  We conjecture new uncertainty relations which restrict correlations between
results of measurements performed by two separated parties on a shared quantum
state. The first uncertainty relation bounds the sum of two mutual informations
when one party measures a single observable and the other party measures one of
two observables. The uncertainty relation does not follow from Maassen-Uffink
uncertainty relation and is much stronger than Hall uncertainty relation
derived from the latter. The second uncertainty relation bounds the sum of two
mutual informations when each party measures one of two observables. We provide
numerical evidence for validity of conjectured uncertainty relations and prove
them for large classes of states and observables.
