this difficulty by relativising
the definition of CJP mappings to a particular class $\Cat$ of embedded JC
algebras. In what follows, assume that $\Cat$ is closed under $\odot$
and contains the tensor unit $I = (\mathbb{R},\mathbb{C})$.

\begin{definition}{\em Let $(A,\mathbf{M}_A)$ and $(B,\mathbf{M}_B)$ belong to $\Cat$.
 A positive linear mapping $\phi : \mathbf{M}_A \rightarrow \mathbf{M}_B$ is {\em relatively CJP} with respect to $\Cat$
if, for all $(C,\mathbf{M}_C) \in \Cat$, the mapping $\phi \otimes \mbox{id}_{\mathbf{M}_{C}}$ is positive and maps
$A \odot C$ into $B \odot C$. We denote the set of all such maps by $\mbox{\bf CJP}_{\Cat}(A,B)$. }
\end{definition}

{}%\color{red} Note that $\phi$'s being relatively CJP with respect to $\Cat$ does not imply that $\phi$ is CP, unless
$\Cat$ contains $C_n$ for every $n \in \mathbf{N}$. Nevertheless,}
exactly as in the proof of Proposition \ref{prop: CJP composes and tensors},
 we see that if $A,B,C \in \Cat$ and $\phi \in \mbox{\bf CJP}_{\Cat}(A,B)$ and $\psi \in \mbox{\bf CJP}_{\Cat}(B,C)$, then $\psi \circ \phi \in \mbox{\bf CJP}_{\Cat}(A,C)$, and also that if $A,B,C,D \in \Cat$ and
$\phi \in \mbox{\bf CJP}_{\Cat}(A,B)$ and $\psi \in \mbox{\bf CJP}_{\Cat}(C,D)$,
then $\phi \otimes \psi \in \mbox{\bf CJP}_{\Cat}(A \otimes C,  B \otimes D)$. In other words, $\Cat$ becomes a symmetric
monoidal category with relatively CJP mappings as morphisms. We denote this category by $\mbox{\bf CJP}_{\Cat}$.
Here are three important ones.

\begin{example}[The category $\mbox{$\mathbb C$QM}$]\label{ex: CQM}
{\em Let $\Cat$ be the class of hermitian parts of
complex $*$-algebras with standard embeddings. Then $\phi$
belongs to $\mbox{CJP}_{\Cat}$
iff $\phi$ is CP.  Evidently, {}%\color{red} this category is} essentially orthodox, mixed-state QM with
  superselection rules. From now on, we shall call this category $\mbox{$\mathbb C$QM}$.
  }
\end{example}

\begin{example}[The category $\mbox{\bf RSE}$] \label{ex: RSE}{\em Let $\mbox{\bf RSE} = \mbox{\bf CJP}_{\Cat}$, where $\Cat$ is the class of reversible EJCs with standard embeddings. By Table 2, plus the distributivity of $\oplus$ over direct sums, the latter is closed under $\odot$. The category $\mbox{\bf RSE}$ represents a kind of unification of finite-dimensional real, complex and quaternionic quantum mechanics, in so far as its objects are the Jordan algebras associated with real, complex
  and quaternionic quantum systems, and direct sums of these. Moreover, as restricted to complex (or real) systems,
  its compositional structure is the standard one. However, Example \ref{ex: no states 2} shows that not every quantum-mechanical
  process --- in particular, not even processes that prepare states --- will count as a morphism in $\mbox{\bf RSE}$,
  so this unification comes at a high cost. }
\end{example}

