# Gleason’s theorem for composite systems

Markus Frembs<EMAIL_ADDRESS>Centre for Quantum Dynamics, Griffith
University,
Yugambeh Country, Gold Coast, QLD 4222, Australia  Andreas Döring
<EMAIL_ADDRESS>
###### Abstract

Gleason’s theorem Gleason1975  is an important result in the foundations of
quantum mechanics, where it justifies the Born rule as a mathematical
consequence of the quantum formalism. Formally, it presents a key insight into
the projective geometry of Hilbert spaces, showing that finitely additive
measures on the projection lattice $\mathcal{P}(\mathcal{H})$ extend to
positive linear functionals on the algebra of bounded operators
$\mathcal{B}(\mathcal{H})$. Over many years, and by the effort of various
authors, the theorem has been broadened in its scope from type I to arbitrary
von Neumann algebras (without type $\text{I}_{2}$ factors). Here, we prove a
generalisation of Gleason’s theorem to composite systems. To this end, we
strengthen the original result in two ways: first, we extend its scope to
dilations in the sense of Naimark and Stinespring Naimark1943 ;
Stinespring1955  and second, we require consistency with respect to dynamical
correspondences on the respective (local) algebras in the composition
AlfsenShultz1998 . We show that neither of these conditions changes the result
in the single system case, yet both are necessary to obtain a generalisation
to bipartite systems.

##  I Gleason’s theorem

Gleason’s theorem is a landmark result in quantum theory. It justifies the
Born rule, which originally had the status of an axiom, as a mathematical fact
of the projective geometry of Hilbert spaces.

######  Theorem 1.

_(GleasonGleason1975 )_ Let $\mathcal{H}$ be a separable Hilbert space,
$\mathrm{dim}(\mathcal{H})\geq 3$. Then every countably additive probability
measure $\mu:\mathcal{P}(\mathcal{H})\rightarrow[0,1]$ over the projections on
$\mathcal{H}$ is of the form $\mu(p)=\mathrm{tr}[\rho_{\mu}p]$ for all
$p\in\mathcal{P}(\mathcal{H})$, with
$\rho_{\mu}:\mathcal{H}\rightarrow\mathcal{H}$, $\rho_{\mu}\geq 0$,
$\mathrm{tr}[\rho_{\mu}]=1$ a density operator.

Here, a _countably additive probability measure_ is a map
$\mu:\mathcal{P}(\mathcal{H})\rightarrow[0,1]$, $\mu(1)=1$ and such that
$\mu(\sum_{i=1}^{\infty}p_{i})=\sum_{i=1}^{\infty}\mu(p_{i})$ for all
$p_{i}\in\mathcal{P}(\mathcal{H})$ such that $p_{i}p_{j}=0$ whenever $i\neq
j$. Note that when $\mathcal{N}=M_{2}(\mathbb{C})$, there exist measures that
fail to extend to linear functionals.

While the original argument was for type I factors only, Thm. 1 was later
extended to type II and type III von Neumann algebras in Christensen1982 ;
Yeadon1983 ; Yeadon1984  (see also Maeda1989 ). In this setting, $\mu$ is
further called completely additive if $\mu(\sum_{i\in I}p_{i})=\sum_{i\in
I}\mu(p_{i})$ for every family of pairwise orthogonal projections
$(p_{i})_{i\in I}$, $p_{i}\in\mathcal{P}(\mathcal{N})$.111For
$\mathcal{N}=\mathcal{B}(\mathcal{H})$ with $\mathcal{H}$ separable, complete
additivity reduces to countable additivity. Recall that a state
$\sigma\in\mathcal{S}(\mathcal{N})$ is a positive, normalised linear
functional, it is _normal_ if $\sigma(\sum_{i\in I}p_{i})=\sum_{i\in
I}\sigma(p_{i})$ for all families of pairwise orthogonal projections
$(p_{i})_{i\in I}$, $p_{i}\in\mathcal{P}(\mathcal{N})$ (Thm. 7.1.12 in
KadisonRingroseII ). Clearly, a (normal) state $\sigma$ defines a finitely
(completely) additive probability measure on $\mathcal{P}(\mathcal{N})$,
conversely:

######  Theorem 2.

_(Gleason-Christensen-YeadonChristensen1982 ; Yeadon1983 ; Yeadon1984 )_ Let
$\mathcal{N}$ be a von Neumann algebra with no summand of type $I_{2}$ and let
$\mu:\mathcal{P}(\mathcal{N})\rightarrow\mathbb{R}$ be a finitely additive
probability measure on the projections of $\mathcal{N}$. There exists a unique
state $\sigma_{\mu}\in\mathcal{S}(\mathcal{N})$ such that
$\mu(p)=\sigma_{\mu}(p)$ for all $p\in\mathcal{P}(\mathcal{N})$. If $\mu$ is
completely additive then $\sigma_{\mu}$ is normal and of the form
$\mu(p)=\sigma_{\mu}(p)=\mathrm{tr}[\rho_{\mu}p]$ for all
$p\in\mathcal{P}(\mathcal{N})$ with $\rho_{\mu}$ a positive trace-class
operator.

##  II Gleason’s theorem in context

###  II.1 Partial order of contexts and probabilistic presheaf.

Note that finite (complete) additivity only assumes that
$\mu:\mathcal{P}(\mathcal{N})\rightarrow[0,1]$ is _quasi-linear_ , i.e.,
linear in commuting von Neumann subalgebras. Remarkably, Thm. 1 and its
generalisation Thm. 2 show that this already implies linearity on all of
$\mathcal{N}$.

We emphasise the passage from quasi-linearity to linearity as follows. Note
that $\mu$ restricts to a probability measure $\mu_{V}$ in every commutative
von Neumann subalgebra $V\subset\mathcal{N}$, i.e., $\mu_{V}=\mu|_{V}$. $\mu$
thus defines a collection of probability measures
$(\mu_{V})_{V\subset\mathcal{N}}$, one for every commutative von Neumann
subalgebra, and such that whenever $\tilde{V}\subset V$ is a von Neumann
subalgebra, the respective probability measures are related by restriction,
$\mu_{\tilde{V}}=\mu_{V}|_{\tilde{V}}$ (viz. marginalisation).

From this perspective, the constraints on measures
$\mu:\mathcal{P}(\mathcal{N})\rightarrow[0,1]$ in Gleason’s theorem arise via
the inclusion relations between commutative von Neumann subalgebras. This
motivates the definition of the following partially ordered set (poset).

######  Definition 1.

Let $\mathcal{N}$ be a von Neumann algebra. The poset of commutative von
Neumann subalgebras of $\mathcal{N}$ is called the _context category of
$\mathcal{N}$_ and is denoted by $\mathcal{V}(\mathcal{N})$.

The name ‘context category’ is motivated as follows. First, note that every
poset can be regarded as a category, whose objects are the elements of the
poset and whose arrows are defined by $a\rightarrow b\Leftrightarrow a\leq b$.
Second, in quantum physics contextuality refers to the fact that not all
observables, represented by the self-adjoint operators in $\mathcal{N}$, can
be measured simultaneously in an arbitrary state.222This is in contrast to
classical physics, where observables are represented by elements in a
(commutative) algebra of functions on a (locally) compact Hausdorff space. The
Kochen-Specker theorem shows that in the noncommutative case such a
description is impossible KochenSpecker1967 . Only commuting operators can be
measured simultaneously, i.e., those that are contained in a commutative von
Neumann subalgebra—a _context_ DoeringFrembs2019a . The constraints in
Gleason’s theorem are therefore _noncontextuality constraints_ : a projection
$p\in\mathcal{P}(\mathcal{N})$ is assigned a probability $\mu(p)$ independent
of the context that $p$ lies in: $\mu_{V}(p)=\mu_{\tilde{V}}(p)=\mu(p)$
whenever $p\in\tilde{V},V$.

Next, we formalise the idea that a measure
$\mu:\mathcal{P}(\mathcal{N})\rightarrow[0,1]$ corresponds to a collection of
probability measures over $\mathcal{V}(\mathcal{N})$.

######  Definition 2.

Let $\mathcal{N}$ be a von Neumann algebra with context category
$\mathcal{V}(\mathcal{N})$. The _(normal) probabilistic presheaf 333A presheaf
$\underline{P}$ over the category $\mathcal{C}$ is a functor
$\underline{P}:\mathcal{C}^{\mathrm{op}}\rightarrow\mathbf{Set}$. We denote
presheaves with an underscore. $\underline{\Pi}$ of $\mathcal{N}$ over
$\mathcal{V}(\mathcal{N})$_ is the presheaf given

  * (i)

on objects: for all $V\in\mathcal{V}(\mathcal{N})$, let

$\underline{\Pi}_{V}:=\\{\mu_{V}:\mathcal{P}(V)\rightarrow[0,1]\mid\mu_{V}\text{
is a finitely (completely) additive probability measure}\\}\;,$

  * (ii)

on arrows: for all $\tilde{V},V\in\mathcal{V}(\mathcal{N})$ with
$\tilde{V}\subset V$, let
$\underline{\Pi}(i_{\tilde{V}V}):\underline{\Pi}_{V}\rightarrow\underline{\Pi}_{\tilde{V}}$
with $\mu_{V}\mapsto\mu_{V}|_{\tilde{V}}$, where
$i_{\tilde{V}V}:\tilde{V}\hookrightarrow V$ denotes the inclusion map between
contexts $\tilde{V}\subset V$.

