Generalized Bell inequalities and frustrated spin systems %\footnote{Dedicated to Marshall Luban on the occasion of his 70th birthday.}
Heinz-J\"urgen Schmidt
We find a close correspondence between generalized Bell inequalities of a special kind and certain frustrated spin systems. For example, the Clauser-Horn-Shimony-Holt inequality corresponds to the frustrated square with the signs $+++-$ for the nearest neighbor interaction between the spins. Similarly, the Pearle-Braunstein-Cave inequality corresponds to a frustrated even ring with the corresponding signs $+\ldots +-$. Upon this correspondence, the violation of such inequalities by the entangled singlet state in quantum mechanics is equivalent to the spin system possessing a classical coplanar ground state, the energy of which is lower than the Ising ground state's energy. We propose a scheme which generates new inequalities and give further examples, the frustrated hexagon with additional diagonal bonds and the frustrated hypercubes in $n=3,4,5$ dimensions. Surprisingly, the hypercube in $n=4$ dimensions yields an inequality which is not violated by the singlet state. We extend the correspondence to other entangled states and XXZ-models of spin systems.\\ \noindent PACS: 03.65.Ud, % Bell inequalities 75.10.Hk % Classical spin models
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\title{Generalized Bell inequalities and frustrated spin systems
%\footnote{Dedicated to Marshall Luban on the occasion of his 70th birthday.}
}
\author{Heinz-J\"urgen Schmidt }

\address{Universit\"at Osnabr\"uck, Fachbereich Physik,
Barbarastr.~7, 49069 Osnabr\"uck, Germany}


\begin{abstract}
We find a close correspondence between generalized Bell
inequalities of a special kind and certain frustrated spin
systems. For example, the Clauser-Horn-Shimony-Holt inequality
corresponds to the frustrated square with the signs $+++-$ for the
nearest neighbor interaction between the spins. Similarly, the
Pearle-Braunstein-Cave inequality corresponds to a frustrated even
ring with the corresponding signs $+\ldots +-$. Upon this
correspondence, the violation of such inequalities by the
entangled singlet state in quantum mechanics is equivalent to the
spin system possessing a classical coplanar ground state, the
energy of which is lower than the Ising ground state's energy. We
propose a scheme which generates new inequalities and give further
examples, the frustrated hexagon with additional diagonal bonds
and the frustrated hypercubes in $n=3,4,5$ dimensions.
Surprisingly, the hypercube in $n=4$ dimensions yields an
inequality which is \emph{not} violated by the singlet state. We
extend the correspondence to other entangled states and XXZ-models
of spin systems.\\

\noindent
PACS: 03.65.Ud,   % Bell inequalities
      75.10.Hk    % Classical spin models
\end{abstract}
\maketitle


\widetext
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\section{Introduction\label{sec:I}}
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Bell's inequality, published more than four decades ago, has not
ceased to invoke keen interest in the physics community. The title
of the seminal paper of J.~Bell \cite{Bell:1964} refers to the
famous article of A.~Einstein, B.~Podolski, and N.~Rosen
\cite{EPR:1935} (EPR) who concluded that, according to their
criteria, quantum theory (QT) is incomplete. Bell proved that the
assumptions of EPR lead to an inequality for measurable
correlations of spin measurements for two particles which is
violated in QT, and, as later work showed, also in experiments.
Thus certain kinds of hidden variable theories are empirically ruled out.
Of course, the exact relation between Bell's assumptions and those of
EPR has to be carefully examined. According to an analysis of
L.~E.~Ballentine and J.~P.~Jarrett \cite{BJ:1987} the assumptions
of Bell, simple locality and predictive completeness, are even
weaker than the EPR assumptions. Hence, in the words of these
authors, the \emph{incompleteness} of QT \emph{is, in some sense, a
property of nature}
\cite{BJ:1987}.\\

There have been many proposals to generalize Bell's inequality. An
important generalization is the Clauser-Horn-Shimony-Holt
inequality \cite{CHSH:1969} (CHSH) which is about a linear
combination of the correlations of two pairs of measurements. A
generalization to $n$ pairs of measurements has been considered by
Pearle \cite{P:1970} and later investigated by Braunstein and Cave
\cite{BC:1990}. Other work generalized Bell's inequality to an
arbitrary number of measurements \cite{GS:1979} or to more than
two particles \cite{AP:1993}. See the textbook of A.~Peres
\cite{AP:1993}
and literature quoted there for more details.\\

In this article we will point out a close correspondence between
possible generalized Bell inequalities (GBI's) and certain
frustrated \emph{classical} spin systems $\Sigma_N^{cl}$.
These terms will be explained in more detail below.
It is important to distinguish the spin system $\Sigma_N^{cl}$ from
the \emph{quantum} spin system $\Sigma_2^q$ on which the
EPR measurements are performed.
The number $N$ of spins in $\Sigma_N^{cl}$ corresponds to
the number of measurements considered in the context of GBI's.
As a by-product of the correspondence we will obtain a procedure
to generate new GBI's including a test whether these inequalities are
violated in QT.\\

This paper is organized as follows: In section \ref{sec:G} we
explain the basic idea of the correspondence using the example of
the CHSH inequality. Then we will give the general definitions for
the spin systems which give rise to a correspondence with GBI's.
The GBI is violated by the singlet state if and only if the
corresponding Heisenberg spin system has a classical ground state
with a lower energy than the corresponding Ising ground state.
These systems thus have necessarily non-collinear ground state
configurations, i.~e.~coplanar or $3$-dimensional ones, although
in all examples considered in this article it is not necessary to
consider $3$-dimensional ground states. In section \ref{sec:C} we
present some methods to calculate classical ground states and
apply these to the construction procedure for GBI's in section
\ref{sec:B}. Section \ref{sec:E} is devoted to a couple of
examples, including the frustrated $2n$-ring leading to the
Pearle-Braunstein-Cave inequality, the frustrated hexagon and the
frustrated hypercubes $H_n$. In all these examples it is possible
to analytically calculate classical Heisenberg ground states and
Ising ground states and to compare their ground state energies.
With the exception of $H_4$ the Ising ground state energy is
higher and hence we obtain GBI's violated in QT by the singlet
state. More general entangled states are considered in section
\ref{sec:S} and are shown to lead to XXZ-models of spin systems.
We close with a conclusion in section \ref{sec:CC}. The
correspondence between GBI's and frustrated spin systems is
summarized in table \ref{tab1}.
\\