\begin{example}[The category $\mbox{\bf URUE}$]\label{ex: URUE} {\em
Let $\Cat$ be the class of universally reversible EJCs with universal embeddings. {}%\color{red} This
is closed under $\odot$ by Corollary 5.4 and Proposition 3.5.} {}%\color{red}
We denote the category $\mbox{\bf CJP}_{\Cat}$ by $\mbox{\bf URUE}$. This is closer to being a legitimate ``unified"
  quantum theory, although it omits the quaternionic bit (which is reversible, but not
  universally so). Even as
  restricted to complex quantum systems, however, it differs from orthodox QM
  in two interesting ways. First, and most conspicuously, the tensor product
  is not the usual one: $C_n \widetilde{\otimes} C_k = C_{nk} \oplus C_{nk}$, rather
  than $C_{nk}$.   Secondly, it allows some processes that orthodox QM does not.
  Any CP mapping $\phi : \Cu(A) \rightarrow \Cu(B)$ intertwining the involutions $\Phi_A$ and $\Phi_B$
(that is, $\Phi_B \circ \phi = \phi \circ \Phi_A$) is relatively CJP, i.e., a morphism in $\mbox{\bf URUE}$.
In particular, the the mapping
  on $\Cu(C_n) = M_n(\mathbb{C}) \oplus M_n(\mathbb{C})$ that swaps the two summands is a morphism. Since
  the image of $C_n$ in $\Cu(C_n)$ consists of pairs $(a,a^T)$, this mapping
  effects the transpose automorphism on $C_n$. This is not
  permitted in orthodox QM, as the transpose is not a CP mapping on
  $M_n(\mathbb{C})$.  (This causes no difficulty with positivity in the context of $\mbox{\bf URUE}$, where the
  tensor product is different.)}
}
\end{example}

  In spite of its divergences from orthodoxy, $\mbox{\bf URUE}$ is in many respects a
  well-behaved probabilistic theory. In the next section, we shall improve on it,
  by considering a category of EJC-algebras having a slightly larger class of objects (in particular,
  it includes the quabit), but a slightly more restricted set of morphisms.

\subsection{\bf The Category $\mbox{}%\color{red} \bf InvQM}$}

Recall that we write $\mathbf{M}^{\Phi}$ for the set of fixed-points of an involution $\Phi$  on a complex $\ast$-algebra $\mathbf{M}$.
Notice that the involution $\Phi$ on $M_n(\mathbb{C})$ with  $R_n = M_n(\mathbb{C})^{\Phi}_{{sa}}$ (the transpose) and
on $M_{2n}(\mathbb{C})$ with $Q_n = M_{2n}(\mathbb{C})^{\Phi}_{{sa}}$, namely $\Phi(a) = -Ja^T J$ where $J$ is the unitary
$J = \left  [ \begin{array}{rc} 0 & \mathbf{1} \\ -\mathbf{1} & 0 \end{array} \right ]$, are both unitary with respect to the
trace inner product $\langle a, b \rangle := \mbox{Tr}(ab^{\ast})$\footnote{}%\color{red} We follow the convention that
inner products are linear in the {\em first} argument.}. The canonical involution on $\Cu(C_n) = M_n(\mathbb{C}) \oplus M_n(\mathbb{C})$, namely,
the mapping $(a,b) \mapsto (b^T, a^T)$, {}%\color{red} is likewise unitary}.

\begin{definition} {\em Call an EJC $(A,\mathbf{M}_A)$ {\em involutive} iff there exists a {\em unitary} involution $\Phi$ on $\mathbf{M}_A$ with $A = (\mathbf{M}_A)^{\Phi}_{{sa}}$.
Let $\mbox{}%\color{red} \bf InvQM}$ denote the category in which objects are involutive EJC-algebras,
and in which a morphism $\phi : A \rightarrow B$ is a CP mapping $\phi : \mathbf{M}_{A} \rightarrow \mathbf{M}_B$ intertwining
$\Phi_{A}$ and $\Phi_{B}$, i.e, $\Phi_{B} \circ \phi = \phi \circ \Phi_{A}$.
}
\end{definition}

As pointed out earlier, the condition that $A$ be the set of
  self-adjoint fixed points of an involution makes $A$ reversible in
  $\mathbf{M}_A$. Thus, the class of involutive EJC-algebras
  contains no ``higher'' ($n = 4$ or $n > 5$)
  spin factors. In fact,

  it contains exactly {}%\color{red} direct sums of} the universally
  embedded UR EJCs $(A,\Cu(A))$ where $A = R_n, C_n$, with $n$ arbitrary,
  or $Q_n$ with $n  > 2$, {\em together with} the {\em standardly} embedded quabit, i.e.,
  $(Q_2,M_4(\mathbb{C}))$, with $Q_2$ the self-adjoint fixed points of the
  involution $\Phi(a) = -Ja^{T}J$ discussed in Section \ref{subsec:
    reps of EJAs}.  In other words, $\mbox{}%\color{red} \bf InvQM}$ includes
     exclusively {\em
    quantum} systems over the three division rings $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$,
  albeit with the complex systems represented in their universal
  embeddings. Note that $(\mathbb{R}, M_1(\mathbb{R}))$ counts as an involutive EJC:
  since $M_1(\mathbb{R}) = \mathbb{R}$ is commutative, the identity map provides the
  necessary involution.

