# Locally Tomographic Shadows (Extended Abstract)

Howard Barnum<EMAIL_ADDRESS>Institute for Quantum Computing
University of Waterloo Susquehanna University Matthew A. Graydon Institute for
Quantum Computing
University of Waterloo Susquehanna University Matthew A. Graydon
<EMAIL_ADDRESS>Susquehanna University Alex Wilce  Susquehanna University
<EMAIL_ADDRESS>
###### Abstract

Given a monoidal probabilistic theory — a symmetric monoidal category $\cal C$
of systems and processes, together with a functor $\bf V$ assigning concrete
probabilistic models to objects of ${\mathcal{C}}$ — we construct a locally
tomographic probabilistic theory $\mbox{LT}({\mathcal{C}},{\mathbb{V}})$ — the
locally tomographic shadow of $({\mathcal{C}},{\mathbb{V}})$ — describing
phenomena observable by local agents controlling systems in ${\mathcal{C}}$,
and able to pool information about joint measurements made on those systems.
Some globally distinct states become locally indistinguishable in
$\mbox{LT}({\mathcal{C}},{\mathbb{V}})$, and we restrict the set of processes
to those that respect this indistinguishability. This construction is
investigated in some detail for real quantum theory.

##  1 Introduction

As is well known, complex quantum theory is distinguished from its real
counterpart by a property called tomographic locality or local tomography [5,
11]: states of a bipartite system are entirely determined by the joint
probabilities they assign to measurements performed separately on the two
component systems. That this fails for finite-dimensional real quantum theory
is evident on dimensional grounds, but it’s more instructive to note that if
${\mathbf{H}}$ and ${\mathbf{K}}$ are real Hilbert spaces, the space
${\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})$ of bounded self-adjoint
operators on ${\mathbf{H}}\otimes{\mathbf{K}}$ decomposes as
${\mathcal{L}}_{ss}\oplus{\mathcal{L}}_{aa}$, where
${\mathcal{L}}_{ss}:={\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{K}})$
and
${\mathcal{L}}_{aa}:={\mathcal{L}}_{a}({\mathbf{H}})\otimes{\mathcal{L}}_{a}({\mathbf{K}})$,
with $L_{a}({\mathbf{H}})$ the space of bounded anti-self adjoint operators on
${\mathbf{H}}$, and similarly for ${\mathbf{K}}$. This is an orthogonal
decomposition with respect to the trace inner product. Thus, a density
operator $\rho$ on ${\mathbf{H}}\otimes{\mathbf{K}}$ has a unique
decomposition $\rho_{ss}+\rho_{aa}$, with $\rho_{ss}\in{\mathcal{L}}_{ss}$ and
$\rho_{aa}\in{\mathcal{L}}_{aa}$. Given effects
$a\in{\mathcal{L}}_{s}({\mathbf{H}})$ and
$b\in{\mathcal{L}}_{s}({\mathbf{K}})$, $a\otimes b$ lives in
${\mathcal{L}}_{ss}$, so $\mbox{Tr}((a\otimes b)\rho_{aa})=0$, and thus
$\mbox{Tr}((a\otimes b)\rho)=\mbox{Tr}((a\otimes b)\rho_{ss})$. In other
words, states with the same ${\mathcal{L}}_{ss}$ component but distinct
${\mathcal{L}}_{aa}$ components are locally indistinguishable.111In contrast,
in CQM, any anti-selfadjoint operator $a$ has the form $ib$ where $b$ is self-
adjoint. Hence, if $a=ib$ and $a^{\prime}=ib^{\prime}$ are anti-self adjoint,
$a\otimes a^{\prime}$ (as an element of
${\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})$) coincides with
$-(b\otimes
b^{\prime})\in{\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{K}})$.
That is:
${\mathcal{L}}_{ss}={\mathcal{L}}_{aa}={\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})$.

This suggests a construction: simply “factor out” the non-locally observable
${\mathcal{L}}_{aa}$ degrees of freedom to obtain a locally tomographic
theory. Here, we not only show that this is feasible, but go much further. By
a monoidal probabilistic theory we mean a pair $({\mathcal{C}},{\mathbb{V}})$
where ${\mathcal{C}}$ is a symmetric monoidal category and
${\mathbb{V}}:{\mathcal{C}}\rightarrow\mbox{\bf Prob}$ is a functor from a
symmetric monoidal category ${\mathcal{C}}$ to a suitable category of concrete
probabilistic models, taking monoidal products in ${\mathcal{C}}$ to (non-
signaling) composites in Prob. We outline this framework in Section
2.222Because this slightly extends the standard GPT framework as described,
e.g., in [6], we go into a bit of detail here Given a such a theory
$({\mathcal{C}},{\mathbb{V}})$, we construct a new, locally tomographic
theory, $\mbox{LT}({\mathcal{C}},{\mathbb{V}})$, that describes what the world
“looks like” to local agents, at least if one restricts attention to processes
that respect local indistinguishability. Taking a cue from Plato, we call this
the locally tomographic shadow of the original theory. This is described in
Section 3. In Section 4, we return to the case of real QM, and show that our
construction leads to a non-trivial theory that differs from both real and
complex QM, but still allows for entangled states and effects.

If one lifts the restriction on processes mentioned above, it is still
possible to construct a “shadow” theory, at the price of accepting an
additional layer of non-determinism on top of that built into the
probabilistic structure of the original model. This is briefly discussed in
the concluding section 5, along with a number of other directions for further
work.

This is a preliminary sketch of a longer work in progress.

##  2 Generalized Probabilistic Theories

For our purposes, a probabilistic model [4, 6] is pair $({\mathbb{V}},u)$
where ${\mathbb{V}}$ is an ordered real vector space and $u$ is a strictly
positive linear functional thereon, referred to as the unit effect of the
model. Elements $\alpha$ of ${\mathbb{V}}_{+}$ with $u(\alpha)=1$ are the
(normalized) states of the system. Effects (measurement results) are positive
functionals $a$ on ${\mathbb{V}}$ with $a\leq u$; $a(\alpha)$ is the
probability of $a$’s of occurring in state $\alpha$. Where we wish to keep
track of several models, we label them $A$, $B$, etc., writing, e.g.,
${\mathbb{V}}(A)$, ${\mathbb{V}}^{\ast}{(}A)$ for the associated ordered
vector spaces and their duals, and $u_{A}$, for the unit effects. A process
from a probabilistic model $({\mathbb{V}}(A),u_{A})$ to a probabilistic model
$({\mathbb{V}}(B),u_{B})$ is a positive linear mapping
$\phi:{\mathbb{V}}(A)\rightarrow{\mathbb{V}}(B)$ with $u_{B}(\phi(\alpha))\leq
u_{A}(\alpha)$ for every $\alpha\in{\mathbb{V}}(A)_{+}$

We write Prob for the category of probabilistic models and processes. In the
broadest sense, a probabilistic theory is simply a functor ${\mathbb{V}}$ from
some category ${\mathcal{C}}$ into Prob.333Note that such a functor
${\mathbb{V}}$ comes with a designated unit functional
$u_{A}\in{\mathbb{V}}(A)^{\ast}$ for every object $A\in{\mathcal{C}}$ with
$u_{A}\circ{\mathbb{V}}(\phi)\leq u_{B}$ for all $\phi\in{\mathcal{C}}(A,B)$.
${\mathcal{C}}$ can be understood to consist of “actual” physical systems and
processes, or of any mathematical proxies for these (classical phase spaces,
Hilbert spaces, open regions of spacetime, spin networks, or what-have-you).
${\mathbb{V}}$ attaches probabilistic apparatus to these systems and processes
in such a way as, e.g., to describe the possible experiences of agents.