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\section{Inequalities and spin systems\label{sec:G}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In order to motivate the correspondence between generalized Bell
inequalities (GBI's) and frustrated spin systems
we consider the CHSH inequality \cite{CHSH:1969},
following \cite{AP:1993}.\\
Let $a,b,c,d$ be four numbers which assume only
the values $\pm 1$. Then either $a+c=0$ or $a-c=0$ and hence
$(a+c)b+(a-c)d$ assumes only the values $\pm 2$,
which yields the inequality
\begin{equation}\label{G1}
-2\le ab+ad+cb-cd \le 2
\;.
\end{equation}
Imagine that the numbers $a_i,b_i,c_i,d_i,\; i=1,\ldots,N$ are the
outcomes of four $N$-times repeated experiments and consider the
mean values of the above products, i.~e.~the correlation,
\begin{equation}\label{G2}
\langle a b \rangle \equiv \frac{1}{N}\sum_{i=1}^N
a_i\, b_i,\;\text{ etc. }
\;.
\end{equation}
Then the inequality (\ref{G1}) also holds for the mean values, i.~e.
\begin{equation}\label{G3}
-2\le \langle ab\rangle +\langle ad\rangle +
\langle cb\rangle -\langle cd\rangle  \le 2
\;.
\end{equation}
This is an equivalent form of the CHSH inequality \cite{CHSH:1969}
which belongs to the family of generalized Bell inequalities GBI.
It bounds the classical correlations between four different
measurements. We have formulated it
without the assumption that the mean values converge against some
expectation values for $N\rightarrow\infty$, following
\cite{AP:1993} since it seems to be more appropriate to consider
this as part of the postulates which connect the CHSC inequality
to measurements or to QT. These postulates are known to lead to a
contradiction. The CHSC inequality alone is a mathematically valid
statement about
$N\times 4$-matrices with entries $\pm 1$.\\
Now consider in QT a pair of particles with spin $s=\frac{1}{2}$
in its entangled singlet $(S=0)$ spin state
\begin{equation}\label{G4}
\phi=\frac{1}{\sqrt{2}}\left( |\uparrow\downarrow\,\rangle -
|\downarrow\uparrow\,\rangle\right) \;.
\end{equation}
Further consider measurements of the single particle spin in
direction of the unit vectors
$\vec{a},\vec{b},\vec{c},\vec{d}$. These are measurements of
observables represented by the Hermitean operators
\begin{equation}\label{G5}
A=\vec{a}\cdot \vec{\sigma} = a_x \sigma_x +
a_y \sigma_y +a_z \sigma_z
\;,
\end{equation}
where $\sigma_x,\sigma_y,\sigma_z$ are the Pauli matrices and the
observables $B,C,D$ for the other directions
$\vec{b},\vec{c},\vec{d}$ are analogously defined. It is possible
to combine any two of these measurements and to measure, say, $A$
at the left-hand particle and $B$ at the other one. The two
experimenters doing these measurements are traditionally called
``Alice" and ``Bob". According to the rules of QT, the combined
measurement is represented by the tensor product operator
$A\otimes B$. If $a_i, b_i$ are the outcomes of $N$ repetitions of
this combined measurements, the mean values $\langle a b \rangle$
according to (\ref{G2}) converge towards the expectation value,
which can be calculated by QT and depends on the state of the
system. For the singlet state (\ref{G4}) the expectation value
turns out to be
\begin{equation}\label{G6}
\langle A B \rangle \equiv \langle \phi| A \otimes B
|\phi\rangle=-\vec{a}\cdot\vec{b}
\;,
\end{equation}
and analogously for $\langle A D \rangle,\langle C B \rangle$
and $\langle C D \rangle$. Hence the correlation term in
the CHSH inequality (\ref{G3}) has the quantum theoretical
counterpart
\begin{equation}\label{G7}
\langle A B \rangle +\langle A D \rangle +\langle C B \rangle -
\langle C D \rangle
=
-\vec{a}\cdot\vec{b} -\vec{a}\cdot\vec{d}-\vec{c}\cdot\vec{b}+
\vec{c}\cdot\vec{d}
\;.
\end{equation}
It is easily seen that the extreme values of (\ref{G7}) exceed
the bounds of the CHSH inequality. The possible values
of (\ref{G7}) are symmetric with respect to $0$, since the
substitution
$\vec{a}\mapsto -\vec{a}, \vec{c}\mapsto -\vec{c}$ changes
the overall sign in (\ref{G7}).
Writing (\ref{G7}) in the form $-\vec{a}\cdot(\vec{b} +\vec{d})-
\vec{c}\cdot(\vec{b}-\vec{d})$
it is obvious that each term is minimal for the choice
$\vec{a}\, |\,|\, (\vec{b} +\vec{d})$ and $\vec{c} \,
|\,|\,(\vec{b}-\vec{d})$. Moreover,
$|\vec{b} +\vec{d}|+|\vec{b} -\vec{d}|$ is maximal for
$\sphericalangle(\vec{b},\vec{d})=90^\circ$.
This is equivalent to the statement: The square
has the maximal circumference among the rectangles
with fixed length of their diagonals. Hence
\begin{equation}\label{G8a}
-2\sqrt{2} \le \vec{a}\cdot\vec{b} +\vec{a}\cdot\vec{d}+
\vec{c}\cdot\vec{b}-\vec{c}\cdot\vec{d}\le 2\sqrt{2}
\;,
\end{equation}
or
\begin{equation}\label{G8b}
-2\sqrt{2} \le
\langle A B \rangle +\langle A D \rangle +\langle C B \rangle -
\langle C D \rangle
\le 2\sqrt{2}
\;,
\end{equation}
and the lower bound is assumed for any configuration with
$\sphericalangle(\vec{b},\vec{d})=90^\circ,
\sphericalangle(\vec{c},\vec{d})=45^\circ,
\sphericalangle(\vec{a},\vec{c})=90^\circ$, see figure \ref{fig04a}.
The upper bound is assumed similarly.
Here and in what follows we always count the
angles $\sphericalangle(\vec{a},\vec{b})$ between two unit vectors
counter-clockwise, beginning with $\vec{a}$.
Equation (\ref{G8b}) expresses the bounds for quantum correlations
between four possible measurements. Since the bounds are attained
the CHSH inequality (\ref{G3}) is violated in QT for suitable
measurements and entangled states.
\\

The violation of the CHSH and similar inequalities,
which are rigorously proven theorems, can only be understood
in the sense that some
of the assumptions leading to (\ref{G3}) must
not hold in QT. Indeed, if $[A,C]\neq 0$ and $[B,D]\neq 0$,
only one of the four possible combinations
of measurements $AB, AD, CB, CD$ can be performed as a joint
measurement and hence only two of the four numbers
$a, b, c, d$ can be actually measured in one single experiment.
If QT is right, it is thus not possible, by whatever means,
to predict the missing two numbers in a consistent way,
i.~e.~in such a way that the mean values of all correlation
measurements
(actual and hypothetical ones) approach the mean values of the actual
correlation measurements alone.
In the words of A.~Peres \cite{AP:1978}:
\emph{ Unperformed experiments have no results}.\\

Next we introduce the correspondence to spin systems, see
table \ref{tab2}.
The basic idea is to re-interpret (\ref{G8a}) as a statement
about the energy
of a classical Heisenberg spin system $\Sigma$ of four spins,
which are represented by the unit vectors
$\vec{a},\vec{b},\vec{c},\vec{d}$. Of course, $\Sigma$ has nothing
to do with the quantum two-spin system for which the EPR type of
measurements are performed. The different signs in (\ref{G8a})
reflect the coupling between the four spins: $\Sigma$ can be
visualized as a square with three anti-ferromagnetic bonds
($J=+1$) and one ferromagnetic bond ($J=-1$), see (\ref{E31}) and
figure \ref{fig03}. Each spin configuration
$\vec{a},\vec{b},\vec{c},\vec{d}$ realizing the lower bound of
(\ref{G8a}) is thus a classical ground state for the
(dimensionless) Hamiltonian

\begin{equation}\label{G9}
H(\vec{a},\vec{b},\vec{c},\vec{d})= \vec{a}\cdot\vec{b} +
\vec{a}\cdot\vec{d}+\vec{c}\cdot\vec{b}-\vec{c}\cdot\vec{d}
\;.
\end{equation}

In this paper we will always denote the ground state energy of a spin system
by $-E_0$ in order to avoid inequalities of the form $E_0\le \ldots \le -E_0$.\\
The ground states do not minimize each term in (\ref{G9})
separately. Indeed, each term in (\ref{G9}) has the form of a spin dimer
with a classical ground state energy of $-1$, irrespective of the sign of the dimer term.
Adding these four ground state energies would result in
$-E_0=-4$ (the same as in the case where all four signs in
(\ref{G9}) are $+1$). Actually, the ground state energy of (\ref{G9})
is only $-E_0=-2\,\sqrt{2}$ since one cannot minimize each term  of (\ref{G9})
separately without regard to the other terms.\\
Spin systems with this property,
namely where not each term in the Heisenberg
(or Ising) Hamiltonian can be minimized simultaneously by a classical ground state
will be called ``frustrated" throughout this article. Sometimes this notion
is used in the literature with slightly different meanings.
For the theory of frustrated spin systems, see, for example, \cite{SRFB:2004},
or, for an introduction, \cite{MR:2006}. One of the simplest
examples is the anti-ferromagnetic (AF) spin triangle where the
local states $\uparrow\downarrow$ minimize one term in the
Heisenberg Hamiltonian but cannot be extended to a global ground
state. The AF triangle has coplanar ground states with angles of
$120^\circ$ between the spin vectors and is said to be
``geometrically frustrated". The frustrated square (\ref{G9}), or
``Bell square", as we will call it,
is an example of non-geometric frustration.
\\

The Bell square is an ``AB-system", that is, the spins can
be divided into two sub-lattices ${\mathcal A}$ and
${\mathcal B}$ such that only ${\mathcal A}$-spins interact with
${\mathcal B}$-spins.
(The nomenclature is again reminiscent of ``Alice" and ``Bob").
A system with this property, non-positive interaction {\emph within} the sub-lattices
and non-negative interaction {\emph between} the sub-lattices
is usually called a ``bi-partite" system and has ground states of the form that, say,
all ${\mathcal A}$-spins are down and all ${\mathcal B}$-spins are up.
The Bell square is an AB-system, but not bi-partite.
This could be taken as a preliminary definition of
``non-geometric frustration". However,
we will not attempt to define ``geometrical frustration" and
``non-geometrical frustration"  more precisely and
will rather use these notions in a somewhat intuitive and vague sense.\\

