# Separation theorems and Bell inequalities in algebraic quantum
mechanics111This paper appeared in Busch, P., Lahti, P.J., Mittelstaedt, P.
(eds.): Proceedings of the Symposium on the Foundations of Modern Physics,
Cologne, 1–5 June 1993, pp. 29–37. Since it is not otherwise available online,
I am archiving it now (with minor edits).

Guido Bacciagaluppi

###### Abstract

The paper discusses the concept of separation of quantum mechanical systems in
the algebraic approach. We review known theorems, then establish a link
between the $C^{*}$-algebraic and the corresponding $W^{*}$-algebraic
concepts. A characterization of separation in terms of Bell inequalities, due
to Raggio (1988), is given a $C^{*}$-algebraic formulation. Finally, we
comment on the implications for the understanding of the Bell inequalities.

##  1 Introduction

The concept of separation, as we use the term, was introduced by Primas
(1978), and investigated by Raggio (1981, 1983, 1988). It means the absence of
EPR correlations between two subsystems of a composite system, in the strong
sense that the systems are never correlated. According to Primas (see for
instance his (1983)), the conceptual relevance of such a concept of separation
is that separated systems are to lack the holistic features that lead to the
EPR paradox, and the subsystems of a separated system can thus be treated as
individual objects.

From the point of view of the traditional Hilbert-space formalism this is an
unnatural condition. The states of a product space ${\cal H}_{1}\otimes{\cal
H}_{2}$ that lack EPR-correlations are precisely the product states
$\psi_{1}\otimes\psi_{2}$. Requiring that the two systems never be EPR-
correlated is thus a very strong restriction on the dynamics of the system,
indeed, this requirement is satisfied only by noninteracting systems.

However, if one generalises the formalism to algebraic quantum mechanics, this
condition can be met also in the presence of interactions. As an example, take
two concrete von Neumann algebras ${\cal A}_{1}$ and ${\cal A}_{2}$ acting on
the Hilbert spaces ${\cal H}_{1}$ and ${\cal H}_{2}$, and let ${\cal A}_{1}$
be commutative. It is clear that every element $A$ of ${\cal A}_{1}$ commutes
with every element of the product algebra $\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$. Intuitively, this means that the
possible distributions of the values of $A$ are not constrained by the
uncertainty principle, and so that $A$ has a dispersion-free value in every
pure state $\Psi$ of $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$. But
now there are superselection rules separating any two product states
$\psi_{1}\otimes\psi_{2}$ and $\varphi_{1}\otimes\varphi_{2}$ with
$\psi_{1}\neq\varphi_{1}$. Indeed, if a state of the form

$\Psi=c_{1}\psi_{1}\otimes\psi_{2}+c_{2}\varphi_{1}\otimes\varphi_{2}$

were pure, the reduced state for subsystem 1 would be

$\rho_{1}=\mid c_{1}\mid^{2}P_{\psi_{1}}+\mid c_{2}\mid^{2}P_{\varphi_{1}}$

(where the $P_{\psi}$ are the projectors on the corresponding rays), i. e. a
mixture, but then certainly some $A\in{\cal A}_{1}$ would have a non-zero
dispersion in the state $\Psi$, contradicting the above. Thus, if ${\cal
A}_{1}$ or ${\cal A}_{2}$ is commutative, all pure states are product states,
so indeed it is possible that systems 1 and 2 are never EPR-correlated,
independently of the dynamics.

Separation was investigated by Raggio (1981, 1983, 1988) in the framework of
$W^{*}$-algebras. He derived two structure theorems (Th. 2.6 and 2.7 below)
that characterise separated systems completely. In fact, the above example is
generic: a $W^{*}$-system composed of two subsystems is separated if and only
if at least one of the subsystems is classical (i. e. if its algebra is
commutative). As Primas (1983) puts it: quantum objects exist only in a
classical environment; or better, idealized models of a quantum system in
which the system is to be separated from its environment must treat the
environment of the system classically.