We remark that the study of presheaves over the partial order of contexts is
at the heart of the topos approach to quantum theory IshamButterfieldI ;
IshamButterfieldII ; IshamButterfieldIII ; IshamButterfieldIV ; IshamDoeringI
; IshamDoeringII ; IshamDoeringIII ; IshamDoeringIV ; HLS2009 ; HLS2010 . For
an introduction, see e.g. DoeIsh11 . For the intimate relationship between
contextuality (in the sense of Def. 1) and various key theorems in quantum
theory, see DoeringFrembs2019a .

###  II.2 Gleason’s theorem in presheaf form.

We have seen that a finitely (completely) additive probability measure over
the projections of a von Neumann algebra $\mathcal{N}$ can be regarded as a
collection of finitely (completely) additive probability measures over
contexts $(\mu_{V})_{V\in\mathcal{V}(\mathcal{N})}$, which satisfy the
constraints $\mu_{\tilde{V}}=\mu_{V}|_{\tilde{V}}$ whenever
$\tilde{V},V\in\mathcal{V}(\mathcal{N})$ with $\tilde{V}\subset V$. In terms
of Def. 2, $\mu$ thus becomes a _global section_ of the (normal) probabilistic
presheaf $\underline{\Pi}$. We denote the set of global sections of
$\underline{\Pi}$ by $\Gamma[\underline{\Pi}].$444Here, ‘global’ refers to the
fact that $\mu$ satisfies the restriction constraints in $\underline{\Pi}$
over all of $\mathcal{V}(\mathcal{N})$. In contrast, a _local section_
satisfies the constrains only on a sub-poset of $\mathcal{V}(\mathcal{N})$. In
this terminology, we obtain the following reformulation of Gleason’s theorem
Doering2004 ; DeGroote2007 ; Doering2012 .

######  Theorem 3.

_(Gleason in presheaf form (I))_ Let $\mathcal{N}$ be a von Neumann algebra
with no summand of type $\text{I}_{2}$. There is a bijective correspondence
between (normal) states on $\mathcal{N}$ and global sections of the (normal)
probabilistic presheaf $\underline{\Pi}$ of $\mathcal{N}$ over
$\mathcal{V}(\mathcal{N})$.

###### Proof.

Note that every (normal) state $\sigma:\mathcal{N}\rightarrow\mathbb{C}$
defines a finitely (completely) additive probability measure over the
projections of $\mathcal{N}$ (cf. Thm. 7.1.12 in KadisonRingroseII ), which
corresponds with a global section of $\underline{\Pi}$ by the preceeding
discussion. By the same correspondence, each global section of the (normal)
probabilistic presheaf
$\gamma\in\Gamma[\underline{\Pi}(\mathcal{V}(\mathcal{N}))]$ extends to a
(normal) state $\sigma\in\mathcal{S}(\mathcal{N})$ as a consequence of Thm. 2.
∎

Next, we turn to a generalisation of Gleason’s theorem in presheaf form to
composite systems.

###  II.3 Linearity without positivity.

The canonical product on partial orders, denoted
$\mathcal{V}_{1}\times\mathcal{V}_{2}$, is the Cartesian product with elements
$(V_{1},V_{2})$ for $V_{1}\in\mathcal{V}_{1}$, $V_{2}\in\mathcal{V}_{2}$ and
order relations such that for all $\tilde{V}_{1},V_{1}\in\mathcal{V}_{1}$,
$\tilde{V}_{2},V_{2}\in\mathcal{V}_{2}$:

$(\tilde{V}_{1},\tilde{V}_{2})\subseteq(V_{1},V_{2}):\Longleftrightarrow\tilde{V}_{1}\subseteq_{1}V_{1}\mathrm{\
and\ }\tilde{V}_{2}\subseteq_{2}V_{2}\;.$

It is interesting to ask whether the state space of the composite system can
be recovered from product contexts
$V\in\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2})\subsetneq\mathcal{V}(\mathcal{N}_{1}\otimes\mathcal{N}_{2})$
only. For this to be possible states on the subsystem algebras need to define
unique states on the composite algebra and vice versa. For this reason we will
consider the spatial tensor product between von Neumann algebras
$\bar{\otimes}$ (for details, see section 11.2 in KadisonRingroseII ). Recall
that given normal states $\sigma_{1}\in(\mathcal{N}_{1})_{*}$ and
$\sigma_{2}\in(\mathcal{N}_{2})_{*}$, where $\mathcal{N}_{*}$ denotes the
predual of $\mathcal{N}$, there exists a unique normal product state
$\sigma=\sigma_{1}\bar{\otimes}\sigma_{2}\in(\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2})_{*}$
(Prop. 11.2.7, KadisonRingroseII ). Conversely, every normal linear functional
$\sigma\in\mathcal{N}_{*}$ on the spatial tensor product
$\mathcal{N}=\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}$ is the norm limit of
normal product states $\sigma_{1}\bar{\otimes}\sigma_{2}$ (Prop 11.2.8,
KadisonRingroseII ). It follows that we can identify the product context
$V=(V_{1},V_{2})\in\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2})$
with the commutative von Neumann subalgebra
$V=V_{1}\bar{\otimes}V_{2}\subset\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}$
for every $V_{1}\subset\mathcal{N}_{1}$ and $V_{2}\subset\mathcal{N}_{2}$.

######  Theorem 4.

Let $\mathcal{N}_{1}$, $\mathcal{N}_{2}$ be von Neumann algebras with no
summand of type $\text{I}_{2}$ and let
$\mathcal{N}=\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}$. There is a
bijective correspondence between global sections of the normal probabilistic
presheaf $\underline{\Pi}$ of $\mathcal{N}$ over
$\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2})$ and normal
linear functionals $\sigma:\mathcal{N}\rightarrow\mathbb{C}$ such that
$\sigma(1)=1$ and $\sigma(a\otimes b)\geq 0$ for all
$a\in(\mathcal{N}_{1})_{+}$ and $b\in(\mathcal{N}_{2})_{+}$.

###### Proof.

Clearly, every normal product state
$\sigma=\sigma_{1}\bar{\otimes}\sigma_{2}\in\mathcal{N}_{*}$ yields a product
global section
$\gamma_{\sigma}=\gamma_{\sigma_{1}}\times\gamma_{\sigma_{2}}\in\Gamma[\underline{\Pi}(\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2}))]$,
as a consequence of Thm. 3, applied to $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$
individually. In particular, $\gamma(1)=\sigma(1)=1$ by normalisation, and
$\sigma(a\otimes b)\geq 0$ for all $a\in(\mathcal{N}_{1})_{+}$ and
$b\in(\mathcal{N}_{2})_{+}$ by positivity of the probability measures
$\mu^{\gamma}_{V}$ for all $V\in\mathcal{V}(\mathcal{N})$. Since every normal
state can be approximated by finite linear combinations of normal product
states (Prop 11.2.8, KadisonRingroseII ), the correspondence extends to all
normal states $\sigma\in\mathcal{N}_{*}$.

Conversely, we show that every global section
$\gamma\in\Gamma[\underline{\Pi}(\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2}))]$
defines a unique normal linear functional $\sigma_{\gamma}$ on
$\mathcal{N}=\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}$, which restricts to
$\gamma$ on $\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2})$,
i.e.,
$\gamma=(\sigma|_{\mathcal{P}(V)})_{\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2})}$.
Fix a context $V_{1}\in\mathcal{V}(\mathcal{N}_{1})$ and consider the
corresponding partial order of contexts under inclusion, inherited from
$\mathcal{V}_{1\&2}:=\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2})$
by restriction,

$\mathcal{V}_{1\&2}(V_{1}):=\\{V_{1}\times V_{2}\mid
V_{2}\in\mathcal{V}(\mathcal{N}_{2})\\}\;.$

In every context $V=V_{1}\bar{\otimes}V_{2}\in\mathcal{V}_{1\&2}$, the
probability measure
$\mu^{\gamma}_{V}\in\underline{\Pi}(\mathcal{V}_{1\&2})_{V}$ corresponding to
the global section $\gamma$ can be written (in terms of conditional
probabilities) as follows:

$\forall p\in\mathcal{P}(V_{1}),q\in\mathcal{P}(V_{2}):\
\mu^{\gamma}_{V}(p,q)=\mu_{V_{1}}^{\gamma}(p)\mu_{V_{2}}^{\gamma}(q\mid
p)=\mu_{V_{1}}^{\gamma}(p)\gamma_{2}^{p}(q)=\mu_{V_{1}}^{\gamma}(p)\sigma_{2}^{p}(q)\;.$
(1)

Here, $(\mu_{V_{2}}^{\gamma_{2}}(\cdot\mid
p))_{V_{2}\in\mathcal{V}(\mathcal{N}_{2})}=:\gamma_{2}^{p}\in\Gamma[\underline{\Pi}(\mathcal{V}_{1\&2}(V_{1}))]$
is a global section of the probabilistic presheaf
$\underline{\Pi}(\mathcal{V}_{1\&2}(V_{1}))$, which also depends on
$p\in\mathcal{P}(V_{1})$. Since
$\mathcal{V}_{1\&2}(V_{1})\cong\mathcal{V}(\mathcal{N}_{2})$, such global
sections correspond with normal states on $\mathcal{N}_{2}$ by Thm. 3, hence,
$\sigma_{2}^{p}\in(\mathcal{N}_{2})_{*}$ for all $p\in\mathcal{P}(V_{1})$.
Moreover, since $V_{1}\in\mathcal{V}(\mathcal{N}_{1})$ was arbitrary, Eq. (1)
holds for all $p\in\mathcal{P}(\mathcal{N}_{1})$.