It is easy to see that CP mappings $\mathbf{M}_A \rightarrow \mathbf{M}_B$ intertwining $\Phi_A$ and $\Phi_B$ are automatically
relatively CJP for the class of involutive EJCs. By \ref{cor: canonical composites with involutions}
the canonical tensor product of involutive
EJCs is again involutive, as
\[A \odot B = (\mathbf{M}_{A} \otimes \mathbf{M}_{B})^{\Phi_A \otimes \Phi_B}_{{sa}}\]
for all $A, B \in \mbox{}%\color{red} \bf InvQM}$. Since invoutive EJCs are reversible, Proposition 5.3 implies
that $A \odot B$ is a dynamical composite of $A$ and $B$.
Composites and tensor products of intertwining CP maps
are also such {}%\color{red} (as are associators, unit-introductions and the swap mapping)}, so $\mbox{}%\color{red} \bf InvQM}$ is a symmetric monoidal category --- indeed, a {}%\color{red} monoidal} subcategory of the category
of involutive EJC-algebras and relatively CJP maps\footnote{}%\color{red} We leave open the question of whether
there exist relatively CJP mappings for the class of involutive EJC-algebras that are not intertwiners.}.

In the special case in which $A$ and $B$ are universally embedded complex qantum systems, say
$A = C_n$ and $B = C_k$, we have $\mathbf{M}_{A} = \Cu(C_n) = M_n(\mathbb{C}) \oplus M_n(\mathbb{C})$ and
similarly $\mathbf{M}_{B} = M_k(\mathbb{C})\oplus M_k(\mathbb{C})$. The involutions $\Phi_A$
are given by $\Phi_{A}(a,b) = (b^T, a^T)$, and similarly for $\Phi_B$. In
this case, the interwining CP-maps are sums of mappings of
of the two forms: $(a, b) \mapsto (\phi(a), \phi^T(b))$ and $(a, b) \mapsto
(\phi^T(b), \phi(a))$, where $\phi$ is a CP mapping $M_n(\mathbb{C}) \rightarrow M_k(\mathbb{C})$
and $\phi^T$ is determined by the condition
$\phi^T(x^T) = (\phi(x))^T$, i.e. $\phi^T := T \circ \phi \circ T$.

Three special cases of $\mbox{}%\color{red} \bf InvQM}$-morphisms are worth emphasising:

\begin{corollary}\label{cor: things that are InvQM morphisms} Let $(A,\mathbf{M}_A)$ belong to $\mbox{}%\color{red} \bf InvQM}$. Then
\begin{itemize}
\item[(a)] for every $a \in A$, the corresponding linear mapping $a : \mathbb{R} \rightarrow \mathbf{M}_A$
determined by $a(1) = a$,
belongs to $\mbox{}%\color{red} \bf InvQM}(I,A)$;
\item[(b)] every positive linear functional (in particular, every state) on $\mathbf{M}_A$ of the form
$| a \rangle$,  $a \in A_+$, belongs to $\mbox{}%\color{red} \bf InvQM}(A,I)$;
\end{itemize}	 \end{corollary}

\noindent{\em Proof:} {}%\color{red} Part (a) is immediate from the fact that $a$ is fixed by
$\Phi_{A}$}; part (b) {}%\color{red} also} follows from this, plus the unitarity of $\Phi_A$. $\Box$ \\

The category $\mbox{}%\color{red} \bf InvQM}$ provides a unification of
  finite-dimensional real, complex and quaternionic quantum mechanics,
  but with the same  important caveats that apply to $\mbox{\bf URUE}$: the representation of orthodox, complex
  quantum systems $C_n$ in $\mbox{}%\color{red} \bf InvQM}$ is through the universal embedding
  $\psi : C_n \mapsto \Cu(C_n) = M_n(\mathbb{C}) \oplus M_n(\mathbb{C})$, $a \mapsto
  (a,a^T)$.  As a consequence, the composite of two complex quantum
  systems in $\mbox{}%\color{red} \bf InvQM}$ is a direct sum of two copies of the their
  standard composite --- equivalently, is the standard composite,
  combined with a classical bit.  Moreover, the mapping that swaps the direct summands of
  $\Cu(C_n)$, a perfectly good morphism in $\mbox{}%\color{red} \bf InvQM}$, acts as the transpose on
  $\psi(a) = (a, a^T)$.
  \\

