  The limitation on obtaining precise outcomes of measurements performed on two
non-commuting observables of a particle as set by the uncertainty principle in
its entropic form, can be reduced in the presence of quantum memory. We derive
a new entropic uncertainty relation based on fine- graining, which leads to an
ultimate limit on the precision achievable in measurements performed on two
incompatible observables in the presence of quantum memory. We show that our
derived uncertainty relation tightens the lower bound set by entropic
uncertainty for members of the class of two-qubit states with maximally mixed
marginals, while accounting for the recent experimental results using maximally
entangled pure states and mixed Bell-diagonal states. An implication of our
uncertainty relation on the security of quantum key generation protocols is
pointed out.