In what follows, we denote such a probabilistic theory as a pair
$({\mathcal{C}},{\mathbb{V}})$. We assume that ${\mathbb{V}}$ is injective on
objects, which makes ${\mathbb{V}}({\mathcal{C}})$ a subcategory of Prob. This
assumption holds for all of the examples we wish to consider. Where
${\mathbb{V}}$ is also injective on morphisms, we say that
$({\mathcal{C}},{\mathbb{V}})$ is process tomographic [8], in which case, we
can simply identify ${\mathcal{C}}$ with the corresponding subcategory of
Prob. Note that the latter — more exactly, the probabilistic theory
$({\mathbb{V}}({\mathcal{C}}),I)$, where
$I:{\mathbb{V}}({\mathcal{C}})\rightarrow\mbox{\bf Prob}$ is the inclusion
functor — is automatically process tomographic, regardless of whether or not
$({\mathcal{C}},{\mathbb{V}})$ is so.

The example of primary interest here takes ${\mathcal{C}}=\mbox{\bf
CPM}_{{\mathbb{R}}}$, the category of finite-dimensional real Hilbert spaces
with morphisms ${\mathbf{H}}\rightarrow{\mathbf{K}}$ given by completely
positive linear mappings
${\mathcal{L}}({\mathbf{H}})\rightarrow{\mathcal{L}}({\mathbf{K}})$. In
contrast to the complex case, there exist linear mappings
${\mathcal{L}}_{s}({\mathbf{H}})\rightarrow{\mathcal{L}}_{s}({\mathbf{K}})$
that preserve positivity but not adjoints [10]. We reserve the term positive
for those linear mappings that preserve both. Equivalently,
$\phi:{\mathcal{L}}({\mathbf{H}})\rightarrow{\mathcal{L}}({\mathbf{K}})$ is
positive iff $\phi$ maps positive operators to positive operators, hence
mapping ${\mathcal{L}}_{s}({\mathbf{H}})$ to
${\mathcal{L}}_{s}({\mathbf{K}})$, and also maps
${\mathcal{L}}_{a}({\mathbf{H}})$ to ${\mathcal{L}}_{a}({\mathbf{K}})$. The
probabilistic theory we have in mind when we speak of real QM is then
$(\mbox{\bf CPM}_{{\mathbb{R}}},{\mathbb{V}})$ where
${\mathbb{V}}({\mathbf{H}})={\mathcal{L}}_{s}({\mathbf{H}})$, the latter
understood as an order unit space with $u_{{\mathbf{H}}}=\mbox{Tr}(\cdot)$. In
contrast to the complex case, the restriction of $\phi$ to the self-adjoint
part, ${\mathcal{L}}_{s}({\mathbf{H}})$, of ${\mathcal{L}}({\mathbf{H}})$ does
not determine $\phi$, so $\boldsymbol{{\mathbb{R}}}$QM is not process
tomographic.

Composite Models A (non-signaling) composite of models ${\mathbb{V}}(A)$ and
${\mathbb{V}}(B)$ is a model ${\mathbb{V}}(AB)$, together with a pair of
bilinear mappings

$m:{\mathbb{V}}(A)\times{\mathbb{V}}(B)\rightarrow{\mathbb{V}}(AB)\
\mbox{and}\ \
\pi:{\mathbb{V}}(A)^{\ast}\times{\mathbb{V}}(B)^{\ast}\rightarrow{\mathbb{V}}(AB)^{\ast}$

such that (i) $\omega\circ\pi$ is a joint state on $A$ and $B$ for all
$\omega\in{\mathbb{V}}(AB)$, and (ii)
$\pi(a,b)m(\alpha,\beta)=a(\alpha)b(\beta)$ for all
$a\in{\mathbb{V}}^{\ast}(A),b\in{\mathbb{V}}^{\ast}(B)$ and all
$\alpha\in{\mathbb{V}}(A)$, $\beta\in{\mathbb{V}}(B)$.

We think of $\pi(a,b)$ as representing the joint outcome $(a,b)$ of a pair of
experiments performed on $A$ and $B$, and $m(\alpha,\beta)$ as the result of
preparing states $\alpha$ and $\beta$ independently on $A$ and $B$. We refer
to $\pi(a,b)$ and $m(\alpha,\beta)$ as product effects and states,
respectively.

If ${\mathbf{H}}_{A}$ and ${\mathbf{H}}_{B}$ are two complex (finite-
dimensional) Hilbert spaces, the obvious choice for a composite model is
${\mathbb{V}}(AB)={\mathbb{V}}({\mathbf{H}}_{A}\otimes{\mathbf{H}}_{B})={\mathcal{L}}_{s}({\mathbf{H}}_{A}\otimes{\mathbf{H}}_{B})$,
with $\pi(a,b)(\omega)=\mbox{Tr}((a\otimes b)\omega)$ and
$m(\alpha,\beta)=\alpha\otimes\beta$.

If every state $\omega\in{\mathbb{V}}(AB)$ is determined by the joint
probabilities $\omega(\pi(a,b))$, we say that $AB$ is locally tomographic.
Given arbitrary models ${\mathbb{V}}(A)$, ${\mathbb{V}}(B)$, there are two
extremal locally tomographic composites, obtained by endowing
${\mathbb{V}}(A)\otimes{\mathbb{V}}(B)$ with the minimal and maximal tensor
cones [6, 13, 15]. The former is generated by separable states, i.e., convex
combinations of product states. The latter consists of all bilinear forms that
are positive on products of effects. We write
${\mathbb{V}}(A)\otimes_{\mbox{min}}{\mathbb{V}}(B)$ and
${\mathbb{V}}(A)\otimes_{\mbox{max}}{\mathbb{V}}(B)$ for
${\mathbb{V}}(A)\otimes{\mathbb{V}}(B)$ as ordered by these minimal and
maximal cones, respectively. A composite ${\mathbb{V}}(AB)$ is locally
tomographic iff
$m:{\mathbb{V}}(A)\otimes{\mathbb{V}}(B)\rightarrow{\mathbb{V}}(AB)$ is a
linear isomorphism, and in this case it’s usual simply to identify the two
spaces. One then has

$({\mathbb{V}}(A)\otimes_{\mbox{min}}{\mathbb{V}}(B))_{+}\leq{\mathbb{V}}(AB)_{+}\leq({\mathbb{V}}(A)\otimes_{\mbox{max}}{\mathbb{V}}(B))_{+}.$

If $A$ and $B$ are quantum models, the inclusions are proper:
$({\mathbb{V}}(A)\otimes_{\mbox{min}}{\mathbb{V}}(B))_{+}$ contains only
separable states, while
$({\mathbb{V}}(A)\otimes_{\mbox{max}}{\mathbb{V}}(B))_{+}$ contains states
corresponding to non-positive operators [12, 13]. More generally,
${\mathbb{V}}(A)\otimes_{\mbox{min}}{\mathbb{V}}(B)$ allows only separable
states, but arbitrarily entangled effects; the maximal tensor product
${\mathbb{V}}(A)\otimes_{\mbox{max}}{\mathbb{V}}(B)$ allows the reverse. Both
tensor products are naturally associative, and extending straightforwardly to
tensor products of more than two factors.