\subsection{Compact Closure}

A {\em compact structure} on a symmetric monoidal category $\Cat$ is a choice, for every object $A \in \Cat$, of a
{\em dual object}: A triple $(A',\eta_A,f_A)$ consisting of an object $A' \in \mathbb{C}$, a {\em co-unit} $\eta_A : A \otimes A' \rightarrow I$ and a {\em unit} $f_{A} : I \rightarrow A' \otimes A$
such that
\[A \rightarrow A \otimes I \stackrel{\mbox{id}_A \otimes f_{A}}{\longrightarrow}  A \otimes (A' \otimes A) \longrightarrow (A \otimes A') \otimes A \stackrel{\eta_{A} \otimes \mbox{id}_A}{\longrightarrow} I \otimes A \rightarrow A\]
and
\[A' \rightarrow I \otimes A' \stackrel{f_{A} \otimes \mbox{id}_{A'}}{\longrightarrow} (A' \otimes A) \otimes A' \longrightarrow A' \otimes (A \otimes A') \stackrel{\mbox{id}_{A'} \otimes \eta_{A}}{\longrightarrow} A' \otimes I \rightarrow A'\]
give the identity morphisms on $A$ and $A'$, respectively (and where the unlabeled arrows are the obvious {}%{\color{magenta} unit introductions and associators}).\footnote{Our usage is slightly perverse. The usual convention is to denote the {\em unit} by $\eta_A$ and the
co-unit by $\epsilon_A$. Our choice is motivated in part by the desire to represent states as morphisms $A \rightarrow I$
and effects as morphisms $I \rightarrow A$, rather than the reverse, together with the convention that
takes the unit to correspond to the maximally entangled state.}
The standard example is the category $\mbox{\bf FinVec}$ of finite-dimensional vector spaces (say, over $\mathbb{C}$) and linear mappings. Here there is a canonical linear functional $\eta_{A} : A \otimes A^{\ast} \rightarrow \mathbb{C}$, namely, the trace. A canonical unit is supplied by picking a basis $E$ for $A$, and setting $f_{A} = \sum_{x \in E} f_x \otimes f_x$, where $\{f_x\}$ is the dual basis for $A^{\ast}$; one then shows that this is independent of $E$, {}%\color{red} and that the  identities above hold.}

In \cite{Abramsky-Coecke}, it is shown that a large number of
information-processing protocols, including
{}%{\color{magenta} in particular} conclusive
teleportation and entanglement-swapping, hold in any compact closed
symmetric monoidal category, if we interpret objects as systems and
morphisms as physically allowed processes. In this section, we shall
see that our category $\mbox{}%\color{red} \bf InvQM}$ is compact closed. More exactly, we
shall show that it inherits a compact structure from the {}%{\color{magenta}
  natural compact structure on the} category $\mbox{\bf $\ast$-Alg}$ of
finite-dimensional complex $\ast$-algebras, {}%\color{red} which we now review}.

If $\mathbf{M}$ is a finite-dimensional complex $\ast$-algebra, let $\mbox{Tr}$ denote the
canonical trace on $\mathbf{M}$, regarded as acting on itself by left
multiplication (so that $\mbox{Tr}(a) = \mbox{tr}(L_a)$, $L_a : \mathbf{M} \rightarrow \mathbf{M}$
being $L_a(b) = ab$ for all $b \in \mathbf{M}$). This induces an inner product\footnote{}%\color{red} double check!! Checked.}
on $\mathbf{M}$, given by $\langle a , b \rangle_{\mathbf{M}} = \mbox{Tr}(ab^{\ast})$\footnote{Again, we are following the convention
that complex inner products are linear in the first argument.}.  Note
that this inner product is self-dualizing, i.e,. $a \in \mathbf{M}_+$ iff
$\langle a , b \rangle \geq 0$ for all $b \in \mathbf{M}_+$.

