Entropic uncertainty relations are analyzed for the case of N-dimensional Hilbert space and two orthogonal measurements performed in two generic bases, related by a Haar random unitary matrix U. We derive estimations for the average norms of truncations of U of a given size, which allow us to study state-independent lower bounds for the sum of two entropies describing the measurements outcomes. In particular, we show that the Maassen{\textendash}Uffink bound asymptotically behaves as lnN-lnlnN-ln2, while the strong entropic majorization relation yields a nearly optimal bound, lnN-const. Analogous results are also obtained for a more general case of several orthogonal measurements performed in generic bases.

}, doi = {10.1063/1.4944425}, url = {http://dx.doi.org/10.1063/1.4944425}, author = {R. Adamczak and R. Lata{\l}a and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} }