Strong majorization entropic uncertainty relations

We analyze entropic uncertainty relations in a finite-dimensional Hilbert space and derive several strong bounds for the sum of two entropies obtained in projective measurements with respect to any two orthogonal bases. We improve the recent bounds by Coles and Piani P. Coles and M. Piani, Phys. Rev. A 89, 022112 (2014), which are known to be stronger than the well-known result of Maassen and Uffink H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988). Furthermore, we find a bound based on majorization techniques, which also happens to be stronger than the recent results involving the largest singular values of submatrices of the unitary matrix connecting both bases. The first set of bounds gives better results for unitary matrices close to the Fourier matrix, while the second one provides a significant improvement in the opposite sectors. Some results derived admit generalization to arbitrary mixed states, so that corresponding bounds are increased by the von Neumann entropy of the measured state. The majorization approach is finally extended to the case of several measurements.