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Uncertainty relations capture the essence of the inevitable randomnessassociated with the outcomes of two incompatible quantum measurements.Recently, Berta et al. have shown that the lower bound on the uncertainties ofthe measurement outcomes depends on the correlations between the observedsystem and an observer who possesses a quantum memory. If the system ismaximally entangled with its memory, the outcomes of two incompatiblemeasurements made on the system can be predicted precisely. Here, we obtain anew uncertainty relation that tightens the lower bound of Berta et al., byincorporating an additional term that depends on the quantum discord and theclassical correlations of the joint state of the observed system and thequantum memory. We discuss several examples of states for which our new lowerbound is tighter than the bound of Berta et al. On the application side, wediscuss the relevance of our new inequality for the security of quantum keydistribution and show that it can be used to provide bounds on the distillablecommon randomness and the entanglement of formation of bipartite quantumstates.

Quantum discord and classical correlation can tighten the uncertainty principle in the presence of quantum memory