Define a map
$\widetilde{\varrho}_{\gamma}:\mathcal{P}(\mathcal{N}_{1})\rightarrow(\mathcal{N}_{2})_{*}$
by $\widetilde{\varrho}_{\gamma}(p):=\mu_{V_{1}}^{\gamma}(p)\sigma_{2}^{p}$.
Clearly, $\widetilde{\varrho}_{\gamma}$ is bounded and $(\mathcal{N}_{2})_{*}$
is a Banach space (as a closed subspace of the continuous dual of
$\mathcal{N}_{2}$). Moreover, for $p=p_{1}+p_{2}$ with
$p_{1},p_{2}\in\mathcal{P}(\mathcal{N}_{1})$ orthogonal, i.e., $p_{1}p_{2}=0$,

$\widetilde{\varrho}_{\gamma}(p)=\mu_{V_{1}}^{\gamma}(p)\sigma_{2}^{p}=\mu_{V_{1}}^{\gamma}(p_{1})\sigma_{2}^{p_{1}}+\mu_{V_{1}}^{\gamma}(p_{2})\sigma_{2}^{p_{2}}=\widetilde{\varrho}_{\gamma}(p_{1})+\widetilde{\varrho}_{\gamma}(p_{2})\;,$

by additivity of $\gamma$. Similarly, $\widetilde{\varrho}_{\gamma}$ inherits
complete additivity from $\gamma$. By Thm. A in BunceWright1992 ,
$\widetilde{\varrho}_{\gamma}$ uniquely extends to a normal linear map
$\widetilde{\phi}_{\gamma}:\mathcal{N}_{1}\rightarrow(\mathcal{N}_{2})_{*}$.
Hence, $\sigma_{\gamma}(p\otimes q):=\widetilde{\phi}_{\gamma}(p)(q)$ is a
normal linear functional such that $\sigma_{\gamma}(1)=\gamma(1)=1$ and
$\sigma_{\gamma}(a\otimes b)\geq 0$ for all $a\in(\mathcal{N}_{1})_{+}$,
$b\in(\mathcal{N}_{2})_{+}$ by positivity of the (product) measures
$\mu^{\gamma}_{V}$ for all
$V\in\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2})$. ∎

We emphasise that the reduction to products of commutative von Neumann
algebras in $\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2})$
is already sufficient to lift quasi-linear measures to linear functionals. In
finite dimensions this has been observed before KlayRandallFoulis1987 ;
Wallach2002 . Yet, unlike Thm. 3 the linear functionals thus obtained are not
necessarily positive, hence, Thm. 4 does not provide a classification of
states on $\mathcal{N}=\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}$. From this
perspective, it seems unjustified to call Thm. 4 a generalisation of Gleason’s
theorem (Thm. 2) to composite systems.

##  III A composite Gleason theorem

Thm. 4 makes it clear that in order to obtain a generalisation of Gleason’s
theorem, which singles out the state spaces of composite systems, one needs to
strengthen Def. 2. In particular, one needs to characterise positivity of the
linear functional
$\sigma:\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}\rightarrow\mathbb{C}$ in
Thm. 4. We will derive positivity of $\sigma$ from complete positivity of an
associated map $\phi:\mathcal{N}_{1}\rightarrow\mathcal{N}^{*}_{2}$ (where
$\mathcal{N}_{2}^{*}$ denotes the algebra under the adjoint, see Sec. III.2
below).

Recall that in the proof of Thm. 4 we constructed a map
$\widetilde{\phi}:\mathcal{N}_{1}\rightarrow(\mathcal{N}_{2})_{*}$, into
normal states on $\mathcal{N}_{2}$ (up to normalisation). We may therefore
identify the normal states $\widetilde{\phi}(p)\in(\mathcal{N}_{2})_{*}$ for
every $p\in\mathcal{P}(\mathcal{N}_{1})$ with trace-class operators in
$\mathcal{N}_{2}$. More precisely, consider a faithful representation of the
von Neumann algebra $\mathcal{N}_{2}$ as bounded operators on a Hilbert space
$\mathcal{H}_{2}$.555We may take $\mathcal{H}_{2}$ to be the unique Hilbert
space defined by the standard form of $\mathcal{N}_{2}$ Haagerup1975 . With
respect to the latter, every normal state $\sigma\in(\mathcal{N}_{2})_{*}$ is
of the form
$\sigma(b)=\mathrm{tr}_{\mathcal{H}_{2}}[\rho^{*}_{\sigma}b]=\mathrm{tr}_{\mathcal{H}_{2}}[\rho_{\sigma}b]$
for all $b\in\mathcal{N}_{2}$ (Thm. 7.1.9 in KadisonRingroseII ); in physics
parlance, $\rho_{\sigma}$ is also called a _density matrix_. Throughout, we
will indicate this identification in maps by writing
$\phi:\mathcal{N}_{1}\rightarrow\mathcal{N}^{*}_{2}$ as opposed to
$\widetilde{\phi}:\mathcal{N}_{1}\rightarrow(\mathcal{N}_{2})_{*}$, and
similarly $\varrho:\mathcal{P}(\mathcal{N}_{1})\rightarrow\mathcal{N}^{*}_{2}$
as opposed to
$\widetilde{\varrho}:\mathcal{P}(\mathcal{N}_{1})\rightarrow(\mathcal{N}_{2})_{*}$.

Under this identification we seek conditions that ensure that $\phi$ is not
only positive but also completely positive. We will achieve this in two steps:
first, we extend the constraints in Gleason’s theorem to dilations in Sec.
III.1; second, we enforce a consistency condition with respect to dynamical
correspondences in Sec. III.2. Finally, in Sec. III.3 we prove our main
result: a generalisation of Gleason’s theorem to composite systems. Along the
way we show that neither of these additional constraints jeopardises
applicability in the single system case. Indeed, we prove sharper versions of
Gleason’s theorem, thereby addressing a number of subtleties that only arise
for composite systems.

###  III.1 Dilations.

First, we require that measures $\mu_{V}$ admit dilations in every commutative
von Neumann subalgebra $V\in\mathcal{V}(\mathcal{N})$. More precisely, by
Gelfand duality we may identify every context $V\in\mathcal{V}(\mathcal{N})$
with a compact Hausdorff $X$ space such that $\mu_{V}$ becomes a measure on
$X$. In particular, we identify the projections $\mathcal{P}(V)$ with the
clopen subsets of the Gelfand spectrum of $V$. In this way, we can interpret
$\mu_{V}$ as a $\mathcal{B}(\mathcal{H})$-valued measure for
$\mathcal{H}=\mathbb{C}$ (thus also
$\mathcal{B}(\mathcal{H})\cong\mathbb{C}$). By Naimark’s theorem Naimark1943 ;
Stinespring1955 , the latter admits a spectral dilation of the form
$\mu_{V}=v^{*}\varphi_{V}v$, where
$\varphi_{V}:\mathcal{P}(V)\hookrightarrow\mathcal{P}(\mathcal{K})$ is an
embedding for some Hilbert space $\mathcal{K}$ and
$v:\mathbb{C}\rightarrow\mathcal{K}$ is a linear map (equivalently, a vector
$v\in\mathcal{K}$). By choosing $\mathcal{K}$ sufficiently large, we can
choose $\mathcal{K}$ independent of contexts $V\in\mathcal{V}(\mathcal{N})$.
Moreover, we take $v$ to be independent of contexts
$V\in\mathcal{V}(\mathcal{N})$, since we can absorb any context dependence on
$v$ into $\varphi_{V}$.

