Correlations in open quantum systems and associated uncertainty relations

We study how correlations in a state of an open quantum system affect the intrinsic uncertainties of the expectation values of an arbitrary pair of noncommuting observables. We show that for such observables, there exist Heisenberg-type uncertainty relations that take fully into account correlations in the state of the system. If the quantum system is in a pure state, such uncertainty relations reduce to the conventional one. We obtain an equation for the density operator of a general state that minimizes the new uncertainty relations, and demonstrate that in the important case of coordinate and momentum operators, the minimum-uncertainty states are displaced squeezed thermal states.