Now we can also re-interpret the CHSH inequality, or, rather, its
precursor (\ref{G1}), as a statement about the Bell square spin
system: It simply says that its energy according to the Ising
model is bounded by $\pm 2$. In the Ising model, the individual
spin is not represented by a $3$-dimensional unit vector, but, so
to speak, by an $1$-dimensional unit vector $\uparrow$ or
$\downarrow$ , or, equivalently, by numbers $\pm 1$. The
bi-partite systems mentioned above have ground states of the form
$\uparrow,\; \downarrow$, i.~e.~collinear or Ising ground states.
But the Bell square Heisenberg spin system has a coplanar ground
state with a ground state energy $-E_0=-2\sqrt{2}$ below its Ising
model ground state energy $-E_0^{(I)}=-2$. This is the violation
of the CHSH inequality in QT,
translated into the language of spin systems.\\

We will try to exploit the described correspondence in order to
construct GBI's violated by the entangled singlet state in QT by considering
the corresponding non-geometrically frustrated
spin systems with only coplanar or $3$-dimensional ground states.
To this end we need some more general notation
which will be adapted to the described correspondence.\\

The state of the spin system $\Sigma$
is described by $N$ unit vectors $\vec{s}_\mu,\;\mu=1,\ldots,N$.
Moreover, the set of spins is divided into two
disjoint subsets, $\{1,\ldots,N\}={\mathcal A}\,
\stackrel{.}{\cup}\,{\mathcal B}$, such that the
Hamiltonian of $\Sigma$
can be written in the form
\begin{equation}\label{G10}
H=\sum_{\mu\in{\mathcal A}} \sum_{\nu\in{\mathcal B}}
J_{\mu\nu} \vec{s}_\mu\cdot\vec{s}_\nu
\;,
\end{equation}
where the $J_{\mu\nu}$ are real coupling coefficients.
The minimum of the Hamiltonian (\ref{G10}) is called
the ground state energy $-E_0$.\\
The corresponding Ising model has states described by
numbers $s_\mu=\pm 1,\;\mu=1,\ldots,N$ and an Ising
Hamiltonian
\begin{equation}\label{G10a}
H^{(I)}=\sum_{\mu\in{\mathcal A}}
\sum_{\nu\in{\mathcal B}}J_{\mu\nu} s_\mu s_\nu
\end{equation}
with the minimum $-E_0^{(I)}$, the Ising ground state energy.
In both cases, the values of the Hamiltonian
change their sign under the spin flip transformation
$\vec{s}_\mu\mapsto - \vec{s}_\mu$, resp. $s_\mu\mapsto -s_\mu$,
$\mu\in{\mathcal A}$.\\

In the context of the EPR situation, the
$s_\mu=\pm 1,\,\mu=1,\ldots, N$
are the possible outcomes of one
experiment and the GBI assumes the form
\begin{equation}\label{G11}
-E_0^{(I)}\le \sum_{\mu\in{\mathcal A}}
\sum_{\nu\in{\mathcal B}}J_{\mu\nu}
\langle s_\mu s_\nu \rangle \le E_0^{(I)}
\;.
\end{equation}
The $N$ unit vectors $\vec{s}_\mu,\;\mu\in{\mathcal A}$
describe the directions of spin measurements
at, say, the left-hand particle (done by Alice),
and analogously
$\vec{s}_\mu,\;\mu\in{\mathcal B}$ for measurements
at the right-hand particle (done by Bob).
The quantum theoretical counter-part of (\ref{G11})
is the inequality
\begin{equation}\label{G12}
-E_0\le - \sum_{\mu\in{\mathcal A}}
\sum_{\nu\in{\mathcal B}} J_{\mu\nu}
\langle A_\mu B_\nu \rangle
=
\sum_{\mu\in{\mathcal A}} \sum_{\nu\in{\mathcal B}}
J_{\mu\nu} \vec{s}_\mu\cdot\vec{s}_\nu \le E_0
\;,
\end{equation}
where all correlations are calculated in the
singlet state (\ref{G4}). The GBI (\ref{G11})
is hence violated in QT if and only if $E_0^{(I)} < E_0$.\\

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\section{Calculation of the ground state\label{sec:C}}
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For the calculation of the classical ground state
of a spin system $\Sigma$ there exists no straightforward
method. However, sometimes the following considerations are useful.
For more details see \cite{SM:2003}.
We write the coupling constants $J_{\mu\nu}$ as the entries
of a symmetric $N\times N$-matrix $\mathbb{J}$.
Let $j_{\text{\scriptsize min}}$ denote its lowest eigenvalue.
Then the Rayleigh-Ritz variation principle yields
\begin{equation}\label{C1}
2 H = \sum_{\mu,\nu=1}^N J_{\mu\nu}
\vec{s}_\mu\cdot\vec{s}_\nu \ge
j_{\text{\scriptsize min}}\sum_{\mu=1}^N
(\vec{s}_\mu)^2 = N j_{\text{\scriptsize min}}
\;,
\end{equation}
whence
\begin{equation}\label{C1a}
\frac{1}{2} N j_{\text{\scriptsize min}}\le -E_0
\;.
\end{equation}
In general, this is only a lower bound and we may have
$\frac{1}{2} N j_{\text{\scriptsize min}}< -E_0$.
However, if $(s_\mu^{(i)})_{\mu=1,\ldots,N},\; i=1,2$ are
two linearly independent eigenvectors of $\mathbb{J}$
with eigenvalue $j_{\text{\scriptsize min}}$ such that
$(s_\mu^{(1)})^2 + (s_\mu^{(2)})^2 =1$ for all
$\mu=1,\ldots,N$ then the inequality (\ref{C1}) shows that we
have found a coplanar ground state
\begin{equation}\label{C2}
\vec{s}_\mu =\left(\begin{array}{c}s_\mu^{(1)}\\
s_\mu^{(2)}\end{array}\right),\;
\mu=1,\ldots,N
\end{equation}
with ground state energy
$-E_0=\frac{1}{2} N j_{\text{\scriptsize min}}$.
Analogously we can argue for
three linearly independent eigenvectors which
yield a $3$-dimensional ground state of $H$. \\

To find the Ising ground state $(s_\mu)_{\mu=1,\ldots,N}$, the
simplest method would be to check all $2^N$ Ising spin
configurations. Since we can choose, say, $s_1=1$ without loss of
generality, it would suffice to check $2^{N-1}$ states. But also
this can be a forbidding large number if $N$ is not too small.
Assume, for example, that we have $N=32$ spins as in section
\ref{sec:E3} and that the calculation of the Ising energy of a
single state and the comparison with the minimum previously
obtained requires approximately $0.01$ seconds for a program on a
desktop computer. Then the total number of $2^{31}$ calculations
would already last longer than eight months. There exist
sophisticated methods to find Ising states which represent a
local, rather low energy minimum, see e.~g.~\cite{BP:2001}, but
for these methods we cannot be sure that we have found the global
minimum. In our case of weakly bi-partite systems the following
simplification is possible: It suffices to check all Ising states
of a subsystem, say, $s_\mu,\;\mu\in{\mathcal A}$. If the
${\mathcal A}$-spins are fixed, the remaining spins
$s_\nu,\;\nu\in{\mathcal B}$ are calculated according to
\begin{equation}\label{C3}
s_\nu =-\text{sign} \left( \sum_{\mu\in{\mathcal A}} J_{\mu\nu} s_\mu \right)
\text{ for all } \nu\in{\mathcal B}
\;.
\end{equation}
The sign according to (\ref{C3}) minimizes the energy of the
interaction of the $\nu$-th Ising spin with its neighbors and
hence must be assumed for the total Ising ground state. This
simplification reduces in our above example $2^{31}$ calculations
to $2^{15}$ ones and thus the time for the total
calculation from months to minutes.\\

An alternative method to calculate the exact Ising ground state is
the so-called``branch and bound" method, see \cite{KH:1978}.
According to this method the ground state problem is viewed as the
problem of finding a minimum (or maximum) of a function defined on
the binary tree given by the possible signs of the individual
Ising spins, such that good trial state energy or ``bound" for the
minimum is available. When scanning through the various
possibilities of the tree, one can ``cut" those branches of the
tree which will never reach the bound, even if the most optimistic
expectation for the remaining energies is adopted. This results in
a considerable reduction of the calculation time for finding the
absolute minimum of the energy, see \cite{KH:1978}.

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\section{Generating generalized Bell inequalities\label{sec:B}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we describe a recipe how to construct
new GBI's and provide a test whether they are
violated by the singlet state in QT. We proceed by
constructing frustrated spin systems.\\

\begin{itemize}
\item
We start by choosing an integer $N$ and a partition
of $\{1,\ldots,N\}$ into two disjoint finite sets
${\mathcal A}$ and ${\mathcal B}$, not necessarily
with the same number of elements.
These sets correspond to the bipartition of the spin system
or to possible measurements performed by Alice and Bob.