More specifically, by a finitely (completely) additive embedding we mean
$\varphi_{V}(0)=0$, $\varphi_{V}(1-p)=1-\varphi_{V}(p)$, and
$\varphi_{V}(p_{1}+p_{2})=\varphi_{V}(p_{1})+\varphi_{V}(p_{2})$ for all
$p_{1},p_{2}\in\mathcal{P}(V)$, $p_{1}p_{2}=0$ ($\varphi_{V}(\sum_{i\in
I}p_{i})=\sum_{i\in I}\varphi_{V}(p_{i})$ for every family of pairwise
orthogonal projections $(p_{i})_{i\in I}$,
$p_{i}\in\mathcal{P}(\mathcal{N})$). We further require that embeddings
preserve spatial tensor products, i.e., whenever $V=V_{1}\bar{\otimes}V_{2}$,
then $\varphi_{V}=\varphi_{V_{1}}\bar{\otimes}\varphi_{V_{2}}$, were
$\varphi_{V_{1}}:\mathcal{P}(V_{1})\rightarrow\mathcal{P}(\mathcal{K}_{1})$
and
$\varphi_{V_{2}}:\mathcal{P}(V_{2})\rightarrow\mathcal{P}(\mathcal{K}_{2})$
are embeddings and $\mathcal{K}=\mathcal{K}_{1}\otimes\mathcal{K}_{2}$. Taken
together, this suggests to strengthen Def. 2 as follows.

######  Definition 3.

Let $\mathcal{N}$ be a von Neumann algebra with context category
$\mathcal{V}(\mathcal{N})$. The _(normal) dilated probabilistic presheaf_
$\underline{\Pi}_{D}$ of $\mathcal{N}$ over $\mathcal{V}(\mathcal{N})$ is the
presheaf given

  * (i)

on objects: for all $V\in\mathcal{V}(\mathcal{N})$, let

$\displaystyle\underline{\Pi}_{D}(V):=\\{\mu_{V}:\mathcal{P}(V)\rightarrow[0,1]\mid\mu_{V}\text{a
probability measure of the form }\mu_{V}=v^{*}\varphi_{V}v\text{ with}$
$\displaystyle\ v:\mathbb{C}\rightarrow\mathcal{K}\text{ linear},\text{ and
}\varphi_{V}:\mathcal{P}(V)\hookrightarrow\mathcal{P}(\mathcal{K})\text{ a
finitely (completely) additive embedding}\\}\;,$

  * (ii)

on arrows: for all $V,\tilde{V}\in\mathcal{V}(\mathcal{N})$, if
$\tilde{V}\subseteq V$, let

$\underline{\Pi}_{D}(i_{\tilde{V}V}):\underline{\Pi}_{D}(V)\rightarrow\underline{\Pi}_{D}(\tilde{V})\text{
with }\varphi_{V}\mapsto\varphi_{V}|_{\tilde{V}}\;.$

As in Def. 2, the condition that $\varphi_{V}$ preserves orthogonality is
required in commutative von Neumann subalgebras only. Consequently, Def. 3
only requires global sections of $\underline{\Pi}_{D}$ to be quasi-linear. The
key difference to the probabilistic presheaf $\underline{\Pi}$ in Def. 2 is
that we require the constraints to hold also with respect to dilations.

If Def. 3 is to generalise the single system case, it should be consistent
with it. We therefore start by analysing global sections of the dilated
probabilistic presheaf in the original setting. We have the following refined
version of Thm. 3.

######  Theorem 5.

_(Gleason in contextual form (II))_ Let $\mathcal{N}$ be a von Neumann algebra
with no summand of type $I_{2}$. There is a bijective correspondence between
(normal) states on $\mathcal{N}$ and global sections of the (normal) dilated
probabilistic presheaf $\underline{\Pi}_{D}$ of $\mathcal{N}$ over
$\mathcal{V}(\mathcal{N})$.

###### Proof.

Let $\sigma\in\mathcal{S}(\mathcal{N})$ be (normal) state and consider the
corresponding representation
$\pi:\mathcal{N}\rightarrow\mathcal{B}(\mathcal{K})$ arising via the GNS
construction from the inner product $(a,b):=\sigma(a^{*}b)$. In this
representation, $\sigma$ is a vector state $\sigma(a)=(\pi(a)v,v)$ with
$v\in\mathcal{K}$ for all $a\in\mathcal{N}$. Interpreting
$v:\mathbb{C}\rightarrow\mathcal{K}$ as a linear map, this reads
$\sigma(a)=v^{*}\pi(a)v$ for all $a\in\mathcal{N}$, and since $\pi$ is a
representation, $\varphi:=\pi|_{\mathcal{P}(\mathcal{N})}$ is an
orthomorphism. It follows that
$(\sigma|_{\mathcal{P}(V)})_{V\in\mathcal{V}(\mathcal{N})}$ defines a global
section of the (normal) dilated probabilistic probabilistic
$\underline{\Pi}_{D}(\mathcal{V}(\mathcal{N}))$.

Conversely, recall that
$\gamma=(v^{*}\varphi_{V}v)_{V\in\mathcal{V}(\mathcal{N})}\in\Gamma[\underline{\Pi}(\mathcal{V}(\mathcal{N}))]$
defines a (normal) state $\sigma_{\gamma}\in\mathcal{S}(\mathcal{N})$ by Thm.
2. In fact, the embeddings $(\varphi_{V})_{V\in\mathcal{V}(\mathcal{N})}$
define an orthomorphism
$\varphi_{\gamma}:\mathcal{P}(\mathcal{N})\rightarrow\mathcal{P}(\mathcal{K})$,
which by Cor. 2 in BunceWright1993  extends to a (normal) Jordan
$*$-homomorphism
$\Phi_{\gamma}:\mathcal{N}\rightarrow\mathcal{B}(\mathcal{K})$ such that
$\sigma_{\gamma}=v^{*}\Phi_{\gamma}v$ and
$\varphi_{\gamma}=\Phi_{\gamma}|_{\mathcal{P}(\mathcal{N})}$. This will become
important in Lm. 1 below. ∎

Despite the fact that, by Thm. 3 and Thm. 5, global sections of probabilistic
and dilated probabilistic presheaf correspond with states on
$\mathcal{N}$,666Since two global sections
$\gamma,\gamma^{\prime}\in\Gamma[\underline{\Pi}_{D}(\mathcal{V}(\mathcal{N}))]$
are identified if and only if $\mu^{\gamma}_{V}=\mu^{\gamma^{\prime}}_{V}$ for
all $V\in\mathcal{V}(\mathcal{N})$. the dilated probabilistic presheaf encodes
more constraints than the probabilistic presheaf. To see this, note that we
deduced linearity in Thm. 4 from a theorem due to Bunce and Wright in
BunceWright1992 . Under the conditions of Def. 3 a stronger version applies
BunceWright1993 , leading to the following refinement of Thm. 4.

######  Lemma 1.

Let $\mathcal{N}_{1}$, $\mathcal{N}_{2}$ be von Neumann algebras with no
summand of type $I_{2}$,
$\mathcal{N}=\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}$, and let
$\underline{\Pi}_{D}(\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2}))$
be the normal dilated probabilistic presheaf over the composite context
category $\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2})$.
Then every global section
$\gamma\in\Gamma[\underline{\Pi}_{D}(\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2}))]$
uniquely extends to a normal linear functional
$\sigma_{\gamma}\in\mathcal{N}_{*}$, given for all $a\in\mathcal{N}_{1}$ and
$b\in\mathcal{N}_{2}$ by

$\sigma_{\gamma}(a\otimes b)=\widetilde{\phi}_{\gamma}(a)(b)\;,$ (2)

where $\phi_{\gamma}:\mathcal{N}_{1}\rightarrow\mathcal{N}_{2}$ is of the form
$\phi_{\gamma}=w^{*}\Phi_{\gamma}w$ with
$\Phi_{\gamma}:\mathcal{N}_{1}\rightarrow\mathcal{B}(\mathcal{K})$ a normal
Jordan $*$-homomorphism and $w:\mathcal{H}_{2}\rightarrow\mathcal{K}$ a
bounded linear map for some Hilbert space $\mathcal{K}$ and
$\mathcal{N}_{2}\subset\mathcal{B}(\mathcal{H}_{2})$.

###### Proof.

Recall that in the proof of Thm. 4 we lifted the map
$\widetilde{\varrho}_{\gamma}:\mathcal{P}(\mathcal{N}_{1})\rightarrow(\mathcal{N}_{2})_{*}$
to a positive bounded linear map
$\widetilde{\phi}_{\gamma}:\mathcal{N}_{1}\rightarrow(\mathcal{N}_{2})_{*}$
(cf. BunceWright1992 ). Similarly, for
$\gamma\in\Gamma[\underline{\Pi}_{D}(\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2}))]$
we construct a completely additive measure
$\widetilde{\varrho}_{\gamma}:\mathcal{P}(\mathcal{N}_{1})\rightarrow(\mathcal{N}_{2})_{*}$,
which by the correspondence between normal states and positive trace-class
operators (Thm. 7.1.9 in KadisonRingroseII ) yields a map
$\varrho_{\gamma}:\mathcal{P}(\mathcal{N}_{1})\rightarrow(\mathcal{N}_{2})_{+}$
into positive trace-class operators (viz. density matrices).