Later, Raggio seized upon a result of Baez (1987) about Bell inequalities in
the algebraic setting, and established a remarkable link between Bell
inequalities and separation. This is a particularly interesting aspect of
Raggio (1988).

In this paper we examine separation in the framework of $C^{*}$-algebras, and
in particular the connection between separation and the Bell inequalities. In
algebraic quantum mechanics, $W^{*}$-algebras often arise in connection with
representations of a $C^{*}$-algebra, and the abstract $C^{*}$-algebra is
regarded as more fundamental.

The definition of a separated $C^{*}$-system is straightforward, and the
structure theorem analogous to Raggio’s theorems is already contained in
Takesaki (1958). In section 2, we give the definitions and structure theorems
for the $C^{*}$\- and the $W^{*}$-algebraic case. We also make the connection
between separation for $C^{*}$\- and $W^{*}$-systems. In section 3 we turn to
the relation between separation and the Bell inequalities. We prove the
converse of Baez’s theorem, and derive the $C^{*}$-analogue of Raggio’s
results. We conclude (section 4) with some comments on the significance of
these results for the understanding of the Bell inequalities.

##  2 Separation theorems

In the following, when ${\cal A}_{1}$ and ${\cal A}_{2}$ are $C^{*}$-algebras
(which are always supposed to be unital), $\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ denotes the injective tensor product of
${\cal A}_{1}$ and ${\cal A}_{2}$. When ${\cal A}_{1}$ and ${\cal A}_{2}$ are
$W^{*}$-algebras, $\mbox{${\cal A}_{1}$}\overline{\otimes}\mbox{${\cal
A}_{2}$}$ denotes their $W^{*}$-tensor product.

We now give the exact definitions of separation in the $C^{*}$\- and in the
$W^{*}$-case.

###### Definition 2.1

Let ${\cal A}_{1}$, ${\cal A}_{2}$ be $C^{*}$-algebras (respectively,
$W^{*}$-algebras). A state (a normalized positive linear functional) $\omega$
on $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ (respectively,
$\mbox{${\cal A}_{1}$}\overline{\otimes}\mbox{${\cal A}_{2}$}$) is said to be
a product state iff for all $(A_{1},A_{2})$ in $\mbox{${\cal
A}_{1}$}\times\mbox{${\cal A}_{2}$}$,

$\omega(A_{1}\otimes A_{2})=\omega(A_{1}\otimes{\bf 1})\omega({\bf 1}\otimes
A_{2}).$

In the case of $C^{*}$-algebras we require all pure states to be product
states.

###### Definition 2.2

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $C^{*}$-algebras. We say
$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated (in the
$C^{*}$-sense) if and only if:

for all pure states $\omega$ on $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal
A}_{2}$}$, $\omega$ is a product state.

Notice that if $\omega$ is a pure product state of $\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$, then
$\omega_{1}:=\omega\mid_{\mbox{${\cal A}_{1}$}}$ and
$\omega_{2}:=\omega\mid_{\mbox{${\cal A}_{2}$}}$ are pure (where ${\cal
A}_{1}$ and ${\cal A}_{2}$ are identified with $\mbox{${\cal
A}_{1}$}\otimes{\bf 1}$ and ${\bf 1}\otimes\mbox{${\cal A}_{2}$}$,
respectively). This follows easily from Prop. 2.8 below.

The structure theorem for separated $C^{*}$-systems is the following.

######  Theorem 2.3 (Takesaki 1958)

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $C^{*}$-algebras. Then:

$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated
$\Leftrightarrow$ ${\cal A}_{1}$ or ${\cal A}_{2}$ is commutative.

The proof is immediate from Takesaki (1979), Th. IV 4.14, together with the
above remark.

In the case of $W^{*}$-algebras there are two different notions of separation.
Raggio (1981) originally considered separation with respect to pure states (as
we have done for $C^{*}$-algebras). The second definition (Raggio 1983) is
more appropriate to the $W^{*}$-setting, since it refers only to normal
states.

