Improved Uncertainty Relation in the Presence of Quantum Memory

In Coles-Piani's recent remarkable version of the entropic uncertainty principle, the entropic sum is controlled by the first and second maximum overlaps between the two projective measurements. We generalize the entropic uncertainty relation in the presence of quantum memory and find the exact dependence on all $d$ largest overlaps between two measurements on any $d$-dimensional Hilbert space. The corresponding entropic uncertainty principle in the absence of quantum memory is also derived. Our bounds are strictly tighter than previous entropic bounds in the presence of quantum memory, which have potential applications to quantum cryptography with entanglement witnesses and quantum key distributions.