Furthermore, note that by assumption there exists a Hilbert space
$\mathcal{K}=\mathcal{K}_{1}\otimes\mathcal{K}_{2}$, a map
$v:\mathbb{C}\rightarrow\mathcal{K}$ and orthomorphisms
$\varphi_{1}:\mathcal{P}(\mathcal{N}_{1})\rightarrow\mathcal{P}(\mathcal{K}_{1})$
and
$\varphi_{2}:\mathcal{P}(\mathcal{N}_{2})\rightarrow\mathcal{P}(\mathcal{K}_{2})$
(uniquely defined by $\varphi_{1}|_{V_{1}}=\varphi_{V_{1}}$ and
$\varphi_{2}|_{V_{2}}=\varphi_{V_{2}}$ for all
$V_{1}\in\mathcal{V}(\mathcal{N}_{1})$,
$V_{2}\in\mathcal{V}(\mathcal{N}_{2})$, respectively) such that

$\displaystyle\sigma_{\gamma}(p\otimes
q)=v^{*}(\varphi_{1}(p)\otimes\varphi_{2}(q))v$

for all $p\in\mathcal{P}(\mathcal{N}_{1})$ and
$q\in\mathcal{P}(\mathcal{N}_{2})$. Without loss of generality, we may
identify $\mathcal{K}_{2}=\mathcal{H}_{2}$ (for
$\mathcal{N}_{2}\subset\mathcal{B}(\mathcal{H}_{2})$) and take
$\varphi_{2}(q)=q$ to be the identity. Then let
$v=\sum_{ij}c_{ij}v_{1i}\otimes v_{2j}\in\mathcal{K}$, where
$c_{ij}\in\mathbb{C}$ and $\\{v_{1i}\\}_{i}$, $\\{v_{2j}\\}_{j}$ are
orthonormal bases in $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$, respectively.
Since $\sigma_{\gamma}(p\otimes\cdot)$ corresponds to a normal state on
$\mathcal{N}_{2}$ for all $p\in\mathcal{P}(\mathcal{N}_{1})$ (by Thm. 2), we
write

$\displaystyle\sigma_{\gamma}(p\otimes q)$
$\displaystyle=(\sum_{ij}c^{*}_{ij}v^{*}_{1i}\otimes v^{*}_{2j})\
(\varphi_{1}(p)\otimes q)\ (\sum_{kl}c_{kl}v_{1k}\otimes v_{2l})$
$\displaystyle=\sum_{ijkl}c^{*}_{ij}c_{kl}\ (v^{*}_{1i}\varphi_{1}(p)v_{1k})\
(v^{*}_{2j}qv_{2l})$
$\displaystyle=\mathrm{tr}_{\mathcal{H}_{2}}\bigg{[}\bigg{(}\sum_{ijkl}c_{ij}c^{*}_{kl}\
(v^{*}_{1k}\varphi_{1}(p)v_{1i})\ v_{2j}v^{*}_{2l}\bigg{)}^{*}q\bigg{]}\;.$

where we used that $\varphi^{*}_{1}(p)=\varphi_{1}(p)$ for all
$p\in\mathcal{P}(\mathcal{N}_{1})$ in the last step. Note that under the
Hermitian adjoint in the last line the inner product on $\mathcal{N}_{1}$ is
changed to its complex conjugate. To reflect this, let
$\mathcal{B}(\mathcal{K}_{1})^{*}$ denote the algebra
$\mathcal{B}(\mathcal{K}_{1})$ under the Hermitian adjoint, such that for all
pairs $v_{1i},v_{1k}\in\mathcal{K}_{1}$,
$\tilde{v}_{1i},\tilde{v}_{1k}\in\widetilde{\mathcal{K}}_{1}\cong\mathcal{K}^{*}_{1}$
(the dual of $\mathcal{K}_{1}$) and $a\in\mathcal{B}(\mathcal{K}_{1})$ we have

$(\tilde{v}^{*}_{1k}a^{*}\tilde{v}_{1i})_{\mathcal{B}(\mathcal{K}_{1})^{*}}:=(v^{*}_{1k}av_{1i})^{*}_{\mathcal{B}(\mathcal{K}_{1})}\;.$
(3)

Define a bounded linear map
$w:\mathcal{K}_{2}\rightarrow\widetilde{\mathcal{K}}_{1}\otimes\mathcal{K}_{2}$
by $w=\sum_{ij}c^{*}_{ij}\tilde{v}_{1i}\otimes v_{20}v^{*}_{2j}$, and let
$\varphi^{\prime}=\varphi_{1}\otimes 1_{\mathcal{B}(\mathcal{K}_{2})}$. This
yields an orthomorphism
$\varphi^{\prime}:\mathcal{P}(\mathcal{N}_{1})\rightarrow\mathcal{P}(\mathcal{K})$
such that

$\displaystyle w^{*}\varphi^{\prime}(p)w$
$\displaystyle=(\sum_{ij}c_{ij}\tilde{v}^{*}_{1i}\otimes v_{2j}v^{*}_{20})\
\varphi^{\prime}(p)\ (\sum_{kl}c^{*}_{kl}\tilde{v}_{1k}\otimes
v_{20}v^{*}_{2l})$ $\displaystyle=\sum_{ijkl}c_{ij}c^{*}_{kl}\
(\tilde{v}^{*}_{1i}\varphi_{1}(p)\tilde{v}_{1k})_{\mathcal{B}(\mathcal{K}_{1})^{*}}\
v_{2j}v^{*}_{2l}$ $\displaystyle=\sum_{ijkl}c_{kl}c^{*}_{ij}\
(\tilde{v}^{*}_{1k}\varphi_{1}(p)\tilde{v}_{1i})_{\mathcal{B}(\mathcal{K}_{1})^{*}}\
v_{2l}v^{*}_{2j}$ $\displaystyle=\left(\sum_{ijkl}c_{ij}c^{*}_{kl}\
(v^{*}_{1k}\varphi_{1}(p)v_{1i})_{\mathcal{B}(\mathcal{K}_{1})}\
v_{2j}v^{*}_{2l}\right)^{*}\;,$

where we changed labels $i\leftrightarrow k$, $j\leftrightarrow l$ in the
third line, and used Eq. (3) together with $\varphi^{*}_{1}(p)=\varphi_{1}(p)$
for all $p\in\mathcal{P}(\mathcal{N}_{1})$ in the last line. Consequently,

$\widetilde{\varrho}_{\gamma}(p)(q)=\mathrm{tr}_{\mathcal{H}_{2}}[w^{*}\varphi^{\prime}(p)wq]\;.$

Finally, by Cor. 1 in BunceWright1993  the completely additive orthomorphism
$\varphi^{\prime}:\mathcal{P}(\mathcal{N}_{1})\rightarrow\mathcal{P}(\mathcal{K})$
extends to a normal Jordan $*$-homomorphism
$\Phi_{\gamma}:\mathcal{J}(\mathcal{N}_{1})\rightarrow\mathcal{B}(\mathcal{K})$.
∎

Recall that a map $\phi:\mathcal{N}\rightarrow\mathcal{B}(\mathcal{H})$ is
called _decomposable_ if it is of the form $\phi=v^{*}\Phi v$ for
$v:\mathcal{H}\rightarrow\mathcal{K}$ a bounded linear map and
$\Phi:\mathcal{N}\rightarrow\mathcal{B}(\mathcal{K})$ a Jordan
$*$-homomorphism. Such maps satisfy a weaker notion of positivity than
complete positivity Stormer1982 , owing to their close resemblance with
completely positive maps $\phi=v^{*}\Phi v$, for which $\Phi$ is a
$C^{*}$-homomorphism by the classification due to Stinespring Stinespring1955
. Importantly, by a generalisation of Choi’s theorem Choi1975  (see Lm. 2
below), completely positive maps $\phi_{\gamma}$ correspond with positive
linear functionals under Eq. (2). In contrast, linear functionals
$\sigma_{\gamma}$ corresponding to decomposable maps under Eq. (2) are
generally not positive. We infer that for $\sigma_{\gamma}$ to be a state,
$\phi_{\gamma}$ in Lm. 1 further needs to lift to a completely positive map,
equivalently the Jordan $*$-homomorphism in Lm. 1 needs to lift to a
$C^{*}$-homomorphism.