Now let $\bar{\mathbf{M}}$ be the conjugate algebra, writing $\bar{a}$ for $a \in \mathbf{M}$ when regarded as belonging to $\bar{\mathbf{M}}$ (so that $\bar{ca} =
\bar{c}~\bar{a}$ for scalars $c \in \mathbb{C}$ and $\bar{a} \bar{b} =
\bar{ab}$ for $a, b \in \mathbf{M}$).  Note that $\langle \bar{a}, \bar{b} \rangle = \langle b ,
a \rangle$.  Now define
\[f_{\mathbf{M}} = \sum_{e \in E} e \otimes \bar{e} \in \mathbf{M} \otimes \bar{\mathbf{M}}\]
where $E$ is any orthonormal basis for $\mathbf{M}$ with respect to
$\langle \  | \ \rangle_{\mathbf{M}}$. {}%\color{red} Then straightforward computations show that $f_{\mathbf{M}} \in (\mathbf{M} \otimes \bar{\mathbf{M}})_{+}$, and that,
for all $a, b \in \mathbf{M}$,
\[\langle a \otimes \bar{b}, f_{\mathbf{M}} \rangle = \langle a, b \rangle = \mbox{Tr}(ab^{\ast}),\]
where the inner product on the left is the trace inner product on $\mathbf{M} \otimes \mathbf{M}^{\ast}$. Now define
$\eta_{\mathbf{M}} : \mathbf{M} \otimes \bar{\mathbf{M}} \rightarrow \mathbb{C}$ by $\eta_{\mathbf{M}} = | f_{\mathbf{M}} \rangle$, noting that this functional
is positive (so, up to normalization, a state) since $f_{\mathbf{M}}$ is positive in $\mathbf{M} \otimes \bar{\mathbf{M}}$. }

\tempout{
Then a computation shows  that
\[ \langle (a \otimes \bar{b}) f_{\mathbf{M}} | f_{\mathbf{M}} \rangle_{\mathbf{M} \otimes \bar{\mathbf{M}}} = \langle a | b \rangle_{\mathbf{M}}.\]
Since the left-hand side defines a positive linear functional on $\mathbf{M}
\otimes \bar{\mathbf{M}}$, so does the right (remembering here that pure
tensors generate $\mathbf{M} \otimes \bar{\mathbf{M}}$, as we're working in finite
dimensions).  Call this functional $\eta_{\mathbf{M}}$. That is,
\[\eta_{\mathbf{M}} : \mathbf{M} \otimes \bar{\mathbf{M}} \rightarrow \mathbb{C} \ \ \mbox{is given by} \ \ \eta_{\mathbf{M}}(a \otimes \bar{b}) = \langle a | b \rangle = \mbox{Tr}(ab^{\ast})\]
and is, up to normalization, a state on $\mathbf{M} \otimes \bar{\mathbf{M}}$. A further computation now shows that
\[\langle a \otimes \bar{b} | f_{\mathbf{M}} \rangle_{\mathbf{M} \otimes \bar{\mathbf{M}}} = \eta(a \otimes \bar{b}).\]
 It follows that $f_{\mathbf{M}}$ belongs to the positive cone of
$\mathbf{M} \otimes \bar{\mathbf{M}}$, by self-duality of the latter.
}

A final
computation  shows that, for any states $\alpha$ and
$\bar{\alpha}$ on $\mathbf{M}$ and $\bar{\mathbf{M}}$, respectively, and any $a \in
\mathbf{M}, \bar{a} \in \bar{\mathbf{M}}$, we have
\[(\eta_{\mathbf{M}} \otimes \alpha)(a \otimes f_{\bar{\mathbf{M}}}) = \alpha(a) \ \ \mbox{and} \ \ (\bar{\alpha} \otimes \eta_{\mathbf{M}})(f_{\bar{M}} \otimes \bar{a}) = \bar{\alpha}(\bar{a}).\]
Thus, $\eta_{\mathbf{M}}$ and $f_{\bar{\mathbf{M}}}$ define a compact structure on
$\mbox{\bf $\ast$-Alg}$, for which the dual object of $\mathbf{M}$ is given by $\bar{\mathbf{M}}$.

\begin{definition} {\em The {\em conjugate} of a EJC-algebra $(A,\mathbf{M}_{A})$ is $(\bar{A},\bar{\mathbf{M}}_{A})$, where
$\bar{A} = \{ \bar{a} | a \in A\}$. We write $\eta_{A}$ for $\eta_{\mathbf{M}_A}$ and $f_A$ for $f_{\mathbf{M}_A}$. }\end{definition}