###### Definition 2.4

Let ${\cal A}$ be a $W^{*}$-algebra, and $\omega$ a normal state on ${\cal
A}$. $\omega$ is said to be decomposable into normal product states if and
only if:

$\omega$ lies in the norm-closure of the convex hull of the normal product
states.

Intuitively, $\omega$ is a ‘continuous convex combination’ of normal product
states.

###### Definition 2.5

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $W^{*}$-algebras.

  * i)

$\mbox{${\cal A}_{1}$}\overline{\otimes}\mbox{${\cal A}_{2}$}$ is separated
with respect to pure states if and only if:

for all pure states $\omega$ on $\mbox{${\cal
A}_{1}$}\overline{\otimes}\mbox{${\cal A}_{2}$}$, $\omega$ is a product state.

  * ii)

$\mbox{${\cal A}_{1}$}\overline{\otimes}\mbox{${\cal A}_{2}$}$ is separated
with respect to normal states if and only if:

for all normal states $\omega$ on $\mbox{${\cal
A}_{1}$}\overline{\otimes}\mbox{${\cal A}_{2}$}$, $\omega$ is decomposable
into normal product states.

The second definition needs to be formulated in terms of decomposability,
because pure normal states need not exist in a general $W^{*}$-algebra. This
makes the two definitions apparently independent of each other. In both cases,
however, analogues of Takesaki’s theorem hold; and the definitions turn out to
be equivalent.

######  Theorem 2.6 (Raggio 1981)

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $W^{*}$-algebras. Then the following
are equivalent:

  * i)

$\mbox{${\cal A}_{1}$}\overline{\otimes}\mbox{${\cal A}_{2}$}$ is separated
w.r.t. pure states;

  * ii)

${\cal A}_{1}$ or ${\cal A}_{2}$ is commutative.

######  Theorem 2.7 (Raggio 1983, 1988)

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $W^{*}$-algebras. Then the following
are equivalent:

  * i)

$\mbox{${\cal A}_{1}$}\overline{\otimes}\mbox{${\cal A}_{2}$}$ is separated
w.r.t. normal states;

  * ii)

${\cal A}_{1}$ or ${\cal A}_{2}$ is commutative.

The first of these two theorems is proved directly in Raggio (1981). One can
also derive it from Th. 2.3. For the proof of the second theorem, see Raggio
(1988) and the references therein. (In Raggio (1983) the theorem is proved
under more restrictive assumptions). Since the two definitions of separation
for $W^{*}$-algebras coincide, we shall simply talk of separation in the
$W^{*}$-sense.

We now establish the link between the concept of separation for
$C^{*}$-algebras and the concept of separation for $W^{*}$-algebras via the
representations of a $C^{*}$-algebra. We shall need it in the proof of Th.
3.7.

###### Proposition 2.8

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $C^{*}$-algebras, and $\pi_{1}$ and
$\pi_{2}$ two representations of ${\cal A}_{1}$ and ${\cal A}_{2}$,
respectively. Then:

$(\pi_{1}\otimes\pi_{2}(\mbox{$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal
A}_{2}$}$}))^{\prime\prime}=\pi_{1}(\mbox{${\cal
A}_{1}$})^{\prime\prime}\>\overline{\otimes}\>\pi_{2}(\mbox{${\cal
A}_{2}$})^{\prime\prime}.$

This is Takesaki (1979), Prop. IV 4.13.