\item Then we choose some real coefficients
$J_{\mu\nu},\;\mu\in{\mathcal A},\;\nu\in{\mathcal B}$.
They can be arbitrary but it is not advisable
to choose all coefficients with the same sign since we
are seeking for frustrated spin systems.


\item Next we find an Ising ground state, i.~e.~a sequence
$s_\mu=\pm 1, \; \mu=1,\ldots,N$ minimizing
the energy
$H^{(I)}=\sum_{\mu\in{\mathcal A}}\sum_{\nu\in{\mathcal B}}
J_{\mu\nu}\,s_\mu\, s_\nu$.
This can be done
by using the procedure described in the previous
section \ref{sec:C}. Let $-E_0^{(I)}$ denote the Ising
ground state energy. We thus obtain a GBI of the form
\begin{equation}\label{B1}
-E_0^{(I)} \le \sum_{\mu\in{\mathcal A}}
\sum_{\nu\in{\mathcal B}}J_{\mu\nu}\,
\langle s_\mu \,s_\nu \rangle \le E_0^{(I)}
\;.
\end{equation}
\item We perform a spin flip transformation
(\ref{E11a}),(\ref{E11b}) such that the Ising ground state becomes
$\uparrow\uparrow\ldots\uparrow$. Denote the transformed
coefficients again by $J_{\mu\nu}$.

\item We define a symmetric matrix $\mathbb{J}$ with
$J_{\mu\nu}=J_{\nu\mu}$ as non-diagonal elements.
The diagonal elements of $\mathbb{J}$ are chosen in such a way
that $\mathbb{J}$ will have constant row sums $j$ and vanishing
trace, see \cite{SM:2003}.
This leaves the Hamiltonian (\ref{G10}) unchanged.
Consequently, the Ising ground state $(1,1,\ldots,1)$ will be an
eigenvector of $\mathbb{J}$ with eigenvalue $j$.

\item We calculate the lowest eigenvalue
$j_{\text{\scriptsize min}}$ of
$\mathbb{J}$. If $j_{\text{\scriptsize min}}=j$ the Ising ground
state is already the Heisenberg ground state of the spin system
and our GBI will not be violated in QT. We have to start the
search anew. If, however, $j_{\text{\scriptsize min}}<j$ we are
done: We have found a GBI which is violated by the singlet state
in QT.
\end{itemize}

The single steps of this recipe are more or less obvious
except the last one. We know from (\ref{C1a}) that
$j_{\text{\scriptsize min}}<j$ is a necessary
condition for a spin system to have a ground state
energy $-E_0<-E_0^{(I)}$, but is it also sufficient? \\

In order to prove this we choose the fully polarized
Ising ground state to point into the $3$-direction
of our coordinate frame. Let $(x_\mu)_{\mu=1\ldots,N}$
denote an eigenstate of $\mathbb{J}$
with eigenvalue $j_{\text{\scriptsize min}}$.
Since, by assumption, $j_{\text{\scriptsize min}}<j$
this eigenstate is orthogonal to the Ising ground state,
i.~e.~$\sum_{\mu=1}^N x_\mu=0$.
Next we consider a smooth curve in the state space of
our spin system, i.~e.~a set of unit vector functions
$t\mapsto \vec{s}_\mu(t),\;\mu=1\ldots,N$ satisfying
\begin{equation}\label{B2}
\vec{s}_\mu(0)=
\left( \begin{array}{l}0\\0\\1\end{array}\right)
\text{  and    }
\frac{d}{d\,t}\vec{s}_\mu(0)=\left(
\begin{array}{l}x_\mu\\0\\0\end{array}
\right),\;\mu=1\ldots,N
\;.
\end{equation}
The Taylor expansion at $t=0$ of the Hamiltonian
evaluated at the states of this curve, $H(t)$, yields,
after a straightforward calculation,
using $\sum_{\mu=1}^N x_\mu=0$ and
$\frac{d^2}{d\,t^2}\vec{s}_\mu(0)\cdot\vec{s}_\mu(0)+
\frac{d}{d\,t}\vec{s}_\mu(0)\cdot\frac{d}{d\,t}
\vec{s}_\mu(0) =0$,
\begin{equation}\label{B3}
H(t)=H(0) + \frac{t^2}{2}(j_{\text{\scriptsize min}}-j)
\sum_{\mu=1}^N x_\mu^2 +{\mathcal O}(t^3)
\;.
\end{equation}
Hence $H(t)<H(0)$ for sufficiently small $t$ and
$-E_0^{(I)}$ cannot be the ground state energy.
This concludes the proof of the above statement.\\
Note, that we didn't prove that the energy
$\frac{N}{2}j_{\text{\scriptsize min}}$ is assumed by
some ground state. In general, this will not be the case.
The proof only shows that
$\frac{N}{2}j_{\text{\scriptsize min}}\le -E_0 <-E_0^{(I)}=\frac{N}{2}j$.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Examples\label{sec:E}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

According to section \ref{sec:B} arbitrary many
GBI's can be constructed. Nevertheless,
it will be instructive to show how known
examples of GBI's fit into our scheme and to
consider further examples for which the
ground states can be calculated in closed form.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The frustrated $2n$-ring\label{sec:E1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As a generalization of the frustrated Bell square
with $N=4$ we consider a spin ring with an even
number $N=2n$ of spins, cyclic boundary condition
$N+1\equiv 1$ and nearest neighbor interaction.
Let us for a moment also consider the bi-partite
AF system with $J=1$. It has a ground state
$\uparrow \downarrow\uparrow \downarrow\ldots\downarrow$
which is also an Ising ground state.
The energy is unchanged by a spin flip transformation
\begin{eqnarray}\label{E11a}
\vec{s}_\mu &\mapsto& \delta_\mu \vec{s}_\mu\\ \label{E11b}
J_{\mu\nu} &\mapsto& \delta_\mu \delta_\nu J_{\mu\nu}
\;,
\end{eqnarray}
where $ \mu,\nu=1,\ldots,N$ and $\delta_\mu=\pm 1$.
Hence the transformed ground state will be the
ground state of the transformed Hamiltonian.
The spin flip transformation does not alter
the number of $-1$-signs in the ring mod $2$:
After the transformation there exists
an even number of $-1$-signs, compared to zero $-1$-signs
before the transformation.
Obviously, any distribution of an even number of $-1$-signs
in the ring can be obtained by a suitable spin flip transformation
starting from the the bi-partite AF system with $J=1$.
Similarly, all distributions of an odd number of $-1$-signs
are equivalent up to spin flip transformations.
Hence it will be enough to consider only one of these,
say, the ring with one negative coupling constant
between $N$ and $1$:
\begin{equation}\label{E12}
J_{\mu\nu}=
\left\{
\begin{array}{rcl}
1 & \text{if} &\nu=\mu+1 \text{ and } \mu\neq N\\
-1 &\text{if} & \mu=N \text{ and } \nu=1\\
0 & \text{else}  &
\end{array}
\right.
\end{equation}

