Uncertainty relations capture the essence of the inevitable randomness
associated with the outcomes of two incompatible quantum measurements.
Recently, Berta et al. have shown that the lower bound on the uncertainties of
the measurement outcomes depends on the correlations between the observed
system and an observer who possesses a quantum memory. If the system is
maximally entangled with its memory, the outcomes of two incompatible
measurements made on the system can be predicted precisely. Here, we obtain a
new uncertainty relation that tightens the lower bound of Berta et al., by
incorporating an additional term that depends on the quantum discord and the
classical correlations of the joint state of the observed system and the
quantum memory. We discuss several examples of states for which our new lower
bound is tighter than the bound of Berta et al. On the application side, we
discuss the relevance of our new inequality for the security of quantum key
distribution and show that it can be used to provide bounds on the distillable
common randomness and the entanglement of formation of bipartite quantum
states.Comment: v1: Latex, 4 and half pages, one fig; v2: 9 pages including 4-page
appendix; v3: accepted into Physical Review A with minor change