If $\pi_{1}$ and $\pi_{2}$ are the representations of ${\cal A}_{1}$ and
${\cal A}_{2}$ induced by a representation $\pi$ of $\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$, we have

$\pi(\mbox{$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal
A}_{2}$}$})=\pi_{1}\otimes\pi_{2}(\mbox{$\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$}),$

and we obtain:

###### Corollary 2.9

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $C^{*}$-algebras, $\pi$ a
representation of $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$, and
$\pi_{1}$ and $\pi_{2}$ the induced representations of ${\cal A}_{1}$ and
${\cal A}_{2}$. Then

$\pi(\mbox{$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal
A}_{2}$}$})^{\prime\prime}=\pi_{1}(\mbox{${\cal
A}_{1}$})^{\prime\prime}\>\overline{\otimes}\>\pi_{2}(\mbox{${\cal
A}_{2}$})^{\prime\prime}.$

Thus, the von Neumann algebra associated with a representation of a tensor
product $C^{*}$-algebra has a canonical factorization as a $W^{*}$-tensor
product. This motivates the following definition:

###### Definition 2.10

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $C^{*}$-algebras, and $\pi$ a
representation of $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$. We say
that $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated in the
representation $\pi$ if and only if:

$\pi(\mbox{${\cal A}_{1}$}\otimes{\bf
1})^{\prime\prime}\>\overline{\otimes}\>\pi({\bf 1}\otimes\mbox{${\cal
A}_{2}$})^{\prime\prime}$ is separated (in the $W^{*}$-sense).

We now prove our first theorem.

###### Theorem 2.11

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $C^{*}$-algebras. The following are
equivalent:

  * i)

$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated;

  * ii)

$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated in all
irreducible representations;

  * iii)

$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated in the
universal representation.

Proof:‘i) $\Rightarrow$ ii)’
Take any irreducible representation $\pi$ of $\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$. By assumption $\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated, so, by Th. 2.3, ${\cal
A}_{1}$ or ${\cal A}_{2}$ is commutative. It follows that
$\pi_{1}(\mbox{${\cal A}_{1}$})$ or $\pi_{2}(\mbox{${\cal A}_{2}$})$ is
commutative, and so that $\pi_{1}(\mbox{${\cal A}_{1}$})^{\prime\prime}$ or
$\pi_{2}(\mbox{${\cal A}_{2}$})^{\prime\prime}$ is commutative. By the second
half of Th. 2.7 (or by another direct argument), $\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated in the representation
$\pi$.

‘ii)$\Leftarrow$i)’
Assume $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is not separated.
By Th. 2.3, ${\cal A}_{1}$ and ${\cal A}_{2}$ are both noncommutative. Then
there exist irreducible representations $(\pi_{1},{\cal H}_{1})$ of ${\cal
A}_{1}$ and $(\pi_{2},{\cal H}_{2})$ of ${\cal A}_{2}$, with $\dim{\cal
H}_{1},\dim{\cal H}_{2}\geq 2$. Irreducibility means that

$\pi_{1}(\mbox{${\cal A}_{1}$})^{\prime\prime}={\cal B}({\cal H}_{1}),\mbox{
and }\pi_{2}(\mbox{${\cal A}_{2}$})^{\prime\prime}={\cal B}({\cal H}_{2}).$

But ${\cal B}({\cal H}{{}_{1}})$ and ${\cal B}({\cal H}{{}_{2}})$ are both
noncommutative, and it follows by the first half of Th. 2.7 that

$\pi_{1}(\mbox{${\cal
A}_{1}$})^{\prime\prime}\>\overline{\otimes}\>\pi_{2}(\mbox{${\cal
A}_{2}$})^{\prime\prime}$

is not separated. Consequently, $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal
A}_{2}$}$ is not separated in the representation $\pi:=\pi_{1}\otimes\pi_{2}$.

‘i) $\Leftrightarrow$ iii)’
Let $\pi$ be the universal representation of $\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$. By Th. 2.3, $\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated iff ${\cal A}_{1}$ or
${\cal A}_{2}$ is commutative. This is the case iff $\pi_{1}({\cal A}_{1})$ or
$\pi_{2}({\cal A}_{2})$ is commutative, iff $\pi_{1}({\cal
A}_{1})^{\prime\prime}$ or $\pi_{2}({\cal A}_{2})^{\prime\prime}$ is
commutative. By Th. 2.7, this is equivalent to separation in the
representation $\pi$.