Any linear mapping $\phi : \mathbf{M} \rightarrow \mathbf{N}$ between $\ast$-algebras $\mathbf{M}$ and $\mathbf{N}$ gives rise to a
linear mapping $\bar{\phi} : \bar{\mathbf{M}} \rightarrow \bar{\mathbf{N}}$, given by $\bar{\phi}(\bar{a}) = \bar{\phi(a)}$
for $a \in \mathbf{M}$. It is straightforward {}%\color{red} to show} that if $\Phi$ is a unitary involution on $\mathbf{M}_A$ with $A = {\mathbf{M}_{A}}^{\Phi}_{{sa}}$, then $\bar{\Phi} : \bar{\mathbf{M}} \rightarrow \bar{\mathbf{M}}$ is also a unitary involution with ${\bar{\mathbf{M}}_{A}}^{\bar{\Phi}}_{{sa}} = \bar{A}$. Thus, the class of involutive EJCs is closed under the formation of conjugates.

\begin{lemma} Let $(A,\mathbf{M}_A)$ belong  {}%\color{red} to $\mbox{}%\color{red} \bf InvQM}$.} Then
$f_{A} \in A \odot \bar{A}$.
\end{lemma}

\noindent{\em Proof:}  {}%\color{red} By assumption,} there is a unitary involution
$\Phi$ on $\mathbf{M}_A$ such that $A = (\mathbf{M}_{A})^{\Phi}_{{sa}}$; by Corollary \ref{cor: canonical composites with involutions},
$A \odot \bar{A}$ is then the set of self-adjoint fixed points of $\Phi \otimes \bar{\Phi}$.
Since $\Phi$ is unitary,  if $E$ is an orthonormal basis for $\mathbf{M}_A$, then so
is $\{\Phi(e) | e \in E\}$; hence, as $f_A$ is independent of the choice of
orthonormal basis, $f_A$ is invariant under $\Phi \otimes \bar{\Phi}$.
Since $f_A$ is also self-adjoint, it belongs to $(\mathbf{M}_A \otimes \mathbf{M}_{\bar{A}})^{\Phi \otimes \bar{\Phi}}_{{sa}}$,
i.e., to $A \odot \bar{A}$. $\Box$ \\

It follows now from part (b) of Corollary \ref{cor: things that are InvQM morphisms} that the
functional
\[{}%\color{red} \eta_{A}} = | f_{\bar{A}} \rangle : \mathbf{M}_{\bar{A}} \otimes \mathbf{M}_{A} \rightarrow \mathbb{R}\]
is an $\mbox{}%\color{red} \bf InvQM}$ morphism.  Hence, $\mbox{}%\color{red} \bf InvQM}$ inherits the compact structure from $\mbox{\bf $\ast$-Alg}$, as promised. This gives
us

\begin{theorem}\label{thm: InvQM is compact closed} $\mbox{}%\color{red} \bf InvQM}$ is compact closed. \end{theorem}

\noindent{\bf Dagger compactness} In fact, we can do a bit better. {}%\color{red} A {\em dagger} on a category $\Cat$
is an involutive contravariant functor $\dagger : \Cat \rightarrow \Cat$ that fixes objects; that is, $A^{\dagger} = A$
for all $A \in \Cat$, and $f^{\dagger} \in \Cat(B,A)$ for all $f \in \Cat(A,B)$ with $f^{\dagger \dagger} = f$.
If $\Cat$ is a symmetric
monoidal category and $(f \otimes g)^{\dagger} = f^{\dagger} \otimes g^{\dagger}$ for all
morphisms $f$ and $g$, and also $\sigma_{A,B}^{\dagger} = \sigma_{B,A}$, where $\sigma_{A,B} : A \otimes B
\rightarrow B \otimes A$ is the ``swap" morphism, then $\Cat$ is said to be
{\em dagger-monoidal.}  Finally, if $\Cat$ has a compact structure such that $\eta_{A}^{\dagger} = f_{A}$,
then $\Cat$ is said to be {\em dagger-compact.} }

