Multipartite minimum-uncertainty products

摘要

In a previous work [Shchukin, Phys. Rev. A 84, 052325 (2011)] a lower bound was found for the multipartitenuncertainty product of the position and momentum observables over all separable states. In this work we arentrying to minimize this uncertainty product over a broader class of states to find the fundamental limits imposednby nature on the observable quantities. We show that it is necessary to consider pure states only and find theninfimum of the uncertainty product over a special class of pure states (states with spherically symmetric wavenfunctions). It is shown that this infimum is not attained. We also explicitly construct a parametrized family ofnstates that approaches the infimum by varying the parameter. Since the constructed states beat the lower boundnfor separable states, they are entangled. We thus show that there is a gap that separates the values of a simplenmeasurable quantity for separable states from entangled ones and we also try to find the size of this gap.