##  3 Bell inequalities and separation

Separation means that two systems never present EPR-correlations, in other
words, that any possible correlations between the two can be explained in
terms of classical mixtures. It is well-known that the problem of whether
given sets of numbers that look like probabilities and correlations can be
fitted into a classical probabilistic scheme is related to the Bell
inequalities. Results of this kind are contained in particular in Pitowsky
(1989); see also Fine (1982 and references therein) and Prof. Beltrametti’s
contribution to the present volume. It is thus natural to ask whether there is
a way of characterising separation using Bell inequalities. In fact there is:
Baez (1987) obtained some partial results for $C^{*}$-algebras, and Raggio
(1988) showed that in the case of $W^{*}$-algebras, a composite system
$\mbox{${\cal A}_{1}$}\overline{\otimes}\mbox{${\cal A}_{2}$}$ is separated if
and only if all normal states on $\mbox{${\cal
A}_{1}$}\overline{\otimes}\mbox{${\cal A}_{2}$}$ satisfy the Bell inequalities
(in the sense of Def. 3.1 below). We prove the analogous results for
$C^{*}$-algebras.

In 1987, Baez published a short note on Bell inequalities in the
$C^{*}$-algebraic formalism. His main results are the following.

###### Definition 3.1

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $C^{*}$\- ($W^{*}$-) algebras. We say
a state $\omega$ on $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$
($\mbox{${\cal A}_{1}$}\overline{\otimes}\mbox{${\cal A}_{2}$}$) satisfies the
Bell inequalities iff $\forall A,A^{\prime}\in\mbox{${\cal A}_{1}$}$, $\forall
B,B^{\prime}\in\mbox{${\cal A}_{2}$}$ with norm less or equal to $1$,

$\mid\omega(A\otimes(B-B^{\prime}))\mid+\mid\omega(A^{\prime}\otimes(B+B^{\prime}))\mid\leq
2.$

###### Proposition 3.2

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $C^{*}$-algebras, and let $\omega$ be
a product state on $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$. Then
$\omega$ satisfies the Bell inequalities.

The proof is completely analogous to the standard proofs of the Bell
inequalities: see Baez (1987).

###### Definition 3.3

Let ${\cal A}$ be a $C^{*}$-algebra, and let $\omega$ be a state on ${\cal
A}$. $\omega$ is said to be decomposable into product states if and only if:

$\omega$ lies in the weak∗-closure of the convex hull of the product states.

###### Proposition 3.4

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $C^{*}$-algebras, and let
$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ be separated. Then:

for all states $\omega$ on $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal
A}_{2}$}$, $\omega$ is decomposable into product states.

Proof: By the Krein-Milman theorem, every state on $\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ lies in the weak∗-closure of the convex
hull of the pure states. By the assumption of separation, these are product
states.

From this and Th. 2.3 Baez obtains:

######  Theorem 3.5 (Baez 1987)

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be two $C^{*}$-algebras, and let ${\cal
A}_{1}$ or ${\cal A}_{2}$ be commutative. Then:

for all states $\omega$ on $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal
A}_{2}$}$, $\omega$ satisfies the Bell inequalities.

Proof: For a product state, the theorem reduces to Prop. 3.2. For an arbitrary
state, it follows by convexity and continuity using Prop. 3.4.

Baez states explicitly that for algebras of type I also the converse holds,
because in this case one can reproduce the Bohm-Bell situation, which gives
violation of the Bell inequalities. We shall now prove more generally that for
arbitrary $C^{*}$-algebras ${\cal A}_{1}$ and ${\cal A}_{2}$, the product
$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated if and only
if ${\cal A}_{1}$ or ${\cal A}_{2}$ is commutative. This is the analogue of
Raggio’s results for $W^{*}$-algebras (Th. 3.6), from which in fact our
theorem follows.