It is not difficult to show that $\mbox{\bf $\ast$-Alg}$ is dagger-compact, where, if $\mathbf{M}$ and $\mathbf{N}$ are finite-dimensional $\ast$-algebras and $\phi : \mathbf{M} \rightarrow \mathbf{N}$ is a linear mapping, $\phi^{\dagger}$ is the hermitian adjoint
of $\phi$ with respect to the natural trace inner products on $\mathbf{M}$ and $\mathbf{N}$. If $(A,\mathbf{M}_A)$ and $(B,\mathbf{M}_B)$
are involutive
EJC-algebras with given unitary involutions $\Phi_A$ and $\Phi_B$, then for any intertwiner $\phi : \mathbf{M}_A \rightarrow \mathbf{M}_B$,
$\phi^{\dagger}$ also intertwines $\Phi_A$ and $\Phi_B$. Hence, we have

\begin{corollary}\label{cor: InvQM is dagger compact} $\mbox{}%\color{red} \bf InvQM}$ is dagger-compact.\end{corollary}

\section{Conclusion}
\label{sec: conclusion}

We have found two {}%\color{red} categories of probabilistic models} ---the categories $\mbox{\bf RSE}$ and
$\mbox{}%\color{red} \bf InvQM}$---that, in different ways, unify finite-dimensional real,
complex and quaternionic quantum mechanics.  In each case, there is a
price to be paid for this unification.  For $\mbox{\bf RSE}$, this price is
steep: $\mbox{\bf RSE}$ is a monoidal category, but one in which states (for
instance) on complex systems don't count as physical processes.
In particular, $\mbox{\bf RSE}$ is
very far from being compact closed.\footnote{}%\color{red} In \cite{BGraW}, we erroniously claimed
that the category $\mbox{\bf URSE}$ of  universally reversible, but standardly embedded, EJC algebras,
with relatively CJP mappings, is compact closed. That this is not the case is clear from
\ref{ex: no states 2}.}

{}%{\color{magenta} In contrast,
$\mbox{}%\color{red} \bf InvQM}$ is clearly
a well-behaved --- indeed, dagger-compact --- probabilistic theory, in which the states, as well
as the effects, of real,
complex, and quaternionic Euclidean Jordan algebras appear as morphisms.
  On the other hand, $\mbox{}%\color{red} \bf InvQM}$
 admits the transpose
  automorphism on the complex Hermitian Jordan algebra,} and requires
complex quantum systems to compose in a non-standard way.
Nevertheless, by virtue of being dagger compact, ${}%\color{red} \mbox{}%\color{red} \bf InvQM}}$
continues
to enjoy many of the information-processing properties of standard
complex QM, e.g., the existence of conclusive teleportation and
entanglement-swapping protocols \cite{Abramsky-Coecke}. {}%{\color{magenta} Also, composites in $\mbox{}%\color{red} \bf InvQM}$
satisfy the Cirel'son bound on correlations owing to the way that,
by construction, these composites are embedded within a tensor product
of complex matrix algebras.}

All of this is in spite of the fact that composites in $\mbox{}%\color{red} \bf InvQM}$ are
not locally tomographic:
the canonical composite $A \odot B$ is larger than the vector
space tensor product $A \otimes B$.
Local tomography is well known to separate complex QM from its
real and quaternionic variants, so its failure in $\mbox{\bf URUE}$ and $\mbox{\bf RSE}$
is hardly surprising,  but it is noteworthy that we are able to construct
(non-locally tomographic) composites in $\mbox{\bf URUE}$ in all of the non-real cases, and
certain composites involving quaternions even in $\mbox{\bf RSE}$.
Even more interesting is the fact that, for
quaternionic systems $A$ and $B$, the {\em information capacity} ---
the number of sharply distinguishable states --- of $A \odot B$ is {\em larger} than the
product of the capacities of $A$ and $B$. A related point is that, for
quaternionc quantum systems $A$ and $B$, the product of a pure state
of $A$ and a pure state of $B$ will generally be a {\em mixed} state
in $A \odot B$.

The category $\mbox{}%\color{red} \bf InvQM}$ contains interesting compact closed
subcategories.  In particular, real and quaternionic quantum systems,
taken together, form a (full) monoidal sub-category of $\mbox{}%\color{red} \bf InvQM}$ closed under
composition.  We  {}%{\color{magenta} conjecture}
that this is exactly what one gets by applying Selinger's CPM construction \cite{Selinger} to Baez'
(implicit) category of pairs $(\mathbf{H},J)$, $H$ a finite-dimensional
Hilbert space and $J$ an anti-unitary with $J^2 = \pm 1$ \cite{BaezDivAlg}.