Definition: A monoidal probabilistic theory is a structure
$({\mathcal{C}},{\mathbb{V}},m,\pi)$ where ${\mathcal{C}}$ is a symmetric
monoidal category, ${\mathbb{V}}$ is a functor
${\mathcal{C}}\rightarrow\mbox{\bf Prob}$, and $m$ and $\pi$ are assignments,
to every pair of objects $A,B\in{\mathcal{C}}$, of bilinear mappings

$m_{A,B}:{\mathbb{V}}(A)\times{\mathbb{V}}(B)\rightarrow{\mathbb{V}}(AB)\
\mbox{and}\
\pi_{A,B}:{\mathbb{V}}^{\ast}(A)\times{\mathbb{V}}^{\ast}(B)\rightarrow{\mathbb{V}}^{\ast}(AB)$

such that

  * (i)

$m_{A,B},\pi_{A,B}$ make ${\mathbb{V}}(AB)$ a composite of ${\mathbb{V}}(A)$
and ${\mathbb{V}}(B)$,

  * (ii)

${\mathbb{V}}(I)={\mathbb{R}}$ with

$m_{I,A}:{\mathbb{V}}(I)\times{\mathbb{V}}(A)\rightarrow{\mathbb{V}}(A),\
\pi_{I,A}:{\mathbb{V}}^{\ast}(I)\times{\mathbb{V}}^{\ast}(A)\rightarrow{\mathbb{V}}^{\ast}(A)$

the bilinear mappings uniquely defined by $m(1,\alpha)=\alpha$ and
$\pi_{I,A}(1,a)=a$, and similarly for $m_{I,A}$ and $\pi_{A,I}$;

  * (iii)

${\mathbb{V}}(\sigma_{A,B})\circ\pi_{A,B}=\pi_{B,A}$, and similarly
$m_{A,B}\circ{\mathbb{V}}(\sigma_{A,B})=m_{B,A}$; and

  * (iv)

for all morphisms $\phi:A\rightarrow A^{\prime}$ and $\psi:B\rightarrow
B^{\prime}$ in ${\mathcal{C}}$, the diagram

${{\mathbb{V}}(A)\otimes{\mathbb{V}}(B)}$${{\mathbb{V}}(AB)}$${{\mathbb{V}}(A^{\prime})\otimes{\mathbb{V}}(B^{\prime})}$${{\mathbb{V}}(A^{\prime}B^{\prime})}$$\scriptstyle{{\mathbb{V}}(\phi)\otimes{\mathbb{V}}(\psi)}$$\scriptstyle{\pi_{A,B}^{\ast}}$$\scriptstyle{{\mathbb{V}}(\phi\otimes\psi)}$$\scriptstyle{\pi_{A^{\prime},B^{\prime}}^{\ast}}$
(1)

commutes.

If $\alpha\in{\mathcal{C}}(I,A)$, then by condition (ii),
${\mathbb{V}}(\alpha):{\mathbb{R}}\rightarrow{\mathbb{V}}(A)$ defines an
element of ${\mathbb{V}}(A)$, namely ${\mathbb{V}}(\alpha)(1)$. We make it a
standing assumption in what follows that every normalized state in
${\mathbb{V}}(A)$ arises in this way.

Remark: On the left side of (1), ${\mathbb{V}}(\phi)\otimes{\mathbb{V}}(\psi)$
is the usual tensor product of linear maps, while on the right,
$\phi\otimes\psi$ is the monoidal composite of the morphisms $\phi$ and $\psi$
in ${\mathcal{C}}$. An equivalent statement is that, for all states
$\omega\in{\mathbb{V}}(AB)$ and all effects
$a^{\prime}\in{\mathbb{V}}^{\ast}(A^{\prime})$ and
$b^{\prime}\in{\mathbb{V}}^{\ast}(B^{\prime})$, we have

$\omega({\mathbb{V}}(\phi)^{\ast}(a^{\prime}),{\mathbb{V}}(\psi)^{\ast}(b^{\prime}))={\mathbb{V}}(\phi\otimes\psi)(\omega)(a^{\prime},b^{\prime}).$

More compactly: $m$ and $\pi^{\ast}$ are natural transformations from the
functor ${\mathbb{V}}\otimes_{\mbox{min}}{\mathbb{V}}$ to the functor
${\mathbb{V}}\circ\otimes$, and from ${\mathbb{V}}\circ\otimes$ to
${\mathbb{V}}\otimes_{\mbox{max}}{\mathbb{V}}$, respectively. When no
confusion seems likely, we will henceforth write
$({\mathcal{C}},{\mathbb{V}})$ for a monoidal probabilistic theory
$({\mathcal{C}},{\mathbb{V}},m,\pi)$.

Evidently, $\boldsymbol{{\mathbb{R}}}$QM, with its standard compositional
structure, is an example of such a monoidal probabilistic theory.

All of the foregoing applies to composites of more than two models. One can
show that both $\otimes_{\mbox{min}}$ and $\otimes_{\mbox{max}}$ are both
naturally associative. If one has an $n$-partite composite $A=A_{1}\cdots
A_{n}$, built up recursively so that $A=(A_{1}\cdots A_{n-1})A_{n}$, then
arguing inductively one has canonical positive mappings
$\bigotimes_{\text{min}}{\mathbb{V}}(A_{i})\stackrel{{\scriptstyle
m}}{{\longrightarrow}}{\mathbb{V}}(A)\stackrel{{\scriptstyle\pi^{\ast}}}{{\longrightarrow}}\bigotimes_{\text{max}}{\mathbb{V}}(A_{i})$.

We call a monoidal probabilistic theory $({\mathcal{C}},{\mathbb{V}})$ locally
tomographic iff, for every pair of objects $A,B\in{\mathcal{C}}$, the
composite ${\mathbb{V}}(AB)$ is locally tomographic.