######  Theorem 3.6 (Raggio 1988)

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $W^{*}$-algebras. Then the following
are equivalent:

  * i)

${\cal A}_{1}$ or ${\cal A}_{2}$ is commutative;

  * ii)

for all normal states $\omega$ on $\mbox{${\cal
A}_{1}$}\overline{\otimes}\mbox{${\cal A}_{2}$}$, $\omega$ satisfies the Bell
inequalities.

For the proof, see Raggio (1988). The trick is to construct a Bohm-Bell type
counterexample not using finite-dimensional projections, which might not exist
(type III), but constructing subalgebras isomorphic to the two-by-two matrices
using pairs of generally infinite-dimensional noncommuting projections.

Finally, we prove the converse of Prop. 3.4 and Th. 3.5.

###### Theorem 3.7

Let ${\cal A}_{1}$ and ${\cal A}_{2}$ be $C^{*}$-algebras. The following are
equivalent:

  * i)

${\cal A}_{1}$ or ${\cal A}_{2}$ is commutative;

  * ii)

$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated;

  * iii)

for all states $\omega$ on $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal
A}_{2}$}$, $\omega$ is decomposable into product states;

  * iv)

for all states $\omega$ on $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal
A}_{2}$}$, $\omega$ satisfies the Bell inequalities.

Proof: We will prove that if all $\omega$ satisfy the Bell inequalities,
$\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated in all $\pi$
(the proof using the universal representation is analogous). From Th. 2.11,
the theorem will follow. Take an irreducible representation $(\pi,{\cal H})$
of $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$, and any $\psi$ in
${\cal H}$. $\psi$ is a state on $\pi(\mbox{$\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$})^{\prime\prime}$ and defines a (pure)
state on $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ via

$\omega_{\psi}(A):=(\psi,\pi(A)\psi).$

By assumption, $\omega_{\psi}$ satisfies the Bell inequalities, i. e. $\forall
A,A^{\prime}\in\mbox{${\cal A}_{1}$}$, $\forall B,B^{\prime}\in\mbox{${\cal
A}_{2}$}$ with norm less or equal to 1,

$\begin{array}[]{cl}&\mid(\psi,\pi(A\otimes(B-B^{\prime}))\psi)\mid+\mid(\psi,\pi(A^{\prime}\otimes(B+B^{\prime}))\psi)\mid=\\\
=&\mid(\psi,\pi(A)\otimes(\pi(B)-\pi(B^{\prime}))\psi)\mid+\mid(\psi,\pi(A^{\prime})\otimes(\pi(B)+\pi(B^{\prime}))\psi)\mid\leq
2.\end{array}$

By continuity, we also have $\forall A,A^{\prime}\in\pi(\mbox{${\cal
A}_{1}$})^{\prime\prime}$, $\forall B,B^{\prime}\in\pi(\mbox{${\cal
A}_{2}$})^{\prime\prime}$ with norm less or equal to 1,

$\mid(\psi,A\otimes(B-B^{\prime})\psi)\mid+\mid(\psi,A^{\prime}\otimes(B+B^{\prime})\psi)\mid\leq
2.$

Thus, $\psi$ as a state on $\pi(\mbox{$\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$})^{\prime\prime}$ satisfies the Bell
inequalities. But now, any normal state on $\pi(\mbox{$\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$})^{\prime\prime}$ lies in the norm-
closure of the convex hull of states of the form $\psi$. So, by convexity and
continuity, it satisfies the Bell inequalities.

By Th. 3.6, $\pi(\mbox{${\cal
A}_{1}$})^{\prime\prime}\>\overline{\otimes}\>\pi(\mbox{${\cal
A}_{2}$})^{\prime\prime}$ is separated, i. e. $\mbox{${\cal
A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated in the representation
$\pi$. Since $\pi$ was arbitrary, $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal
A}_{2}$}$ is separated in all irreducible representations.

By Th. 2.11, $\mbox{${\cal A}_{1}$}\otimes\mbox{${\cal A}_{2}$}$ is separated.