Another compact-closed subcateory of $\mbox{}%\color{red} \bf InvQM}$, which we might call $\mbox{}%\color{red} \bf InvCQM}$,
consists of universally embedded {\em complex} quantum systems $C_n$.
It is interesting to note that,
in an hypothetical universe described by $\mbox{}%\color{red} \bf InvQM}$, the subcategory $\mbox{}%\color{red} \bf InvCQM}$
acts as a kind of ``ideal", in that
if $A \in \mbox{}%\color{red} \bf InvQM}$ and $B \in \mbox{}%\color{red} \bf InvCQM}$, then $A \odot B \in \mbox{}%\color{red} \bf InvCQM}$ as well.
This is provocative, as it suggests that such a universe, initially
consisting of many systems of all three types, {}%\color{red} might} eventually evolve
into one in which complex systems greatly predominate.

Although it is not compact closed, the
category $\mbox{\bf RSE}$ of reversible, standardly embedded EJCs remains of interest. This is
still a monoidal category, and contains, in addition to real and quaternionic quantum systems, orthodox complex
quantum systems in their standard embedding (and composing in the normal way).
Indeed, these form a monoidal subcategory, $\mbox{$\mathbb C$QM}$, which, again,
functions as an ``ideal".

It is worth noting that the set of quaternionic quantum systems
does {\em not} form a monoidal subcategory of either $\mbox{\bf RSE}$ or
$\mbox{}%\color{red} \bf InvQM}$, as the composite of two quaternionic systems is {\em real}.
Efforts to construct a free-standing quaternionic quantum theory
have had to contend with the absence of a suitable {\em quaternionic}
composite of quaternionic systems. For instance, as pointed out by Araki
\cite{Araki80}, the obvious candidate for the composite
of $A = M_m(\mathbb{H})_{{sa}}$ and $B = M_{n}(\mathbb{H})_{{sa}}$, $M_{mn}(\mathbb{H})_{{sa}}$,
does not have a large enough dimension to accommodate the
real tensor product $A \otimes B$,
{}%{\color{magenta} causing difficulty for}  the
representation of product effects.\footnote{Attempts to interpret
  the quaternionic ``Hilbert space'' $\mathbb{H}^{mn}$ as a tensor
  product of $\mathbb{H}^m$
  and $\mathbb{H}^n$ raise at least the possibility of signaling via the
  noncommutativity of scalar multiplication.  This noncommutativity
  underlies the the argument in \cite{McKague} that
  stronger-than-quantum correlations are achievable in such a model.}
 In our approach, the issue simply doesn't arise. It seems that ``quaternionic quantum mechanics" is best seen as an
inextricable part of a larger theory. Essentially
the same point has also been made by Baez \cite{BaezDivAlg}.

The category $\mbox{}%\color{red} \bf InvQM}$ is somewhat mysterious. It encompasses
real and quaternionic QM in a completely natural way;
however, while it also contains complex quantum systems, these compose
in an exotic way: {}%\color{red} as pointed out above}, the composite of two complex quantum systems
in $\mbox{}%\color{red} \bf InvQM}$ comes with an extra classical bit ---
equivalently, $\{0,1\}$-valued superselection rule.
 This functions to make the transpose automorphism of
$M_n(\mathbb{C})_{sa}$ count as a morphism.
The extra classical bit is flipped by the Jordan transpose
(swap of $C^*$ summands) on either factor of {}%\color{red} such} a composite, but unaffected
if {\em both} parties {}%\color{red} implement} the Jordan transpose (which does, of course,
effect a Jordan transpose on the composite).
The precise physical significance of this is a subject for
further study.

As Example \ref{ex: no states} shows, there is no way to enlarge
  $\mbox{}%\color{red} \bf InvQM}$ so as to include higher spin factors, without either
  sacrificing compact closure {}%{\color{magenta} (and even rendering the set
  ${\cal C}(A,I)$, which might naturally be thought to represent states,
  trivial)} or venturing outside the ambient category of
  EJC-algebras, to make use of morphisms that are not {}%\color{red} (relatively)} completely
  Jordan-preserving maps.  Example \ref{ex: no states 2} shows, more strikingly, that
  there is no way to construct a category {}%\color{red} of the form $\mbox{\bf CJP}_{\Cat}$} that contains
  {\em standardly embedded} complex quantum systems {\em and} real
  systems, without, again, sacrificing compact closure
  (indeed, the representation of states by morphisms).

redpajama_set_name: RedPajamaArXiv