As pointed out above, as long as ${\mathbb{V}}$ is injective on objects, as we
are assuming here, process-tomography can be enforced by passing to the
probabilistic theory. $({\mathbb{V}}({\mathcal{C}}),I)$ For local tomography,
the situation is not so simple. Still, as we show in the next section, it is
possible to construct, from a given probabilistic theory, a kind of locally
tomographic quotient theory.

##  3 Locally tomographic Shadows

If a monoidal probabilistic theory $({\mathcal{C}},{\mathbb{V}})$ is not
locally tomographic, one can still ask what the world it describes would “look
like” to agents having access only to local measurements. As a first step, we
need to assume that objects in ${\mathcal{C}}$ can be carved up in a preferred
way into local pieces. To ensure this, we replace ${\mathcal{C}}$ with the its
strictification, the category ${\mathcal{C}}^{\ast}$ having as objects, finite
lists $\vec{A}=(A_{1},...,A_{n})$ of non-trivial (non-identity) objects
$A_{i}\in{\mathcal{C}}$, with the understanding that this represents the
composite $\Pi_{i=1}^{n}A_{i}$ in ${\mathcal{C}}$, but with the indicated
monoidal decomposition. Morphisms from $\vec{A}=(A_{1},...,A_{n})$ to
$\vec{B}=(B_{1},...,B_{k})$ are simply be morphisms
$\Pi_{i=1}^{n}A_{i}\rightarrow\Pi_{i=1}^{k}B_{i}$ in ${\mathcal{C}}$. This is
a strict symmetric monoidal category, with
$(A_{1},...,A_{n})(B_{1},...,B_{k})=(A_{1},...,A_{n},B_{1},...,B_{k})$, the
empty sequence $(\,)$ serving as the monoidal unit. There is a (strong, but
not strict) monoidal functor
$\Pi:{\mathcal{C}}^{\ast}\rightarrow{\mathcal{C}}$ taking
$\vec{A}=(A_{1},...,A_{n})$ to $\Pi\vec{A}:=\Pi_{i=1}^{n}A_{i}$, with
$\Pi(\,)=I_{{\mathcal{C}}}$, and acting as the identity on morphisms. This is
not generally injective on objects (for instance, $\Pi(A,BC)=\Pi(A,B,C)$).
Composing $\Pi$ with ${\mathbb{V}}$ now gives us a probabilistic theory
$({\mathcal{C}}^{\ast},{\mathbb{V}}\circ\Pi)$ in which we have the desired
canonical decompositions, but in which we’ve lost the desirable injectivity-
on-objects feature. This feature will be recovered when we pass to the locally
tomographic shadow, which we will now construct.

###  3.1 The shadow of a composite

If $AB$ is a composite of probabilistic models $A$ and $B$, then a state
$\omega\in\Omega(AB)$ of the composite system determines a joint state
$\widetilde{\omega}:{\mathbb{V}}(A)^{\ast}\times{\mathbb{V}}(B)^{\ast}\rightarrow[0,1]$,
given by $\widetilde{\omega}(a,b):=\pi_{A,B}(a,b)(\omega)$. This describes the
joint probabilities of measurement outcomes carried out on $A$ and $B$
separately. We may call $\widetilde{\omega}$ the local shadow of $\omega$.
More generally, suppose that $\vec{A}:=(A_{1},...,A_{n})$ is a sequence in
${\mathcal{C}}^{\ast}$ with composite $\Pi\vec{A}:=A$ in ${\mathcal{C}}$: as
we’ve seen, there is a positive linear mapping

$\pi^{\ast}_{\vec{A}}:{\mathbb{V}}(A)\rightarrow{\mathcal{L}}^{n}({\mathbb{V}}^{\ast}(A_{1}),...,{\mathbb{V}}^{\ast}(A_{n}))$

taking $\omega\in\Omega(A)$ to the corresponding joint state
$\widetilde{\omega}$ on $A_{1},...,A_{n}$ — its local shadow — given by

$\widetilde{\omega}(a_{1},...,a_{n}):=\pi^{\ast}_{A}(\omega)(a_{1},...,a_{n})=(a_{1}\otimes\cdots\otimes
a_{n})(\omega)$

for all
$(a_{1},...,a_{n})\in{\mathbb{V}}^{\ast}(A_{1})\times\cdots\times{\mathbb{V}}^{\ast}(A_{n})$.

Definition: For $\vec{A}=(A_{1},...,A_{n})\in{\mathcal{C}}^{\ast}$, let
$\widetilde{{\mathbb{V}}}(\vec{A})$ be the space
$\bigotimes_{i}{\mathbb{V}}(A_{i})$, ordered by the cone
$\pi^{\ast}_{\vec{A}}({\mathbb{V}}(\Pi\vec{A})_{+})$ consisting of local
shadows of elements of ${\mathbb{V}}(\Pi\vec{A})_{+}$, and let
$\widetilde{u}_{A}=u_{A_{1}}\otimes\cdots\otimes u_{A_{n}}$. We call the
probabilistic model
$(\widetilde{{\mathbb{V}}}(\Pi\vec{A}),\widetilde{u}_{\vec{A}})$ the locally
tomographic shadow of $A=\Pi\vec{A}$ with respect to the given decomposition
(that is, with respect to $\vec{A}$).

Going forward, it will be convenient to use the abbreviations
$\mbox{LT}_{\vec{A}}$, or even just LT, for for the mapping
$\pi^{\ast}_{\vec{A}}:{\mathbb{V}}(A)\rightarrow\widetilde{{\mathbb{V}}}(\vec{A})$,
whenever context makes it convenient and relatively unambiguous to do so.

We have canonical mappings

$\pi:{\mathbb{V}}^{\ast}(A_{1})\times\cdots\times{\mathbb{V}}^{\ast}(A_{k})\rightarrow\widetilde{{\mathbb{V}}}(A_{1},...,A_{k})^{\ast},\
\mbox{and}\ \
m:{\mathbb{V}}(A_{1})\times\cdots\times{\mathbb{V}}(A_{k})\rightarrow\widetilde{{\mathbb{V}}}(A_{1},...,A_{k})$

given by

$\pi(a_{1},...,a_{n})(\mu)=\mu(a_{1},...,a_{n})\ \ \mbox{and}\ \
m(\alpha_{1},...,\alpha_{n})(a_{1},...,a_{n})=\Pi_{i=1}^{n}a_{i}(\alpha_{i}).$

It is straightforward that $(\widetilde{{\mathbb{V}}}(A_{1},...,A_{n}),\pi,m)$
is a composite of ${\mathbb{V}}(A_{1}),...,{\mathbb{V}}(A_{n})$.