This system will be called the frustrated $2n$-ring.
Its Ising energy satisfies
\begin{equation}\label{E13}
|s_1 s_2 + s_2 s_3 + \ldots s_{N-1} s_N - s_N s_1 | \le N-2
\;,
\end{equation}
since not all terms in (\ref{E13}) can be positive.
The Ising ground state energy $-E_0^{(I)}=-N+2$
is realized by the alternating state
$\uparrow\downarrow\uparrow\ldots\downarrow$.\\
The Heisenberg ground state energy $-E_0$ is strictly lower.
It is assumed by any spin configuration
satisfying
\begin{equation}\label{E14}
\sphericalangle(\vec{s}_\mu,\vec{s}_{\mu+1})=
\pi\frac{N-1}{N},\; \mu=1,\ldots,N-1
\;.
\end{equation}
It follows that
\begin{equation}\label{E15}
\sphericalangle(\vec{s}_1,\vec{s}_{N})=
\pi\frac{(N-1)^2}{N}=\frac{\pi}{N}\mod 2\pi
\;,
\end{equation}
since $N$ is even. Hence the energy of this configuration is
\begin{eqnarray}\label{E16a}
H_0
&=&
\sum_{\mu=1}^{N-1}\vec{s}_\mu\cdot\vec{s}_{\mu+1}\;
- \vec{s}_N\cdot\vec{s}_1\\ \label{E16b}
&=&
(N-1) \cos\left( \pi\frac{N-1}{N} \right)\;
-\cos\frac{\pi}{N}\\ \label{E16c}
&=&
-N\cos\frac{\pi}{N}=-N+\frac{\pi^2}{2N}-
\frac{\pi^4}{24N^3}\pm\ldots\\ \label{E16d}
&<& 2-N\text{ for }N\ge 4\;.
\end{eqnarray}
According to the remarks in section \ref{sec:C}
it is possible to prove that (\ref{E14}) is a ground state
by calculating the lowest eigenvalue of the matrix
$\mathbb{J}$ corresponding to the coupling constants (\ref{E12}).
To this end we define the orthogonal $N\times N$-matrix
$\mathbb{U}$ by
\begin{equation}\label{E17}
U_{\mu\nu}=
\left\{
\begin{array}{rcl}
1 & \text{if} &\nu=\mu+1 \text{ and } \mu\neq N\\
-1 &\text{if} & \mu=N \text{ and } \nu=1\\
0 & \text{else}  &
\end{array}
\right.
\end{equation}
It follows that $\mathbb{J}=\mathbb{U}+\mathbb{U}^\ast$
and $[\mathbb{J},\mathbb{U}]=0$,
hence the eigenvalues of $\mathbb{J}$ are obtained as
twice the real part of the eigenvalues of $\mathbb{U}$.
The latter are of the form
\begin{equation}\label{E18}
u_k=\exp\left( \frac{i\,\pi\,k}{N}\right),\;
k=1,3,\ldots,2N-1
\;,
\end{equation}
since $\mathbb{U}^N=-\mathbb{I}_N$.
Hence the eigenvalues of $\mathbb{J}$ are
\begin{equation}\label{E19}
j_k=2\cos\left( \frac{\pi\,k}{N}\right),\;
k=1,3,\ldots,2N-1
\;,
\end{equation}
and the lowest eigenvalue $j_{\text{\scriptsize min}}$ is
obtained for $k=N-1$. By (\ref{C1})
this yields the bound $H\ge \frac{N}{2}j_{\text{\scriptsize min}}=
-N\cos\frac{\pi}{N}$,
which confirms that (\ref{E16c}) is the minimal
energy of the frustrated $2n$-ring.\\
The GBI (\ref{E13}) has been found by P.~M.~Pearle \cite{P:1970}
and its violation (\ref{E16d}) by quantum correlations
has been discussed by S.~L.~Braunstein and C.~M.~Caves
\cite{BC:1990} without proving the maximal
violation for the configuration (\ref{E14}).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The frustrated hexagon\label{sec:E2}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Frustrated even rings are not the only AB-systems. We consider as another example
the frustrated hexagon with additional interactions
between even and odd spin sites, see figure \ref{fig01}.
Its coupling constants are chosen as follows:
\begin{eqnarray}\label{E21a}
J_{12}
&=&
J_{14}=J_{25}=-1\\ \label{E21b}
J_{23}
&=&
J_{34}=J_{45}=J_{56}=J_{16}=J_{36}=1
\;.
\end{eqnarray}
The corresponding GBI reads
\begin{equation}\label{E22}
-5\le \langle s_2 s_3\rangle + \langle s_3 s_4\rangle + \langle
s_4 s_5\rangle + \langle s_5 s_6\rangle + \langle s_6 s_1\rangle +\langle
s_3 s_6\rangle -\langle s_1 s_2\rangle -\langle s_1 s_4\rangle
-\langle s_2 s_5\rangle \le 5 \;.
\end{equation}
corresponding to the Ising ground state
$\uparrow\uparrow\uparrow\downarrow\uparrow\downarrow$.
The spin configuration with
\begin{equation}\label{E23}
\vec{s}_1=\vec{s}_2,\;
\sphericalangle(\vec{s}_6,\vec{s}_4)=
\sphericalangle(\vec{s}_4,\vec{s}_2)=
\sphericalangle(\vec{s}_2,\vec{s}_5)=
\sphericalangle(\vec{s}_5,\vec{s}_3)=60^\circ
\;,
\end{equation}
see figure \ref{fig02}, realizes the Heisenberg ground state with
energy $-E_0=-6$ corresponding to a violation of (\ref{E22}) in
QT. Since this result can be confirmed by using the same methods
as in section \ref{sec:E1}, we leave the details to the reader.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Frustrated hypercubes\label{sec:E3}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Another way to generalize the Bell square is to
consider ``frustrated hypercubes" $H_n$ with
$N=2^n$ vertices and $n=1,2,3,\ldots$. The $N$
spins are located at the vertices and are
interacting along the $n 2^{n-1}$ edges of the
hypercube.
Sometimes, the vertices $v_\nu,\;\nu=0,\ldots,N-1$
are most conveniently labelled by binary strings
$(\delta_1,\ldots,\delta_n)$
of length $n$, such that the $\delta_i\in\{0,1\}$
are the binary digits of $\nu$.
Two strings $v,w$
which differ exactly at one position form an edge
$e=\{v,w\}$.
The set of vertices can be divided into the set
${\mathcal A}$ of binary strings with
an even number of ones and the set ${\mathcal B}$
of binary strings with
an odd number of ones. Thus every edge connects an
${\mathcal A}$-vertex with
a ${\mathcal B}$-vertex and the corresponding
spin system is an AB-system.
The sign attached to an edge $e=\{v,w\}$
can be defined as $(-1)^\ell$ where $\ell$ is
the number of ones at the left hand side
of the position where $v$ and $w$ differ.\\

Alternatively, the signs can be
defined recursively according to the following
procedure.\\
If the signs of $H_n$ are already defined, denote by ${H_n^\ast}$
the same hypercube with inverted signs. Define $H_{n+1}$ as the
union of $H_n$ and ${H_n^\ast}$ where $N$ new edges between the
corresponding vertices of $H_n$ and ${H_n^\ast}$ are
added and $+1$-signs are attached to them.\\
For example, if we start with $H_1=$
\unitlength1mm\begin{picture}(17,6)
\put(2,1.5){\circle{3}}\put(14,1.5)
{\circle{3}}\put(3.5,1.5){\line(1,0){9}}\put(7,3){+}
\end{picture}
and apply this procedure we obtain
\vspace{-1cm}
\begin{equation}\label{E31}
H_2=
\unitlength1mm\begin{picture}(20,20)(-4,7)
\put(2,1.5){\circle{3}}\put(14,1.5){\circle{3}}
\put(3.5,1.5){\line(1,0){9}}\put(7,-1){--}
\put(2,13.5){\circle{3}}\put(14,13.5){\circle{3}}
\put(3.5,13.5){\line(1,0){9}}\put(7,15){+}
\put(2,3){\line(0,1){8.5}}\put(-1,6.5){+}
\put(14,3){\line(0,1){8.5}}\put(14,6.5){+}
\end{picture}
\end{equation}
\vspace{5mm}
\\
which is just the Bell square considered before. See figure
\ref{fig03} for the frustrated hypercubes $H_3$ and $H_4$ obtained
by this procedure. Of course, one can apply symmetry
transformations of $H_n$ (suitable rotations and reflections) and
spin flip transformations to obtain modified frustrated
hypercubes. But these will have analogous properties as the $H_n$
and need not be considered further.\\
The recursive procedure $H_n\rightarrow
H_{n+1}$ can also be applied to the $\mathbb{J}$-matrices.
Indeed, $\mathbb{J}_n$ can be recursively defined by
\begin{eqnarray}\label{E32a}
\mathbb{J}_0
&=&
(0)\\ \label{E32b}
\mathbb{J}_{n+1}
&=&
\left(\begin{array}{rr}\mathbb{J}_{n} & \mathbb{I}_{N}\\
\mathbb{I}_{N}&-\mathbb{J}_{n}\end{array}
\right)
\;,
\end{eqnarray}
where $\mathbb{I}_{N}$ denotes the $N\times N$
identity matrix.
By induction we conclude $\mathbb{J}_{n}^2=
n \mathbb{I}_{N}$ and hence the eigenvalues of
$\mathbb{J}_{n}$ are $\pm\sqrt{n}$ with
degeneracies $\frac{N}{2}$. If these eigenvalues
can be realized by spin vector configurations
the corresponding ground state energy
$-E_0^{(n)}=-\frac{N}{2}\sqrt{n}$ must be lower
than the Ising ground state energy $-E_0^{(n,I)}$
which is always an integer. Possible exceptions
are $n=4,9,16,\ldots$ where $E_0^{(n)}$ is an integer
too.\\

We cannot present a general theorem which warrants that the ground
state energy $-E_0^{(n)}=-\frac{N}{2}\sqrt{n}$ will be assumed by
a coplanar state for all $n\ge 2$. Rather we have studied the
cases $n=3,4,5$ in some detail with the following results.\\
For $n=3$ and $n=5$ we have explicitly constructed coplanar spin
configurations realizing the ground state energy
$-E_0^{(n)}=-\frac{N}{2}\sqrt{n}$ which is below the Ising ground
state energy. In these cases we therefore obtain GBI's which are
violated in QT. For $n=4$ the ground state energy $-E_0^{(4)}=-16$
is realized by an Ising state. Hence we can also in this case
write down a GBI, but it is also satisfied in QT and hence of no
use in the context of the EPR discussion. This GBI is also not
violated for other entangles states, see section \ref{sec:S}.\\