###  III.2 Dynamical correspondences.

We recall some basic facts about Jordan algebras. A _Jordan algebra_
$\mathcal{J}$ is a commutative algebra, which satisfies the characteristic
equation $(a^{2}\circ b)\circ a=a^{2}\circ(b\circ a)$ for all
$a,b\in\mathcal{J}$. $\mathcal{J}$ is called a _JB algebra_ , if it is also a
Banach space such that $||a\circ b||\leq||a||\cdot||b||$ and
$||a^{2}||=||a||^{2}$ for all $a,b\in\mathcal{J}$. Finally, $J$ is called a
_JBW algebra_ if it is a JB algebra with a (unique) predual. A Jordan
homomorphism $\Phi:\mathcal{J}_{1}\rightarrow\mathcal{J}_{2}$ is a linear map
such that $\Phi(a\circ b)=\Phi(a)\circ\Phi(b)$ for all
$a,b\in\mathcal{J}_{1}$. For a more details on Jordan algebras, see
McCrimmon_ATasteOfJordanAlgebras .

Note that the self-adjoint part of a von Neumann algebra $\mathcal{N}$
naturally gives rise to a real JBW-algebra under the symmetrised product
$a\circ b=\frac{1}{2}\\{a,b\\}=\frac{1}{2}(ab+ba)$. We denote this algebra by
$\mathcal{J}(\mathcal{N})_{\mathrm{sa}}=(\mathcal{N}_{\mathrm{sa}},\circ)$ and
by $\mathcal{J}(\mathcal{N})$ its complexification.777A JBW algebra isomorphic
to a subalgebra of $\mathcal{J}(\mathcal{N})_{\mathrm{sa}}$ is also called a
JW algebra. In this case we speak of Jordan $*$-homomorphisms if $\Phi$ is a
Jordan homomorphism and $\Phi^{*}(a)=\Phi(a^{*})$.

In general, $\mathcal{J}(\mathcal{N})$ does not determine $\mathcal{N}$
completely, as it lacks compatibility with the antisymmetric part or
commutator of the associative product in $\mathcal{N}$, $[a,b]=ab-ba$. In
particular, for the Jordan $*$-homomorphism $\Phi$ in Lm. 1 to lift to a
$C^{*}$-homomorphism it needs to preserve commutators. On the level of JBW
algebras, this can be expressed in terms of one-parameter groups of Jordan
automorphisms $\mathbb{R}\ni t\mapsto
e^{t\delta(a)}\in\mathrm{Aut}(\mathcal{J})$, where
$\delta\in\mathcal{OD}_{s}(\mathcal{J})$ is called a _skew order derivation_ ,
i.e., a bounded linear map $\delta:\mathcal{J}_{+}\rightarrow\mathcal{J}_{+}$
acting on the positive cone of $\mathcal{J}$ such that $\delta(1)=0$ (for
details, see AlfsenShultz1998a ). For
$\mathcal{J}=\mathcal{J}(\mathcal{N})_{\mathrm{sa}}$, these are of the form
$\delta_{ia}(b)=\frac{i}{2}[a,b]$ for all
$a,b\in\mathcal{J}(\mathcal{N})_{\mathrm{sa}}$. This yields a canonical map
$\psi_{\mathcal{N}}:\mathcal{J}(\mathcal{N})_{\mathrm{sa}}\rightarrow\mathcal{OD}_{s}(\mathcal{J}(\mathcal{N})_{\mathrm{sa}})$,
$\psi_{\mathcal{N}}:a\rightarrow\delta_{ia}$, which gives expression to the
double role of self-adoint operators as observables and generators of
symmetries AlfsenShultz1998 ; Baez2020 . In particular,
$e^{t\psi_{\mathcal{N}}}$ expresses the unitary evolution of the system
described by $\mathcal{J}(\mathcal{N})_{\mathrm{sa}}$. We also define the map
$\psi^{*}_{\mathcal{N}}:=*\circ\psi_{\mathcal{N}}:a\rightarrow\delta_{-ia}$
for all $a\in\mathcal{J}(\mathcal{N})_{\mathrm{sa}}$ (cf. Prop. 15 in
AlfsenShultz1998a ).

Note that $\psi_{\mathcal{N}}(a)(a)=0$ and
$[\delta_{a},\delta_{b}]=-[\psi_{\mathcal{N}}(a),\psi_{\mathcal{N}}(b)]$ for
all $a,b\in\mathcal{J}(\mathcal{N})_{\mathrm{sa}}$, where
$\delta_{a}(b):=a\circ b$. More generally, Alfsen and Shultz define a
_dynamical correspondence_ to be any map
$\psi:\mathcal{J}\rightarrow\mathcal{OD}_{s}(\mathcal{J})$ satisfying these
properties and prove that a JBW algebra $\mathcal{J}$ is a JW algebra if and
only if $\mathcal{J}$ admits a dynamical correspondence.888We add that, unlike
a JBW algebra, a von Neumann algebra is generally not anti-isomorphic to
itself Connes1975 . In this case, associative products bijectively correspond
with dynamical correspondences (Thm. 23 in AlfsenShultz1998 ). We conclude
that a von Neumann algebra $\mathcal{N}$ is determined as the pair
$(\mathcal{J}(\mathcal{N}),\psi_{\mathcal{N}})$ with the canonical dynamical
correspondence
$\psi_{\mathcal{N}}:a\rightarrow\delta_{ia}=\frac{i}{2}[a,\cdot]$ for all
$a\in\mathcal{J}(\mathcal{N})_{\mathrm{sa}}$. Moreover, we set
$\mathcal{N}^{*}:=(\mathcal{J}(\mathcal{N}),\psi^{*}_{\mathcal{N}})$ for the
associative algebra whose order of composition is reversed (under the adjoint
map) with respect to the order of composition in
$\mathcal{N}\cong(\mathcal{J}(\mathcal{N}),\psi_{\mathcal{N}})$.

It follows that for a Jordan $*$-homomorphism
$\Phi:\mathcal{J}(\mathcal{N}_{1})\rightarrow\mathcal{J}(\mathcal{N}_{2})$ to
lift to a homomorphism between the respective von Neumann algebras
$\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ is for it to preserve the respective
dynamical correspondences $\psi_{\mathcal{N}_{1}}$ and
$\psi_{\mathcal{N}_{2}}$. Comparing with the Jordan $*$-homomorphism
$\Phi_{\gamma}$ in Lm. 1, we need to lift the dynamical correspondence
$\psi_{\mathcal{N}_{2}}$ to (a subalgebra of) $\mathcal{B}(\mathcal{K})$. To
do so, note first that the argument reduces to factors in $\mathcal{N}_{2}$
AlfsenShultz1998 , and further that we may choose $v$ in Def. 3 so that it
preserves the factor decomposition. With this choice, let
$\mathcal{N}^{v}_{2}\subseteq\mathcal{B}(\mathcal{K})$ be the largest von
Neumann algebra which restricts to $\mathcal{N}_{2}$ under $v$, i.e.,
$\mathcal{N}_{2}=v^{*}\mathcal{N}_{2}^{v}v$. Clearly,
$\Phi(\mathcal{N}_{1})\subseteq\mathcal{N}^{v}_{2}\subseteq\mathcal{B}(\mathcal{K})$,
and it is easy to see that in this case $\psi_{\mathcal{N}_{2}}$ defines a
unique dynamical correspondence $\psi^{\prime}_{\mathcal{N}_{2}}$ on
$\mathcal{N}^{v}_{2}$ via the linear embedding
$\mathcal{N}_{2}\hookrightarrow\mathcal{N}^{v}_{2}$ induced by $v$. This
entitles us to the following key definition (cf. FrembsDoering2022a ).

######  Definition 4.

Let $\mathcal{N}_{1}=(\mathcal{J}(\mathcal{N}_{1}),\psi_{\mathcal{N}_{1}})$,
$\mathcal{N}_{2}=(\mathcal{J}(\mathcal{N}_{2}),\psi_{\mathcal{N}_{2}})$ be von
Neumann algebras. A global section of the dilated probabilistic presheaf
$\gamma\in\Gamma[\underline{\Pi}_{D}(\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2}))]$
is called _time-oriented_ if the Jordan $*$-homomorphism $\Phi_{\gamma}$ in
Lm. 1 preserves dynamical correspondences,

$\Phi_{\gamma}\circ\psi^{*}_{\mathcal{N}_{1}}=\psi^{\prime}_{\mathcal{N}_{2}}\circ\Phi_{\gamma}\;,$
(4)

where $\psi^{\prime}_{\mathcal{N}_{2}}$ denotes the dynamical correspondence
on $\mathcal{N}_{2}^{v}\subset\mathcal{B}(\mathcal{K})$ uniquely defined by
$\psi_{\mathcal{N}_{2}}$.