For any vector spaces ${\mathbb{V}}_{1},...,{\mathbb{V}}_{k}$, let
${\mathcal{L}}^{k}({\mathbb{V}}_{1},...,{\mathbb{V}}_{k})$ denote the space of
$k$-linear forms on ${\mathbb{V}}_{1}\times\cdots\times{\mathbb{V}}_{k}$.
Using the canonical isomorphism
${\mathcal{L}}^{k}({\mathbb{V}}^{\ast}_{1},...,{\mathbb{V}}^{\ast}_{k})\simeq\bigotimes_{i=1}^{k}{\mathbb{V}}_{i}$,
we can identify $\widetilde{{\mathbb{V}}}(A_{1},...,A_{k})$, as a vector
space, with with ${\mathbb{V}}(A_{1})\otimes\cdots\otimes{\mathbb{V}}(A_{k})$,
now ordered by a cone $\widetilde{{\mathbb{V}}}(A_{1},....,A_{n})_{+}$ with

$\left(\bigotimes_{\text{min}}{\mathbb{V}}(A_{i})\right)_{+}\subseteq\widetilde{{\mathbb{V}}}(A_{1},...,A_{n})_{+}\subseteq\left(\bigotimes_{\text{max}}{\mathbb{V}}(A_{i})\right)_{+}.$

We also have an injective linear mapping from
$\widetilde{{\mathbb{V}}}(A_{1},...,A_{k})\simeq\bigotimes_{i}{\mathbb{V}}(A_{i})$
into ${\mathbb{V}}(A_{1},...,A_{k})$, extending the map $m$ taking
$(\alpha_{1},...,\alpha_{n})\in\Pi_{i}{\mathbb{V}}(A_{i})$ to
$\alpha_{1}\otimes\cdots\otimes\alpha_{n}\in{\mathbb{V}}(\Pi_{i}A_{i})$.
However, this mapping is, as a rule, not positive. This will be illustrated
below in the case of real quantum theory. This shows that
$\widetilde{{\mathbb{V}}}(A_{1},...,A_{n})_{+}$ is generally larger than the
minimal tensor cone. As we’ll also see in the next section, it is generally
strictly smaller than the maximal tensor cone.

Further Notation: In the case of a bipartite system, we will sometimes find it
convenient to use the tensor-like notation
${\mathbb{V}}(A)\boxtimes{\mathbb{V}}(B)$ for $\widetilde{{\mathbb{V}}}(A,B)$.
Thus, ${\mathbb{V}}(A)\boxtimes{\mathbb{V}}(B)$ is the vector space
${\mathbb{V}}(A)\otimes{\mathbb{V}}(B)$, but ordered by the cone
$\widetilde{{\mathbb{V}}}(A,B)$ generated by local shadows
$\widetilde{\omega}$ of states $\omega\in{\mathbb{V}}(AB)$.

The following identifies the effect cone of
$\widetilde{{\mathbb{V}}}(A_{1},...,A_{n})$. We omit the straightforward
proof.

Lemma 1:
$\widetilde{{\mathbb{V}}}(A_{1},...,A_{n})^{\ast}_{+}\simeq{\mathbb{V}}^{\ast}(\Pi_{i}A_{i})_{+}\cap(\bigotimes_{i}{\mathbb{V}}^{\ast}(A_{i}))$.
In the bipartite case:

$({\mathbb{V}}(A)\boxtimes{\mathbb{V}}(B))^{\ast}_{+}\simeq{\mathbb{V}}(AB)^{\ast}\cap({\mathbb{V}}(A)^{\ast}\otimes{\mathbb{V}}(B)^{\ast}).$

###  3.2 Shadows of Processes

Given a probabilistic theory $({\mathcal{C}},{\mathbb{V}})$, we would now like
to construct a locally tomographic “shadow” theory by applying the LT
construction to the objects of ${\mathbb{V}}({\mathcal{C}})$. In order to do
this, we first need to decide what the morphisms should be in this putative
“shadow” theory.

In what follows, $A=\Pi\vec{A}$ and $B=\Pi\vec{B}$ are composite systems, with
specified decompositions $\vec{A}=(A_{1},...,A_{n})$ and
$\vec{B}=(B_{1},...,B_{k})$. The proof of the following is routine.

Lemma 2: Let Let $\Phi:{\mathbb{V}}(A)\rightarrow{\mathbb{V}}(B)$ be a
positive linear mapping. The following are equivalent:

  * (a)

$\Phi$ maps $\mbox{Ker}(\mbox{LT}_{\vec{A}})$ into
$\mbox{Ker}(\mbox{LT}_{\vec{B}})$.

  * (b)

If $\omega,\omega^{\prime}\in{\mathbb{V}}(A)$ are locally indistinguishable,
so are $\Phi(\omega),\Phi(\omega^{\prime})$ in ${\mathbb{V}}(B)$.

  * (c)

There exists a linear mapping
$\phi:\bigotimes_{i}{\mathbb{V}}(A_{i})\rightarrow\bigotimes_{j}{\mathbb{V}}(B_{j})$
such that $\mbox{LT}_{\vec{B}}\circ\Phi=\phi\circ\mbox{LT}_{\vec{A}}$

Definition: With notation as above, call a positive linear mapping
$\Phi:{\mathbb{V}}(A)\rightarrow{\mathbb{V}}(B)$ satisfying any, hence all, of
these conditions locally positive (with respect to the specified
decompositions). The linear mapping $\phi$ in part (c) is then uniquely
determined. We call it the shadow of $\Phi$, writing $\phi=\mbox{LT}(\Phi)$.

As an immediate consequence, we have

Lemma 3: If $\Phi:{\mathbb{V}}(A)\rightarrow{\mathbb{V}}(B)$ is locally
positive, then $\phi=\mbox{LT}(\Phi)$ is positive as a mapping from
$\widetilde{{\mathbb{V}}}(A_{1},...,A_{m})\rightarrow\widetilde{{\mathbb{V}}}(B_{1},...,B_{n})$.

Locally positive maps are reasonably abundant, but do exclude some important
morphisms in $\boldsymbol{{\mathbb{R}}}$QM. For instance, if $\sigma$ and
$\alpha$ are swap and associator morphisms in ${\mathcal{C}}$,
${\mathbb{V}}(\sigma)$ is locally positive, but ${\mathbb{V}}(\alpha)$ need
not be.

For another example, if $\alpha$ is a state on $A=A_{1}\otimes\cdots\otimes
A_{n}$, then the corresponding mapping
$\alpha:{\mathbb{R}}={\mathbb{V}}(I)\rightarrow{\mathbb{V}}(A)$ given by
$\alpha(1)=\alpha$ is trivially locally positive (the kernel of
$\mbox{LT}_{I}$ is trivial). On the other hand, there is generally no
guarantee that an effect $f:{\mathbb{V}}(A)\rightarrow{\mathbb{R}}$ will be
locally positive. Indeed, if $\omega,\omega^{\prime}\in{\mathbb{V}}(A)_{+}$
are distinct, there will certainly exist some positive linear functional $f$
with $f(\omega)\not=f(\omega^{\prime})$, and by re-scaling if necessary, we
can take $f$ to be an effect. If
$\mbox{LT}(\omega)=\mbox{LT}(\omega^{\prime})$, then $f$ is not locally
positive. Thus, the passage from ${\mathbb{V}}(A)$ to
$\mbox{LT}({\mathbb{V}}(A))$ not only identifies previously distinct states,
but also jettisons some effects.