The frustrated hypercubes $H_n$ have an obvious reflection
symmetry $\sigma$ defined by
$\nu\stackrel{\sigma}{\longleftrightarrow} \nu+1, \nu \text{
even}$. We thus have $J_{\sigma(\mu)\sigma(\nu)}=J_{\mu\nu}$. This
can by proven by induction over $n$ using the recursive
construction procedure $H_n\rightarrow H_{n+1}$. Hence the
eigenvectors of $\mathbb{J}$ can be chosen either to be invariant
under $\sigma$ or to change their sign. Recall that the $x$- and
the $y$-components of the coplanar spin configuration realizing
the ground state energy $-E_0^{(n)}=-\frac{N}{2}\sqrt{n}$ are
eigenvectors of $\mathbb{J}$. It turns out that we can always find
ground state configurations
\begin{equation}\label{E33}
\vec{s}_\mu={x_\mu \choose y_\mu} \text{  satisfying   }
x_{\sigma(\mu)}=x_\mu\text{ and }y_{\sigma(\mu)}=-y_\mu \text{ for
}{\mu=1,\ldots,N} \;.
\end{equation}
Hence it will suffice to write down the ground state
configurations $\vec{s}_\mu$ only for even $\mu$.
\\

All components of $\vec{s}_\mu$ can be written as radicals
with a similar structure
consisting of nested square roots and small integers.
In order to simplify table \ref{tab1} containing the
ground state configurations we hence introduce
the following abbreviations:
\begin{eqnarray}\label{E34a}
n=2&:&
V_2(\delta_1,\delta_2,\delta_3)
\equiv
\frac{1}{2}\left(
\begin{array}{l}\delta_1\sqrt{2+\delta_3 \sqrt{2}}\\
\delta_2\sqrt{2-\delta_3 \sqrt{2}}
\end{array}
\right)
\\ \label{E34b}
n=3&:&
V_3(\delta_1,\delta_2,\delta_3,\delta_4)
\equiv
\frac{1}{2}\left(
\begin{array}{l}\delta_1\sqrt{2+\delta_3 \sqrt{\frac{2}{3}}+\delta_4\frac{2}{\sqrt{3}}}\\
\delta_2\sqrt{2-\delta_3 \sqrt{\frac{2}{3}}-\delta_4\frac{2}{\sqrt{3}}}
\end{array}
\right)
\\ \label{E34c}
n=4&:&
V_4(\delta_1,\delta_2)
\equiv
\left(
\begin{array}{l}\delta_1\\
\delta_2
\end{array}
\right)
\\ \label{E34d}
n=5&:&
V_5(\delta_1,\delta_2,\delta_3)
\equiv
\left(
\begin{array}{l}\delta_1\sqrt{\frac{1}{2}+\frac{1}{\delta_3 \sqrt{5}}}\\
\delta_2\sqrt{\frac{1}{2}-\frac{1}{\delta_3 \sqrt{5}}}
\end{array}
\right)
\;.
\end{eqnarray}

The coplanar ground states for some frustrated hypercubes $H_n$
are visualized in the figures \ref{fig04a} ($n=2$), \ref{fig04} ($n=3$) and
\ref{fig05} ($n=5$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{General entangled states\label{sec:S}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the correspondence between GBI's and frustrated AB-systems
outlined in section \ref{sec:G} we have always chosen
the singlet state
$\phi=\frac{1}{\sqrt{2}}\left( |\,\uparrow\downarrow\,\rangle -
|\,\downarrow\uparrow\,\rangle\right)$
as the state relative to which the correlations
$\langle AB \rangle$ etc. have been calculated.
The rotational invariance of $\phi$ then directly
corresponds to the rotational invariance
of the Heisenberg Hamiltonian (\ref{G10}).
This picture changes if other entangled states
$\psi\in{\mathcal H}\equiv \mathbb{C}^2\otimes \mathbb{C}^2$
are taken into account.\\

Such states can be written as a Schmidt bi-orthogonal sum
\begin{equation}\label{S1}
\psi=\sum_{i=1}^2 c_i \mathbf{u}_i\otimes\mathbf{v}_i
\;,
\end{equation}
where $\{\mathbf{u}_i\}_{i=1,2}$ and $\{\mathbf{v}_i\}_{i=1,2}$
are orthonormal bases in $\mathbb{C}^2$,
and $|c_1|^2+|c_2|^2=1$, see \cite{AP:1993}, section 5.~3.
$\psi$ is an entangled state if and only if $c_1, c_2\neq 0$.
Upon choosing appropriate, possibly different,
coordinate frames for Alice and Bob, (\ref{S1}) assumes
the form
\begin{equation}\label{S2}
\psi=\cos\alpha |\uparrow\downarrow\rangle-\sin\alpha |\downarrow\uparrow\rangle
\;,
\end{equation}
where $0<\alpha <\frac{\pi}{2}$.
$\alpha=\frac{\pi}{4}$ corresponds to the singlet state $\phi$.
To put this in another way, $\psi$ can be written in the form
(\ref{S2}) by means of a unitary
transformation in ${\mathcal H}$.\\

After a straightforward calculation we obtain for the correlation of
observables of the form (\ref{G5})
\begin{eqnarray}\label{S3a}
\langle AB \rangle
&\equiv&
\langle\psi | A\otimes B |\psi\rangle\\ \label{S3b}
&=&
-a_3 b_3-\sin(2\alpha) (a_1 b_1 +a_2 b_2)\\ \label{S3c}
&\equiv&
-a_3 b_3-\gamma (a_1 b_1 +a_2 b_2)
\;.
\end{eqnarray}
The corresponding classical spin Hamiltonian for a general
frustrated AB-system reads
\begin{equation}\label{S4}
H(\gamma)=\sum_{\mu\in{\mathcal A}}\sum_{\nu\in{\mathcal B}} J_{\mu\nu}
\left(z_\mu z_\nu +\gamma (x_\mu x_\nu + y_\mu y_\nu )\right)
\;,
\end{equation}
where we have written
\begin{equation}\label{S5}
\vec{s}_\mu\equiv \left(\begin{array}{l}x_\mu\\y_\mu\\z_\mu\end{array}\right)
\text{ for }\mu=1,\ldots,N
\;.
\end{equation}
Such anisotropic spin Hamiltonians are well-known under the name
``XXZ-model", see, for example \cite{FB:2004}. For
$\gamma\rightarrow\infty$ the XXZ-model essentially approaches the
Ising model, which means that, in the context of GBI's,
factorable states will not violate the GBI.\\
It has been proven \cite{GP:1992},\cite{AP:1993} that the CHSH
inequality is also violated for an arbitrary entangled state
$\psi$, even for a general Hilbert space ${\mathcal H}$. This
means, in the language of spin systems, that the XXZ Bell square
has a coplanar ground state for all $\gamma$ with $0<\gamma\le 1$.
Unfortunately, we have not found a proof of the analogous
statement for the class of GBI's considered in this article. We only
remark that, obviously, the GBI is violated for all $\gamma$ with
$E_0^{(I)}/E_0<\gamma<1$, if $E_0^{(I)}<E_0$
and the GBI is violated for $\gamma=1$ with a coplanar ground
state. Indeed, if we evaluate the Hamiltonian $H(\gamma)$ at the
coplanar ground state $(\vec{s}_\mu)_{\mu=1,\ldots,N}$ we obtain
\begin{eqnarray}\label{S6a}
H(\gamma)
&=&
\sum_{\mu\in{\mathcal A}}\sum_{\nu\in{\mathcal B}} J_{\mu\nu}
(\underbrace{z_\mu z_\nu}_{=0} +\gamma (x_\mu x_\nu + y_\mu y_\nu ))
\\ \label{S6b}
&=&
\gamma H(1) =-\gamma E_0< - E_0^{(I)}
\;.
\end{eqnarray}

