  If X and Y are independent, Y and Z are independent, and so are X and Z, one
might be tempted to conclude that X, Y, and Z are independent. But it has long
been known in classical probability theory that, intuitive as it may seem, this
is not true in general. In quantum mechanics one can ask whether analogous
statistics can emerge for configurations of particles in certain types of
entangled states. The explicit construction of such states, along with the
specification of suitable sets of observables that have the purported
statistical properties, is not entirely straightforward. We show that an
example of such a configuration arises in the case of an N-particle GHZ state,
and we are able to identify a family of observables with the property that the
associated measurement outcomes are independent for any choice of 2, 3, ...,
N-1 of the particles, even though the measurement outcomes for all N particles
are not independent. Although such states are highly entangled, the
entanglement turns out to be 'fragile', i.e. the associated density matrix has
the property that if one traces out the freedom associated with even a single
particle, the resulting reduced density matrix is separable.