It follows from part (c) of Lemma 3.3 that if
$\Phi:\Pi_{i}A_{i}\rightarrow\Pi_{j}B_{j}$ and
$\Psi:\Pi_{j}B_{j}\rightarrow\Pi_{k}C_{k}$ are locally positive with shadows
$\phi=\mbox{LT}(\Phi)$ and $\psi=\mbox{LT}(\Psi)$, then $\Psi\circ\Phi$ is
locally positive with shadow $\psi\circ\phi$.

Definition: Let $({\mathcal{C}},{\mathbb{V}})$ be a probabilistic theory. For
objects $\vec{A}=(A_{1},...,A_{n})$ and
$\vec{B}=(B_{1},...,B_{k})\in{\mathcal{C}}^{\ast}$, call a morphism
$\Pi\vec{A}\stackrel{{\scriptstyle f}}{{\longrightarrow}}\Pi\vec{B}$ local iff
${\mathbb{V}}(f):{\mathbb{V}}(\Pi\vec{A})\rightarrow{\mathbb{V}}(\Pi\vec{B})$
is locally positive (relative to the preferred factorizations of $A$ and $B$).
We write $\mbox{Loc}({\mathcal{C}},{\mathbb{V}})$ for the category having the
same objects as ${\mathcal{C}}^{\ast}$ — finite lists of non-identity objects
in ${\mathcal{C}}$ — but only local morphisms.

By the remarks above, $\mbox{Loc}({\mathcal{C}},{\mathbb{V}})$ is a monoidal
sub-category of ${\mathcal{C}}^{\ast}$. By construction, the functor
${\mathbb{V}}\circ\Pi:{\mathcal{C}}^{\ast}\rightarrow\mbox{\bf Prob}$ descends
to a functor $\mbox{Loc}({\mathcal{C}},{\mathbb{V}})\rightarrow\mbox{\bf
Prob}$. However, because $\Pi$ is not injective on objects, neither will
${\mathbb{V}}\circ\Pi$ be. To remedy this, we make the following

Definition: For all objects $\vec{A},\vec{B}\in{\mathcal{C}}^{\ast}$ and local
morphisms $f:\vec{A}\rightarrow\vec{B}$, let
$\widetilde{{\mathbb{V}}}(\vec{A}):=LT_{\vec{A}}({\mathbb{V}}(\Pi\vec{A})$ and
$\widetilde{{\mathbb{V}}}(f):=\mbox{LT}({\mathbb{V}}(f))$.

Lemma 4: $\widetilde{{\mathbb{V}}}$ is a functor
$\mbox{Loc}({\mathcal{C}},{\mathbb{V}})\rightarrow\mbox{\bf Prob}$, and is
injective on objects.

Proof: It is straightforward that $\widetilde{{\mathbb{V}}}(f\circ
g)=\widetilde{{\mathbb{V}}}(f)\circ\widetilde{{\mathbb{V}}}(g)$ where
$f\in{\mathcal{C}}^{\ast}(A,B)$ and $g\in{\mathcal{C}}^{\ast}(B,C)$. That
$\widetilde{{\mathbb{V}}}$ is injective on objects follows from the fact that
there is a canonical isomorphism between $\bigotimes{\mathbb{V}}(A_{i})$ and
${\mathcal{L}}^{k}({\mathbb{V}}(A_{1})^{\ast},...,{\mathbb{V}}(A_{n})^{\ast})$;
the latter is literally a space of functions on
${\mathbb{V}}(A_{1})^{\ast}\times\cdots\times{\mathbb{V}}(A_{n})^{\ast}$, from
which one can read off the spaces ${\mathbb{V}}(A_{1})$, …
${\mathbb{V}}(A_{n})$. As ${\mathbb{V}}$ is injective on objects, these in
turn determine $(A_{1},...,A_{n})$. $\Box$

Definition: The locally tomographic shadow of a monoidal probabilistic theory
$({\mathcal{C}},{\mathbb{V}})$ is the probabilistic theory
$\mbox{LT}({\mathcal{C}},{\mathbb{V}}):=(\mbox{Loc}({\mathcal{C}},{\mathbb{V}}),\widetilde{{\mathbb{V}}})$.

By construction,
$\mbox{LT}({\mathbb{V}},{\mathcal{C}})=(\mbox{Loc}({\mathcal{C}},{\mathbb{V}}),\widetilde{{\mathbb{V}}})$
is locally tomographic. We have the following picture: there are two functors
(probabilistic models) associated with
$\mbox{Loc}({\mathcal{C}},{\mathbb{V}})$: one is given by
${\mathbb{V}}\circ\Pi$, and the other by
$\widetilde{{\mathbb{V}}}:=\mbox{LT}({\mathbb{V}}\circ\Pi)$, i.e., for each
object $\vec{A}=(A_{1},...,A_{n})$ in
$\mbox{Loc}({\mathcal{C}},{\mathbb{V}})$, we have a positive linear mapping
$\mbox{LT}_{\vec{A}}:{\mathbb{V}}(\Pi(\vec{A}))\rightarrow\widetilde{{\mathbb{V}}}(\vec{A})$.
By construction, LT then defines a natural transformation
${\mathbb{V}}\circ\Pi\Rightarrow\widetilde{{\mathbb{V}}}$.

##  4 The Shadow of Real Quantum Theory

We’ll now consider the case of finite-dimensional real quantum theory in some
detail, concentrating on the bipartite case.

###  4.1 The LT map

Suppose ${\mathbf{H}}$ and ${\mathbf{K}}$ are two finite-dimensional real
Hilbert spaces. In what follows, we identify states with the corresponding
density operators, so that if $W$ is a density operator on
${\mathbf{H}}\otimes{\mathbf{K}}$, $\mbox{LT}(W)$ is the unique operator in
${\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{K}})$
satisfying $\mbox{Tr}(\mbox{LT}(W)a\otimes b)=\mbox{Tr}(Wa\otimes b)$ for all
$a\in{\mathcal{L}}_{s}({\mathbf{H}})$ and
$b\in{\mathcal{L}}_{s}({\mathbf{K}})$.

As discussed in the Introduction,
${\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})$ has a natural orthogonal
direct sum decomposition as

${\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})\ =\
\left({\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{K}})\right)\oplus\left({\mathcal{L}}_{a}({\mathbf{H}})\otimes{\mathcal{L}}_{a}({\mathbf{K}})\right)$

Since $\mbox{Ker}(\mbox{LT})$ contains
${\mathcal{L}}_{a}({\mathbf{H}})\otimes{\mathcal{L}}_{a}({\mathbf{K}})$ and
$\mbox{Ran}(\mbox{LT})$ equals
${\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{H}})$, we
have
$\mbox{Ker}(\mbox{LT})={\mathcal{L}}_{a}({\mathbf{H}})\otimes{\mathcal{L}}_{a}({\mathbf{K}})$.

