Majorization entropic uncertainty relations

Abstract

Entropic uncertainty relations in a finite dimensional Hilbert space are\ninvestigated. Making use of the majorization technique we derive explicit lower\nbounds for the sum of R\\'enyi entropies describing probability distributions\nassociated with a given pure state expanded in eigenbases of two observables.\nObtained bounds are expressed in terms of the largest singular values of\nsubmatrices of the unitary rotation matrix. Numerical simulations show that for\na generic unitary matrix of size N = 5 our bound is stronger than the well\nknown result of Maassen and Uffink (MU) with a probability larger than 98%. We\nalso show that the bounds investigated are invariant under the dephasing and\npermutation operations. Finally, we derive a classical analogue of the MU\nuncertainty relation, which is formulated for stochastic transition matrices.\n