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Improved Uncertainty Relation in the Presence of Quantum Memory
Yunlong Xiao, Naihuan Jing, Shao-Ming Fei and Xianqing Li-Jost
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 and find the exact dependence on all first $d$ largest overlaps between two measurements on any $d$-dimensional Hilbert space. The corresponding entropic uncertainty principle in the presence of quantum memory is also derived. Our bounds are strictly tighter than previous entropic bounds.