The terminology reflects the fact that the one-parameter groups $\mathbb{R}\ni
t\mapsto e^{t\psi_{\mathcal{N}}}$ express unitary dynamics in quantum theory.
Consequently, they give physical meaning to $t$ as a time parameter. In
particular, $\psi_{\mathcal{N}}:a\rightarrow\delta_{ia}$ fixes the (canonical)
forward time direction of the system described by $\mathcal{N}$, whereas
$\psi^{*}_{\mathcal{N}}:=*\circ\psi_{\mathcal{N}}:a\rightarrow\delta_{-ia}$
describes the system under time reversal Doering2014  (see also
FrembsDoering2022a ; Frembs2022a ). We emphasise that the appearance of
$\psi^{*}_{1}$ and $\psi_{2}$ in Def. 4 is a consequence of interpreting the
normal linear functional
$\sigma_{\gamma}:\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}\rightarrow\mathbb{C}$
in Lm. 1 as a linear map
$\phi_{\gamma}:\mathcal{N}_{1}\rightarrow\mathcal{N}^{*}_{2}$ (under the
identification between trace-class operators and normal linear functionals),
equivalently $\phi_{\gamma}:\mathcal{N}^{*}_{1}\rightarrow\mathcal{N}_{2}$,
where the order of composition in $\mathcal{N}^{*}_{1}$ is naturally reversed
(under the adjoint map) with respect to the order of composition in
$\mathcal{N}_{1}$.

Once again, we show that Def. 4 poses no additional constraint in the single
system case.

######  Theorem 6.

_(Gleason in contextual form (III))_ Let
$\mathcal{N}=(\mathcal{J}(\mathcal{N}),\psi_{\mathcal{N}})$ be a von Neumann
algebra with no summand of type $\text{I}_{2}$. There is a bijective
correspondence between (normal) states on $\mathcal{N}$ and time-oriented
global sections of the (normal) dilated probabilistic presheaf
$\underline{\Pi}_{D}$ of $\mathcal{N}$ over $\mathcal{V}(\mathcal{N})$.

###### Proof.

By Thm. 5, states on $\mathcal{N}$ correspond with global sections of
$\underline{\Pi}_{D}(\mathcal{V}(\mathcal{N}))$. It is thus sufficient to show
that every state is already time-oriented. This follows from the fact that
every positive linear functional is automatically completely positive
Stinespring1955 . More precisely, recall that the GNS construction for
$\sigma\in\mathcal{S}(\mathcal{N})$ trivially yields a dilation of the form in
Lm. 1 (for $\mathcal{N}_{2}=\mathbb{C}$), namely
$\sigma(a)=v^{*}\pi(a)v=\mathrm{tr}_{\mathcal{K}}[vv^{*}\pi(a)]=\mathrm{tr}_{\mathcal{K}}[(vv^{*})^{*}\pi(a)]$
for all $a\in\mathcal{N}$, where $v:\mathbb{C}\rightarrow\mathcal{K}$ and
$\pi:\mathcal{N}\rightarrow\mathcal{B}(\mathcal{K})$ is a representation of
$\mathcal{N}$.999Note that this construction works for all states
$\sigma\in\mathcal{S}(\mathcal{N})$, not just for normal states. In
particular, $\sigma$ is time-oriented with respect to the algebras
$\mathcal{N}_{1}=\mathcal{N}$ and $\mathcal{N}_{2}=\mathbb{C}$. ∎

###  III.3 Gleason’s theorem for composite systems.

The refinements of Thm. 2 in Thm. 5 and Thm. 6 both reveal extra structure in
the classification of states in the single system case. In turn, having
identified this additional structure in Def. 3 and Def. 4, we may impose it in
order to generalise Gleason’s theorem to composite systems. Then, every
$\gamma\in\Gamma[\underline{\Pi}_{D}(\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2}))]$
yields a map $\phi_{\gamma}(a)=v^{*}\Phi_{\gamma}(a)v$ with $\Phi_{\gamma}$ a
$C^{*}$-homomorphism, which implies that $\phi_{\gamma}$ is completely
positive by Stinespring’s theorem. In the final step, we deduce from this the
existence of a positive linear functional
$\sigma_{\gamma}:\mathcal{N}\rightarrow\mathbb{C}$ with
$\mathcal{N}=\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}$. In finite
dimensions, this follows from Choi’s theorem Choi1975 . However, the latter
hinges on the existence of a finite trace. For the general case, we use the
following generalisation based on a result by Belavkin and Staszewski
Belavkin1986 .101010If $\phi:\mathcal{N}\rightarrow\mathcal{N}$, where
$\mathcal{N}$ has a cyclic and separating vector, the result also follows from
Stormer2014 .

######  Lemma 2.

The normal linear functional $\sigma_{\gamma}(a\otimes
b)=\widetilde{\phi}_{\gamma}(a)(b)$ in Lm. 1 is positive if and only if the
linear map
$\phi_{\gamma}:\mathcal{N}^{*}_{1}=(\mathcal{J}(\mathcal{N}_{1}),\psi^{*}_{\mathcal{N}_{1}})\rightarrow\mathcal{N}_{2}=(\mathcal{J}(\mathcal{N}_{2}),\psi_{\mathcal{N}_{2}})$
is completely positive.

###### Proof.

By Lm. 1, $\phi_{\gamma}:\mathcal{N}^{*}_{1}\rightarrow\mathcal{N}_{2}$ is a
decomposable map (between Jordan algebras $\mathcal{J}(\mathcal{N}_{1})$ and
$\mathcal{J}(\mathcal{N}_{2})$) Stormer1982 . Assume that $\phi_{\gamma}$ is
moreover completely positive. Fix faithful, normal, semi-finite weights
$\omega_{1}:\mathcal{N}_{1}\rightarrow\mathbb{C}$ and
$\omega_{2}:\mathcal{N}_{2}\rightarrow\mathbb{C}$ with respective faithful GNS
representations
$\pi_{\omega_{1}}:\mathcal{N}_{1}\rightarrow\mathcal{B}(\mathcal{H}_{1})$ and
$\pi_{\omega_{2}}:\mathcal{N}_{2}\rightarrow\mathcal{B}(\mathcal{H}_{2})$. In
the product representation
$\pi_{\omega}:\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}\rightarrow\mathcal{B}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})$
induced by the weight $\omega=\omega_{1}\bar{\otimes}\omega_{2}$ we have
$\sigma_{\gamma}(a\otimes b)=\omega(\rho^{*}_{\gamma}(a\otimes b))$ for all
$a\in\mathcal{N}_{1}$, $b\in\mathcal{N}_{2}$ and some trace-class operator
$\rho_{\gamma}\in\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}\subset\mathcal{B}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})$.111111In
other words, we again identify
$\mathcal{N}_{1}\subset\mathcal{B}(\mathcal{H}_{1})$ and
$\mathcal{N}_{2}\subset\mathcal{B}(\mathcal{H}_{1})$ with their standard forms
Haagerup1975 . We show that $\rho_{\gamma}$ is positive and self-adjoint.

Define a normal, completely positive reference map
$\phi_{0}:\mathcal{N}^{*}_{1}\rightarrow\mathcal{N}_{2}\subset\mathcal{B}(\mathcal{H}_{2})$
by
$\phi_{0}=\omega^{*}_{1}1_{\mathcal{B}(\mathcal{H}_{2})}=v_{0}^{*}\pi_{0}v_{0}$,
where $v_{0}=v_{1}\otimes
1_{\mathcal{B}(\mathcal{H}_{2})}:\mathcal{H}_{2}\rightarrow\mathcal{H}_{1}\otimes\mathcal{H}_{2}$
for $v_{1}:\mathbb{C}\rightarrow\mathcal{H}_{1}$ in
$\omega^{*}_{1}=(v^{*}_{1}\pi_{\omega_{1}}v_{1})^{*}=v^{*}_{1}\pi^{*}_{\omega_{1}}v_{1}=v^{*}_{1}(\pi_{\omega_{1}}\circ*)v_{1}$
(cf. Stinespring1955 ) and $\pi_{0}=(\pi_{\omega_{1}}\circ*)\otimes
1_{\mathcal{B}(\mathcal{H}_{2})}$. Recall that complete positivity of
$\phi_{0}$ means $\sum_{i,j=1}^{n}(\phi_{0}(a_{ij})\eta_{i},\eta_{j})\geq 0$
for all $\eta_{i},\eta_{j}\in\mathcal{H}_{2}$, $n\in\mathbb{N}$ whenever
$a_{ij}\in M_{n}(\mathcal{N}_{1})\cong M_{n}(\mathcal{N}^{*}_{1})_{+}$. Let
$(a_{ij})_{m}\in\mathcal{N}^{*}_{1}$ be a family of sequences such that

$\mathrm{lim}_{m\rightarrow\infty}\sum_{i,j=1}^{n}(\phi_{0}((a_{ij})_{m})\eta_{i},\eta_{j})=\sum_{i,j=1}^{n}(\eta_{i},\eta_{j})\
\mathrm{lim}_{m\rightarrow\infty}\omega_{1}((a_{ij})_{m})=0\;.$

Since $\eta_{i},\eta_{j}$ are arbitrary and $\omega_{1}$ is a faithful, normal
weight, we conclude $\mathrm{lim}_{m\rightarrow\infty}(a_{ij})_{m}=0$ in the
ultraweak topology. Since $\phi_{\gamma}$ is also a completely positive and
normal map, we have
$\mathrm{lim}_{m\rightarrow\infty}\sum_{i,j=1}^{n}(\phi_{\gamma}((a_{ij})_{m})\eta_{i},\eta_{j})=0$
for all $\eta_{i},\eta_{j}\in\mathcal{H}_{2}$ (cf. Prop. III.2.2.2 in
Blackadar_OperatorAlgebras ). In the terminology of Belavkin1986 ,
$\phi_{\gamma}$ is thus strongly completely absolutely continuous with respect
to $\phi_{0}$. Consequently, Thm. 2 in Belavkin1986  asserts the existence of
a positive, self-adjoint operator $\rho_{\gamma}$ such that
$\phi_{\gamma}=v_{0}^{*}\rho_{\gamma}\pi_{0}v_{0}$. It follows that
$\sigma_{\gamma}(a\otimes b)=\omega(\rho_{\gamma}(a\otimes b))$ for all
$a\in\mathcal{N}_{1}$, $b\in\mathcal{N}_{2}$ defines a positive and thus
normal state on $\mathcal{N}=\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}$.