##  4 For whom the Bell tolls

The original derivations of the Bell inequalities by Bell and others proceeded
from a hypothesis of local hidden variables that corresponds to a classical
(albeit generally indeterministic and certainly contextual) model for the
probabilities to be expected in EPR-type experiments. Classical examples of
composite systems also trivially satisfy the Bell inequalities. ‘Classicality
implies the Bell inequalities’ seems to be the conclusion. The converse also
appears to be true since the violation of the Bell inequalities by quantum
mechanical systems is what makes them interesting in the first place as a
touchstone for testing local hidden variable theories experimentally.

One sense in which the converse has been, indeed, shown to hold rigorously is
Pitowsky’s analysis (1989) of the Bell inequalities: satisfaction of the Bell
inequalities is a necessary and sufficient condition for the existence of a
classical probability space and probability distributions reproducing the
given data. This is the same situation as we have here: Theorems 3.6 and 3.7
indeed show that all possible correlations between results of measurements on
the systems represented by ${\cal A}_{1}$ and ${\cal A}_{2}$ are purely
classical.

However, Pitowsky’s results are not vindicating any correspondence between
classicality and the Bell inequalities other than in the precise form he gave:
as a matter of fact, it is just the correlations that are classical. Indeed,
the catch-phrase ‘Bell inequalities mean that everything is classical’ is
wrong. Theorems 3.6 and 3.7 show that, contrary to the case of traditional
Hilbert-space quantum mechanics, in algebraic quantum mechanics there are
systems that satisfy the Bell inequalities in all possible states, and
nevertheless are not entirely classical — indeed, one of the subsystems may be
completely quantum mechanical.

Neither is it the case that the system, even though not classical, is
described by ‘classical’ hidden variables. In a hidden variable account, one
tries to explain the EPR-correlations present in traditional Hilbert-space
quantum mechanics by adding ‘hidden variables’ to the theory. Separation in
our sense requires instead that we limit ourselves to systems that are not
EPR-correlated, thus remaining within quantum mechanics, albeit within the
more general formalism of algebraic quantum mechanics.

Thus classicity of behaviour in the sense of the Bell inequalities being
satisfied is not sufficient for the system as a whole to be classical — only a
subsystem is. Further, Aerts (1991) has argued that the Bell inequalities are
not even necessary for a system to be classical: even a classical system can
violate the Bell inequalities if it cannot be separated into individual
subsystems. The relation between separation and the Bell inequalities seems to
suggest that what lies physically — or metaphysically — behind the Bell
inequalities (and the mathematical existence of classical probabilities) is
the individuality of the systems considered: each system has a physical state
of its own, uninfluenced by the states of other systems, a special instance of
this being Bell’s original concept of locality for hidden variables.

## Acknowledgements

I wish to thank first of all Jeremy Butterfield for the endless and immensely
helpful discussions I had with him. My thanks go further to the audiences who
in some form or other heard and commented on this material. In particular I
wish to thank Anton Amann, Harvey Brown, Fiona Harrison, Klaas Landsman, and
Constantin Piron. [I am also very grateful to Adrian Kent for alerting me to
the inaccuracy of some side remarks in the published version, which have been
accordingly edited out of this online version.] This work was supported by the
British Academy and the Arnold Gerstenberg Fund.

## References

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Bell inequalities with $2\cdot\sqrt{2}$, exactly “in the same way” as its
violations by the EPR experiments’ Helv. Phys. Acta, 64, 1-23.

Baez, J. (1987): ‘Bell’s inequality for $C^{*}$-algebras’, Lett. Math. Phys.,
13, 135-136.

Fine, A. (1982): ‘Joint distributions, quantum correlations, and commuting
observables’, J. Math. Phys., 23, 1306-1310.

Pitowsky, I. (1989): Quantum Probability — Quantum Logic. Springer, Berlin and
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Primas, H. (1978): Quantenmechanik und Theoretische Chemie. Lecture course,
ETH Zürich, Summer Term 1978.

Primas, H. (1983): Chemistry, Quantum Mechanics and Reductionism. 2nd edition.
Springer, Berlin.

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