Let
$\mbox{Sym}:{\mathcal{L}}({\mathbf{H}})\rightarrow{\mathcal{L}}_{s}({\mathbf{H}})$
be the symmetrization mapping $\mbox{Sym}(a)=\tfrac{1}{2}(a+a^{t})$. Note that
this is the orthogonal projection on ${\mathcal{L}}({\mathbf{H}})$ (with
respect to the trace inner product) with range
${\mathcal{L}}_{s}({\mathbf{H}})$. A straightforward computation gives us

Lemma 5: $\mbox{LT}(W)=(\mbox{Sym}\otimes\mbox{Sym})(W)$ for all
$W\in{\mathcal{L}}({\mathbf{H}})$.

###  4.2 The locally tomographic cone

Using the notation introduced in Section 3,
${\mathcal{L}}_{s}({\mathbf{H}})\boxtimes{\mathcal{L}}_{s}({\mathbf{H}})$
stands for
${\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{H}})$ as
ordered by the locally tomographic cone

$({\mathcal{L}}_{s}({\mathbf{H}})\boxtimes{\mathcal{L}}_{s}({\mathbf{K}}))_{+}\
:=\ \mbox{LT}({\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{H}})_{+}).$

Let
$({\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{K}}))_{+}$
stand for the cone of positive operators belonging to
${\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{K}})$. That
is,

$({\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{H}}))_{+}\
:=\
({\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{H}}))\cap{\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{H}})_{+}.$

A priori, we have

$\displaystyle({\mathcal{L}}_{s}({\mathbf{H}})\otimes_{\mbox{min}}{\mathcal{L}}_{s}({\mathbf{K}})_{+}$
$\displaystyle\subseteq$
$\displaystyle({\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{K}}))_{+}$
$\displaystyle\subseteq$
$\displaystyle({\mathcal{L}}_{s}({\mathbf{H}})\boxtimes{\mathcal{L}}_{s}({\mathbf{K}}))_{+}\subseteq({\mathcal{L}}_{s}({\mathbf{H}})\otimes_{\mbox{max}}{\mathcal{L}}_{s}({\mathbf{K}}))_{+}.$

We’ll presently see that if ${\mathbf{H}}$ and ${\mathbf{K}}$ are of dimension
two or greater, all four of these inclusions are proper. If
$x,y\in{\mathbf{H}}$, we use the standard operator-theoretic notation $x\odot
y$ (rather than $|x\rangle\langle y|$ as in Dirac notation) for the operator
on ${\mathbf{H}}$ given by $(x\odot y)z\ =\ \langle z,y\rangle x$ for all
$z\in{\mathbf{H}}$. If $\|x\|=1$, then $x\odot x=P_{x}$, the rank-one
projection onto the span of $x$.

Example 1: Let ${\mathbf{H}}={\mathbb{R}}^{2}$ and let $\\{x,y\\}$ be any
orthonormal basis. Let $z$ be the real EPR state $z\ =\
\tfrac{1}{\sqrt{2}}(x\otimes y+y\otimes x)$. Direct computation shows that

$z\odot z\ =\ \tfrac{1}{2}(P_{x}\otimes P_{y}+P_{y}\otimes P_{x}+(x\odot
y)\otimes(y\odot x)+(y\odot x)\otimes(x\odot y))$

Now, $\mbox{Sym}(x\odot y)=\tfrac{1}{2}(x\odot y+y\odot x)=\mbox{Sym}(y\odot
x)=:S$ where $Sx=\tfrac{1}{2}y$ and $Sy=\tfrac{1}{2}x$. So
$W:=\mbox{LT}(z\odot z)\ =\ \tfrac{1}{2}(P_{x}\otimes P_{y}+P_{y}\otimes
P_{x})+S\otimes S$. This is not a positive operator. For instance, if
$v=x\otimes x-y\otimes y$, then $Wv=-\tfrac{1}{4}v$.

This shows that the embedding ${\mathbb{V}}(A\boxtimes
B)\rightarrow{\mathbb{V}}(AB)$ is in general not positive, as mentioned
earlier. Consequently, $({\mathcal{L}}_{s}\otimes{\mathcal{L}}_{s})_{+}$ is
strictly smaller than
$({\mathcal{L}}_{s}({\mathbf{H}})\boxtimes{\mathcal{L}}_{s}({\mathbf{K}}))_{+}$.

Theorem 1: _Let $\dim{{\mathbf{H}}}$ and $\dim{{\mathbf{K}}}$ be $3$ or
greater. The cone
$({\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{K}}))_{+}$
is strictly larger than the cone
$({\mathcal{L}}_{s}({\mathbf{H}})\otimes_{\mbox{min}}{\mathcal{L}}_{s}({\mathbf{K}}))_{+}$,
and the cone
$({\mathcal{L}}_{s}({\mathbf{H}})\boxtimes{\mathcal{L}}_{s}({\mathbf{K}}))_{+}$
is strictly smaller than the cone_
$({\mathcal{L}}_{s}({\mathbf{H}})\otimes_{\mbox{max}}{\mathcal{L}}_{s}({\mathbf{K}}))_{+}$.

_Proof (sketch):_ To simplify notation a bit, let ${\mathbf{H}}={\mathbf{K}}$,
and write ${\mathcal{L}}_{s}({\mathbf{H}})$ and
${\mathcal{L}}_{a}({\mathbf{H}})$ as ${\mathcal{L}}_{s}$ and
${\mathcal{L}}_{a}$. Lemma 1 tells us that
$({\mathcal{L}}_{s}\boxtimes{\mathcal{L}}_{s})_{+}^{\ast}\simeq({\mathcal{L}}_{s}\otimes{\mathcal{L}}_{s})_{+}$.
There exist well-known examples of entangled states in
${\mathcal{L}}_{s}\otimes{\mathcal{L}}_{s}$, namely those arising from
unextendible product bases; see [7].555We thank Giulio Chiribella for drawing
our attention to these. Hence,
$({\mathcal{L}}_{s}\otimes{\mathcal{L}}_{s})\cap{\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})_{+}$
is strictly larger than the minimal tensor cone, which contains only
unentangled states. Dualizing, we see that
$({\mathcal{L}}_{s}({\mathbf{H}})\boxtimes{\mathcal{L}}_{s}({\mathbf{K}}))_{+}$
must be strictly smaller than the maximal tensor cone. $\Box$

Remark: The geometry of the locally tomographic cone
$({\mathcal{L}}_{s}({\mathbf{H}})\boxtimes{\mathcal{L}}_{s}({\mathbf{K}}))_{+}$
appears to be quite interesting. Although the mapping
$\mbox{LT}:{\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})\rightarrow{\mathcal{L}}_{s}({\mathbf{H}})\boxtimes{\mathcal{L}}_{s}({\mathbf{K}})$
is in general many-to-one, remarkably, this is not the case for pure states:
as shown by Lemma 17 of [9], any pure state of
${\mathbf{H}}\otimes{\mathbf{K}}$ can be distinguished from any other state
(pure or mixed) by product effects. Thus, if $\omega$ is a pure state in
${\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})$, it is the only state
with local shadow $\mbox{LT}(\omega)$.666If $\omega$ is an interior point in
the cone ${\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})_{+}$, then the
affine space
$\omega+{\mathcal{L}}_{a}({\mathbf{H}})\otimes{\mathcal{L}}_{a}({\mathbf{K}})$,
whose elements $\mu$ all satisfy $\mbox{Tr}(\mu(a\otimes
b))=\mbox{Tr}(\omega(a\otimes b))$ for all product effects $a\otimes b$, will
certainly intersect the boundary of the positive cone. However, this
intersection will not contain pure states, but only points interior to higher-
dimensional faces of the cone.

