  In our previous work we have found a lower bound for the multipartite
uncertainty product of the position and momentum observables over all separable
states. In this work we are trying to minimize this uncertainty product over a
broader class of states to find the fundamental limits imposed by nature on the
observable quantites. We show that it is necessary to consider pure states only
and find the infimum of the uncertainty product over a special class of pure
states (states with spherically symmetric wave functions). It is shown that
this infimum is not attained. We also explicitly construct a parametrized
family of states that approaches the infimum by varying the parameter. Since
the constructed states beat the lower bound for separable states, they are
entangled. We thus show that there is a gap that separates the values of a
simple measurable quantity for separable states from entangled ones and we also
try to find the size of this gap.