We now consider the case that the Ising ground state is an
eigenstate of $\mathbb{J}$
corresponding to the lowest eigenvalue
$j_{\text{\scriptsize min}}$, and hence
the GBI is {\emph not} violated for the singlet state $\phi$.
This happens
in the example of the frustrated hypercube $H_4$, see table \ref{tab1}.
Then the GBI will
also not be violated for other entangled states $\psi$, i.~e.~,
the Ising ground state will
remain the ground state for all $H(\gamma)$.
To show this we consider
\begin{eqnarray}\label{S7a}
2H(\gamma)
&=&
\sum_{\mu,\nu=1}^N J_{\mu\nu}
(z_\mu z_\nu+\gamma (x_\mu x_\nu + y_\mu y_\nu ))\\ \label{S7b}
&\ge&
j_{\text{\scriptsize min}}(
\underbrace{\sum_{\mu=1}^N z_\mu^2}_{\zeta} +\gamma
\underbrace{\sum_{\mu=1}^N(x_\mu^2 +y_\mu^2)}_{\xi}) \\ \label{S7c}
&\equiv&
j_{\text{\scriptsize min}}(\zeta +\gamma\xi)\ge N j_{\text{\scriptsize min}}
\;,
\end{eqnarray}
since $j_{\text{\scriptsize min}}<0$ and
$\zeta +\gamma\xi$ assumes its maximum under the constraint
$\zeta +\xi=N$ for $\zeta=N$.
The minimum (\ref{S7c}) is assumed by inserting the Ising
ground state
for the $z_\mu$ and setting $x_\mu=y_\mu=0$.\\


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion\label{sec:CC}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
What are the benefits of the proposed correspondence
between the considered class of GBI's and
frustrated AB-systems?\\

For readers which are mainly interested in the EPR discussion and
Bell inequalities the most interesting result might be the recipe
to obtain an arbitrary number of GBI's, see section \ref{sec:B}.
This construction procedure should give us some additional insight
into the structure of GBI's. Moreover, we hope that the proposed
correspondence would lead the reader to think of GBI's in a more
geometric or graphical way: The possible measurements can be
visualized as vertices of a spin system and the signed
correlations occurring in the GBI as the edges or bonds of this
system. As we have shown, some methods from the theory of spin
systems can be employed to tackle questions in the realm of the
foundations of QT.\\

Also for readers which are mainly interested in frustrated spin
systems, the unexpected connection to GBI's might be valuable in its
own right. For problems in the theory of spin systems a transfer
of methods from another field of research will be of some
interest. Moreover, those readers will probably find the class of
frustrated hypercubes, section \ref{sec:E3}, to be a useful set of
toy examples worth while to be further studied. For example, in
the quantum XY-model of frustrated hypercubes there exist
so-called localized multi-magnon states which lead to prominent
properties as huge magnetization jumps and large residual entropy
at $T=0$ resulting in a marked magneto-caloric effect, see, for
example \cite{RSH:2004}, \cite{JS:2004}, and \cite{SR:2006}. \\
Summarizing, the correspondence between GBI's and frustrated spin
systems fosters the insight into both branches of physics and
leads to some new phenomena in non-geometric frustration.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

I thank Klaus B\"arwinkel, Marshall Luban, Johannes Richter, and J\"urgen Schnack
for a critical reading of the manuscript.

\begin{thebibliography}{99}

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correspondence: hschmidt\char'100uos.de


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A. Einstein, B. Podolski, and N. Rosen,
Can quantum-mechanical description be
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(1935).

\bibitem{BJ:1987}
L.~E.~Ballantine and J.~P.~Jarrett, Bell's theorem:
Does quantum mechanics contradict
relativity?, Am. J. Phys. \textbf{55} (8), 696--701 (1987)



\bibitem{CHSH:1969}
J.~F. Clauser, M.~A. Horne, A. Shimony, and R.~A. Holt,
Proposed experiment to test local hidden-variable theories,
Phys. Rev. Lett. \textbf{23}, 880--884 (1969).

\bibitem{P:1970}
P.~M.~Pearle, Hidden-variable example based upon data rejection,
Phys. Rev. D \textbf{8}, 1418--1425 (1970)



\bibitem{BC:1990}
S.~L. Braunstein and C.~M. Caves,
Wringing out better Bell inequalities, Ann. Phys. (NY)
\textbf{202}, 22--56 (1990).

\bibitem{GS:1979}
A.~Garuccio and F.~Selleri, Systematic derivation of all inequalities of Einstein locality,
Found. Phys.\textbf{10}, 209--216, (1979)

\bibitem{AP:1993}
A. Peres, \emph{Quantum Theory: Concepts and Methods},
Kluwer (1993).


\bibitem{AP:1978}
A.~Peres, Unperformed experiments have no results,
Am. J. Phys. \textbf{46}, 745--747 (1978)

\bibitem{MR:2006}
R.~Moessner and A.~Ramirez, Geometrical Frustration,
Physics Today, \textbf{59} (2), 24--29 (2006)

\bibitem{SM:2003}
H.~-~J.~Schmidt and M.~Luban, Classical ground states
of symmetric Heisenberg spin systems,
J. Phys. A:~Math.~Gen.~\textbf{36}, 63514--6378 (2003)

\bibitem{BP:2001}
S.~Boettcher and A.~G.~Percus, Optimization with extremal dynamics,
Phys. Rev. Lett. \textbf{86}, 5211--5214 (2001)


\bibitem{KH:1978}
S.~Kobe and A.~Hartwig, Exact ground states of finite amorphous Ising systems,
Comp. Phys. Commun. \textbf{16} 1, 1--4  (1978)

\bibitem{GP:1992}
N.~Gisin, A.~Peres,
Maximal violation of Bell inequality for arbitrarily large spin,
Phys.Letters A  \textbf{162}, 15--17 (1992)

\bibitem{FB:2004}
D.~J.~J.~Farnell and R.~F.~Bishop, chapter 7 in \cite{SRFB:2004}

\bibitem{RSH:2004}
J.~Richter, J.~Schulenburg, and A.~Honecker,
chapter 2 in \cite{SRFB:2004}

\bibitem{JS:2004}
J.~Schnack, chapter 3 in \cite{SRFB:2004}

\bibitem{SRFB:2004}
U.~Schollw\"ock, J.~Richter, D.~J.~J.~Farnell, and R.~F.~Bishop (Eds.~),
\emph{Quantum Magnetism}, Lecture Notes in Physics 645, Springer (2004)

\bibitem{SR:2006}
H.~-~J.~Schmidt, J.~Richter, Frustrated XY-hypercubes, in preparation

\end{thebibliography}

%-------------------------table-------------------------------
\begin{table}
\caption{\label{tab2}Correspondence between GBI's and
classical spin systems.}
\vspace{5mm} \item[]\begin{tabular}{ll} \hline
EPR experiment & Classical spin system\\
\hline
$N$ possible measurements & $N$ spins \\
$2$-particle states & bi-linear Hamiltonian \\
isotropic singlet state & isotropic Heisenberg Hamiltonian $H(\mathbf{s})$\\
general entangled state & XXZ-Hamiltonian, section \ref{sec:S} \\
$2$ experimenters (Alice and Bob) & AB-systems\\
GBI & inequality for Ising states $s$ \\
$|\sum_{\mu\in{\mathcal A}}\sum_{\nu\in{\mathcal B}}J_{\mu\nu}
\langle s_\mu s_\nu \rangle|\le E_0^{(I)}$
&
$|H^{(I)}(s)|\le E_0^{(I)}$\\
Violation of the GBI for quantum correlations
&
non-collinear ground state state $\mathbf{s}$ \\
$\sum_{\mu\in{\mathcal A}}\sum_{\nu\in{\mathcal B}}J_{\mu\nu}
\vec{s}_\mu \cdot \vec{s}_\nu = - E_0$
&
$H(\mathbf{s})=- E_0$\\
such that $E_0^{(I)}< E_0$
&
such that $E_0^{(I)}< E_0$\\
\hline\hline
\end{tabular}
\end{table}
%-------------------------table-------------------------------