###  4.3 Processes

Let
$\Phi:{\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})\rightarrow{\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})$
be a positive linear mapping. For simplicity of notation, let’s write
${\mathcal{L}}_{s,s}$ for
${\mathcal{L}}_{s}({\mathbf{H}})\otimes{\mathcal{L}}_{s}({\mathbf{K}})$,
${\mathcal{L}}_{s,s}$ and ${\mathcal{L}}_{a,s}$ for
${\mathcal{L}}_{a}({\mathbf{H}})\otimes{\mathcal{L}}_{a}({\mathbf{K}})$, and
so on. We then have an orthogonal decomposition

${\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})={\mathcal{L}}_{s,s}\oplus{\mathcal{L}}_{s,a}\oplus{\mathcal{L}}_{a,s}\oplus{\mathcal{L}}_{a,a},$

with respect to which $\Phi$ has an operator matrix
$\left[\begin{array}[]{cc}\Phi_{s,s}&\Phi_{s,a}\\\
\Phi_{a,s}&\Phi_{a,a}\end{array}\right]$, where, e.g,
$\Phi_{s,s}:{\mathcal{L}}_{s,s}\rightarrow{\mathcal{L}}_{s,s}$,
$\Phi_{a,s}:{\mathcal{L}}_{s,s}\rightarrow{\mathcal{L}}_{a,a}$, etc. A
straightforward translation of Lemma 2 gives us

Lemma 6: Let $\Phi$ be as above. Then $\Phi$ is locally positive iff
$\Phi_{s,a}=0$, and in this case, $\mbox{LT}(\Phi)=\Phi_{s,s}$.

This provides another way to see that effects on
${\mathbf{H}}\otimes{\mathbf{K}}$, as processes

${\mathcal{L}}_{s}({\mathbf{H}}\otimes{\mathbf{K}})\rightarrow{\mathcal{L}}_{s}({\mathbb{R}}),$

are generally not locally positive. The following example is particularly
noteworthy:

Example 2: Consider the functional
$\epsilon:{\mathcal{L}}({\mathbb{R}}^{2}\otimes{\mathbb{R}}^{2})={\mathcal{L}}({\mathbb{R}}^{2})\otimes{\mathcal{L}}({\mathbb{R}}^{2})\rightarrow{\mathbb{R}}$
corresponding to the trace inner product, i.e., the functional uniquely
defined on pure tensors by $\epsilon(a\otimes b)=\mbox{Tr}(ab^{t})$. The
subspace ${\mathcal{L}}_{a,a}\leq{\mathcal{L}}({\mathbb{R}}^{2})$ is one-
dimensional, spanned by the operator $J\otimes J={\mathbb{J}}$, where
$J(x,y)=(-y,x)$. Note that $J^{2}=-{\mathbf{1}}$. Hence
$\epsilon({\mathbb{J}})=\mbox{Tr}(JJ^{t})=-\mbox{Tr}(J^{2})=\mbox{Tr}({\mathbf{1}})=2$.
Since $\epsilon$ does not vanish on ${\mathcal{L}}_{a,a}$, $\epsilon$ is not
locally positive.

##  5 Conclusions and questions

At a minimum, $\mbox{LT}(\mbox{$\boldsymbol{{\mathbb{R}}}${\bf QM}})$ provides
us with an interesting “foil” GPT, related to but distinct from both complex
and real finite-dimensional real quantum theory, and from their Jordan-
algebraic relatives [3] (which, like $\boldsymbol{{\mathbb{R}}}$QM, are not
locally tomographic). Among the many questions that suggest themselves about
this theory, and about the LT construction more generally, the following stand
out to us as particularly interesting.

Compact Closure Example 2 shows that
$\mbox{LT}(\mbox{$\boldsymbol{{\mathbb{R}}}${\bf QM}})$ does not inherit the
usual compact structure from $\boldsymbol{{\mathbb{R}}}$QM. Given a monoidal
probabilistic theory, theory $({\mathcal{C}},{\mathbb{V}})$ with
${\mathcal{C}}$ compact closed, when is
$\mbox{LT}({\mathcal{C}},{\mathbb{V}})$ compact closed?

LT and Complex QM The functor
$R:\mbox{${\mathbb{C}}$QM}\rightarrow\mbox{$\boldsymbol{{\mathbb{R}}}${\bf
QM}}$ given by restriction of scalars does not preserve tensor products. It
would be of interest to understand the functor $\mbox{LT}\circ R$ from
${\mathbb{C}}$QM to $\mbox{LT}(\mbox{$\boldsymbol{{\mathbb{R}}}${\bf QM}})$.
One can ask a parallel question about complexification.

The Shadow of InvQM In [3], we constructed a non-locally-tomographic theory we
called InvQM, which contains finite-dimensional real and quaternionic QM as
sub-theories, and also a relative of complex QM in which the composite of two
complex quantum systems comes with an extra binary superselection rule. As we
will discuss elsewhere, much of what was done above for
$\boldsymbol{{\mathbb{R}}}$QM generalizes readily to InvQM, but the resulting
theory — like $\mbox{LT}(\mbox{$\boldsymbol{{\mathbb{R}}}${\bf QM}})$ —
remains largely unexplored.

Non-deterministic shadows If local agents Alice and Bob agree that their joint
state is $\omega$, this is consistent with the actual, global state being any
element $\mu\in\mbox{LT}_{A,B}^{-1}(\omega)$. If the (unknown) actual state
evolves under a (global) process
$\phi:{\mathbb{V}}(AB)\rightarrow{\mathbb{V}}(CD)$, the result will be one of
the states in $\phi(\mbox{LT}_{A,B}^{-1}(\omega))$. Unless $\phi$ is local,
these will not be confined to a single fibre of $\mbox{LT}_{C,D}$; parties $C$
and $D$ might observe any of the different states in
$\mbox{LT}_{C,D}(\phi(\mbox{LT}_{A,B}^{-1}(\omega)))$, giving the impression
that $\phi$ acts indeterministically. How ought this uncertainty be
quantified? Note that in this situation, the states of $AB$ act as hidden
variables “explaining” this apparent lack of determinism.

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