Conversely, if $\sigma_{\gamma}$ is positive, there exists a positive, self-
adjoint operator $\rho_{\gamma}$ of trace-class such that $\sigma(a\otimes
b)=\omega(\rho_{\gamma}(a\otimes b))$ for all $a\in\mathcal{N}_{1}$,
$b\in\mathcal{N}_{2}$, hence,
$\phi_{\gamma}=v_{0}^{*}\rho_{\gamma}\pi_{0}v_{0}$. In particular,
$\phi_{\gamma}:\mathcal{N}^{*}_{1}\rightarrow\mathcal{N}_{2}$ is completely
positive by complete positivity of $\phi_{0}$. ∎

We finally arrive at a generalisation of Gleason’s theorem to composite
systems.

######  Theorem 7.

Let $\mathcal{N}_{1}=(\mathcal{J}(\mathcal{N}_{1}),\psi_{\mathcal{N}_{1}})$,
$\mathcal{N}_{2}=(\mathcal{J}(\mathcal{N}_{2}),\psi_{\mathcal{N}_{2}})$ be von
Neumann algebras with no summands of type $I_{2}$ and let
$\mathcal{N}=\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}$. Then there is a
bijective correspondence between the set of normal states on $\mathcal{N}$ and
the set of time-oriented global sections of the normal dilated probabilistic
presheaf $\underline{\Pi}_{D}$ over
$\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2})$.

###### Proof.

By Thm. 5, every normal state
$\sigma\in(\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2})_{*}$ restricts to a
global section of the normal dilated probabilistic presheaf,
$\gamma_{\sigma}=(\sigma|_{\mathcal{P}(V)})_{V\in\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2})}\in\Gamma[\underline{\Pi}_{D}(\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2}))]$.
It is thus sufficient to show that every normal state $\sigma$ can be written
in the form $\sigma(a\otimes b)=\widetilde{\phi}(a)(b)$ for all
$a\in\mathcal{N}_{1}$, $b\in\mathcal{N}_{2}$ and
$\phi:\mathcal{N}^{*}_{1}\rightarrow\mathcal{N}_{2}$ completely positive,
which follows from Lm. 2.

Conversely, let
$\gamma\in\Gamma[\underline{\Pi}_{D}(\mathcal{V}(\mathcal{N}_{1})\times\mathcal{V}(\mathcal{N}_{2}))]$.
By Lm. 1, $\gamma$ corresponds to a normal linear functional of the form
$\sigma_{\gamma}(a\otimes b)=\widetilde{\phi}_{\gamma}(a)(b)$ for all
$a\in\mathcal{N}_{1}$, $b\in\mathcal{N}_{2}$ and $\phi_{\gamma}$ a
decomposable map (between the respective Jordan algebras
$\mathcal{J}(\mathcal{N}_{1})$ and $\mathcal{J}(\mathcal{N}_{2})$), i.e.,
$\phi_{\gamma}(a)=w^{*}\Phi_{\gamma}(a)w$ for
$w:\mathcal{H}_{2}\rightarrow\mathcal{K}$ with
$\mathcal{N}_{2}\subset\mathcal{B}(\mathcal{H}_{2})$ and
$\Phi_{\gamma}:\mathcal{J}(\mathcal{N}_{1})\rightarrow\mathcal{J}(\mathcal{B}(\mathcal{K}))$
a Jordan $*$-homomorphism. Finally, since $\gamma$ is time-oriented with
respect to the dynamical correspondences
$\psi^{*}_{\mathcal{N}_{1}}=*\circ\psi_{\mathcal{N}_{1}}$ and
$\psi_{\mathcal{N}_{2}}$,
$\Phi_{\gamma}:\mathcal{N}^{*}_{1}\rightarrow\mathcal{N}^{v}_{2}\subset\mathcal{B}(\mathcal{K})$
is also a $C^{*}$-homomorphism, and $\phi_{\gamma}$ thus completely positive
by Stinespring’s theorem Stinespring1955  (cf. Thm. III.2.2.4 in Belavkin1986
). Lm. 2 asserts that $\sigma_{\gamma}$ is positive and thus a normal state on
$\mathcal{N}=\mathcal{N}_{1}\bar{\otimes}\mathcal{N}_{2}$. ∎

We finish with a few remarks. Note that the key steps in the argument leading
to Thm. 7 apply to $C^{*}$-algebras. The crucial exception is lifting quasi-
linearity to linearity. As such the restriction to von Neumann algebras might
not be optimal. For instance, our result may be strengthened to
$AW^{*}$-algebras, for which Gleason’s theorem holds by Hamhalter2015 .

Comparing with $C^{*}$-algebras, we further remark that we restricted to
normal states, since non-normal product states generally do not have a unique
extension in the spatial tensor product. On the other hand, in the spatial
tensor product of $C^{*}$-algebras every product state has a unique extension
(Prop. 11.1.1 in KadisonRingroseII ). This suggests that for $AW^{*}$-algebras
the bijective correspondence in Thm. 7 further extends beyond normal states.

Moreover, we observe that Thm. 7 can be formulated in an intrinsically Jordan-
algebraic setting. Note that Bunce and Wright prove a generalisation of
Gleason’s theorem for JBW-algebras in BunceWright1985 . The case considered
here is the one where JBW algebras arise as self-adjoint parts of von Neumann
algebra, hence, are JW algebras. Alfsen and Shultz show that this is the case
if and only if the JBW algebra admits a dynamical correspondence
AlfsenShultz_StateSpaceOfOperatorAlgebras . Thm. 7 applies for any choice of
dynamical correspondences on the factor JW algebras.

In turn, in the context of general JBW algebras, the concept of state requires
a weaker notion of positivity, reflecting the generalisation from completely
positive to decomposable maps Stormer1982 . Clearly, the relation with
(preservation of) dynamical correspondences in Def. 4 is lost in this case. We
point out that this is closely analogous to the limited applicability of
Tomita-Takesaki theory in JBW algebras HaagerupHanche-Olsen1984 .

##  IV Conclusion

We proved a generalisation of Gleason’s theorem to composite systems. An ad
hoc attempt establishes a correspondence with normal linear functionals on the
composite algebra that are positive on product operators, but fails to single
out those that are positive (Thm. 4). We remedy this by strengthening the
additivity constraints on measures in commutative von Neumann subalgebras to
dilated systems, together with a consistency condition between dynamical
correspondences on the respective subalgebras (Thm. 7). Neither of these
conditions changes the result in the single system case (Thm. 5 and Thm. 6),
since positive linear functionals are completely positive as a consequence of
Stinespring’s theorem Stinespring1955 .

Apart from its mathematical value, our result also carries physical
significance. Gleason’s theorem (Thm. 2) plays a crucial role in the
foundations of quantum theory, where it justifies Born’s rule. It is all the
more interesting that a similar result extends to composite systems. For
instance, we note that within the framework of algebraic quantum field theory
the observable algebra is naturally composed of local algebras HaagKastler1964
.

Thm. 7 was foreshadowed for finite type I factors, and with a focus on
classifying quantum from non-signalling correlations in FrembsDoering2022a .
Its close connection with entanglement classification will be discussed
elsewhere Frembs2022a . For a broader perspective on the significance of
contextuality in the foundations of quantum mechanics we refer to
DoeringFrembs2019a .

#### Acknowledgements.

This work is supported through a studentship in the Centre for Doctoral
Training on Controlled Quantum Dynamics at Imperial College funded by the
EPSRC, by grant number FQXi-RFP-1807 from the Foundational Questions Institute
and Fetzer Franklin Fund, a donor advised fund of Silicon Valley Community
Foundation, and ARC Future Fellowship FT180100317.

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