\begin{table}
\caption{\label{tab1}Results for the ground states of some frustrated hypercubes $H_n,\;n=2,3,4,5$.
The spin vectors $\vec{s}_\mu$ of the ground state are only given for $\mu=0,2,\ldots,N-2$; the other
spin vectors result from the symmetry (\ref{E33}). Moreover, we utilize the abbreviations
(\ref{E34a}), (\ref{E34b}), (\ref{E34c}), and (\ref{E34d}).
The Ising ground states are degenerate; the table contains only one example of an Ising ground state
for each $n=2,3,4,5$.
We observe that for $n=2,3,5$ the
Heisenberg ground state energy $-E_0^{(n)}$ is below the energy $-E_0^{(n,I)}$ of the Ising ground state.
This corresponds to a violation of the generalized Bell inequalities in quantum theory.
For $n=4$ both ground state energies coincide, hence the corresponding
generalized Bell inequality is \emph{not} violated in quantum theory.
}
\vspace{5mm}
\item[]\begin{tabular}{@{}llp{65mm}p{30mm}ll}
\hline
$n$ & $N=2^n$ &  Heisenberg ground state & $-E_0^{(n)}$ & Ising ground state & $-E_0^{(n,I)}$\\
\hline
$2$
&
$4$
&
$\vec{s}_\mu=V_2(\delta_1,\delta_2,\delta_3)$, where $(\delta_1,\delta_2,\delta_3)=$
&
$-2\sqrt{2}\approx -2.82843$
&
$\uparrow \downarrow\downarrow\uparrow $
&
$-2$\\
&& $(+ + -),(- - +)$
&&&\\
\hline $3$ & $8$ &
$\vec{s}_\mu=V_3(\delta_1,\delta_2,\delta_3,\delta_4)$, where
$(\delta_1,\delta_2,\delta_3,\delta_4)=$ & $-4\sqrt{3}\approx
-6.9282$ &
$\uparrow\uparrow\downarrow\downarrow\downarrow\downarrow\downarrow\uparrow$
&
$-6$\\
&& $(+ - + -),(- + - +),
(- - + +),(- - - -)$
&&&\\
\hline
$4$
&
$16$
&
$\vec{s}_\mu=V_4(\delta_1,\delta_2)$, where $(\delta_1,\delta_2)=$
&
$-8\sqrt{4}=-16$
&
$\downarrow\downarrow\uparrow\uparrow\uparrow\uparrow\downarrow\uparrow\uparrow
\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow$
&
$-16$\\
&& $(-1 0),(+1 0),(+1 0),(0 -1),$\newline $(+1 0),(0 +1),(0 -1),(0 +1)$
&&&\\
\hline
$5$
&
$32$
&
$\vec{s}_\mu=V_5(\delta_1,\delta_2,\delta_3)$, where $(\delta_1,\delta_2,\delta_3)=$
&
$-16\sqrt{5}\approx -35.7771$
&
$\uparrow\uparrow\downarrow\downarrow\downarrow
\downarrow\downarrow\downarrow\downarrow\downarrow
\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow
$
\newline
$\uparrow\uparrow\uparrow\uparrow\uparrow
\uparrow\uparrow\uparrow\uparrow\uparrow
\downarrow\downarrow\uparrow\uparrow\uparrow\uparrow
$
&
$-32$\\
&&
(+, --, -1), (--, --, 2),
(+, +, 2),  (+, +, -2), \newline
(+, +, -1),  (--, +, -1),
(--, +, -1),  (--, --, 2),  \newline
(--, +, 1), (+, +, -2),
(--, --, -2),  (+, +, 2),\newline
(--, --, 1), (+, --, 1),
(+, --, 1), (+, +, -2)
&&&\\
\hline\hline
\end{tabular}
\end{table}
%-------------------------table-------------------------------




%===================    figure   =================================
\begin{figure}
\begin{center}
\includegraphics[width=100mm]{Hexfig}
\caption{\label{fig01}A frustrated hexagon. The thick red lines indicate the edges with a
coupling constant of $-1$, the thin black lines correspond to the coupling $+1$.
}
\end{center}
\end{figure}

%===================    figure   =================================
\begin{figure}
\begin{center}
\includegraphics[width=100mm]{HexGSfig}
\caption{\label{fig02}A coplanar ground state of the frustrated hexagon.
The correlations of measurements of the
spin components according to these vectors in the singlet state violate
the inequality (\ref{E22}).
}
\end{center}
\end{figure}
%===================    figure   =================================

\begin{figure}
\begin{center}
\includegraphics[width=200mm]{SAfig}
\caption{\label{fig03}A sequence of frustrated hypercubes $H_n,\;n=2,3,4$ generated by a recursive procedure,
see section \ref{sec:E3}. The negative bonds are indicated by thick red lines.
}
\end{center}
\end{figure}

%===================    figure   =================================
\begin{figure}
\begin{center}
\includegraphics[width=100mm]{H2GSfig}
\caption{\label{fig04a}A coplanar ground state of the frustrated
Bell square $H_2$ according to table \ref{tab1}. Note the reflection symmetry of the
vectors with label $\nu$ and $\nu+1$, $\nu$ even.
The correlations of measurements of the spin components according to these vectors
in the singlet state violate the CHSH inequality.
}
\end{center}
\end{figure}
%===================    figure   =================================
\begin{figure}
\begin{center}
\includegraphics[width=100mm]{H3GSfig}
\caption{\label{fig04}A coplanar ground state of the frustrated
cube $H_3$ according to table \ref{tab1}. Note the reflection symmetry of the
vectors with label $\nu$ and $\nu+1$, $\nu$ even.
The correlations of measurements of the spin components according to these vectors
in the singlet state violate the corresponding generalized Bell inequality.
}
\end{center}
\end{figure}
%===================    figure   =================================
\begin{figure}
\begin{center}
\includegraphics[width=100mm]{H5GSfig}
\caption{\label{fig05}A coplanar ground state of the frustrated
hypercube $H_5$ according to table \ref{tab1}. Note the reflection symmetry of the
vectors with label $\nu$ and $\nu+1$, $\nu$ even.
The correlations of measurements of the spin components according to these vectors
in the singlet state violate the corresponding generalized Bell inequality.
}
\end{center}
\end{figure}
%===================    figure   =================================









\end{document}

cond-mat0604591.extracted_bib
\begin{thebibliography}{99}

\bibitem[*]{correspondence}
correspondence: hschmidt\char'100uos.de


\bibitem{Bell:1964}
J.~S. Bell, On the Einstein Podolski Rosen paradox, Physics
\textbf{1}, 195--200 (1964).


\bibitem{EPR:1935}
A. Einstein, B. Podolski, and N. Rosen,
Can quantum-mechanical description be
considered complete? Phys. Rev. \textbf{47}, 777--780
(1935).

\bibitem{BJ:1987}
L.~E.~Ballantine and J.~P.~Jarrett, Bell's theorem:
Does quantum mechanics contradict
relativity?, Am. J. Phys. \textbf{55} (8), 696--701 (1987)



\bibitem{CHSH:1969}
J.~F. Clauser, M.~A. Horne, A. Shimony, and R.~A. Holt,
Proposed experiment to test local hidden-variable theories,
Phys. Rev. Lett. \textbf{23}, 880--884 (1969).

\bibitem{P:1970}
P.~M.~Pearle, Hidden-variable example based upon data rejection,
Phys. Rev. D \textbf{8}, 1418--1425 (1970)



\bibitem{BC:1990}
S.~L. Braunstein and C.~M. Caves,
Wringing out better Bell inequalities, Ann. Phys. (NY)
\textbf{202}, 22--56 (1990).

\bibitem{GS:1979}
A.~Garuccio and F.~Selleri, Systematic derivation of all inequalities of Einstein locality,
Found. Phys.\textbf{10}, 209--216, (1979)

\bibitem{AP:1993}
A. Peres, \emph{Quantum Theory: Concepts and Methods},
Kluwer (1993).


\bibitem{AP:1978}
A.~Peres, Unperformed experiments have no results,
Am. J. Phys. \textbf{46}, 745--747 (1978)

\bibitem{MR:2006}
R.~Moessner and A.~Ramirez, Geometrical Frustration,
Physics Today, \textbf{59} (2), 24--29 (2006)

\bibitem{SM:2003}
H.~-~J.~Schmidt and M.~Luban, Classical ground states
of symmetric Heisenberg spin systems,
J. Phys. A:~Math.~Gen.~\textbf{36}, 63514--6378 (2003)

\bibitem{BP:2001}
S.~Boettcher and A.~G.~Percus, Optimization with extremal dynamics,
Phys. Rev. Lett. \textbf{86}, 5211--5214 (2001)


\bibitem{KH:1978}
S.~Kobe and A.~Hartwig, Exact ground states of finite amorphous Ising systems,
Comp. Phys. Commun. \textbf{16} 1, 1--4  (1978)

\bibitem{GP:1992}
N.~Gisin, A.~Peres,
Maximal violation of Bell inequality for arbitrarily large spin,
Phys.Letters A  \textbf{162}, 15--17 (1992)

\bibitem{FB:2004}
D.~J.~J.~Farnell and R.~F.~Bishop, chapter 7 in \cite{SRFB:2004}

\bibitem{RSH:2004}
J.~Richter, J.~Schulenburg, and A.~Honecker,
chapter 2 in \cite{SRFB:2004}

\bibitem{JS:2004}
J.~Schnack, chapter 3 in \cite{SRFB:2004}

\bibitem{SRFB:2004}
U.~Schollw\"ock, J.~Richter, D.~J.~J.~Farnell, and R.~F.~Bishop (Eds.~),
\emph{Quantum Magnetism}, Lecture Notes in Physics 645, Springer (2004)

\bibitem{SR:2006}
H.~-~J.~Schmidt, J.~Richter, Frustrated XY-hypercubes, in preparation

\end{thebibliography}