General entropic uncertainty relations for POVM

BOSYK, GUSTAVO MARTÍN; ZOZOR, STEEVE; PORTESI, MARIELA

Mar del Plata

Congreso; VII Quantum Optics; 2014

We present entropic formulations of the uncertainty principle for arbitrary pairs of positive operator-valued measures (POVM) A and B acting on N-dimensional Hilbert space. We introduce a generalized entropy to measure the lack of information associated to the probability distributions of the POVM depending on the state of the quantum system, and we find a non trivial bound for the sum of generalized entropies. To obtain our bound, we proceed in two steps: (i) evaluation of the minimal entropy of a distribution subject to maximal probability; and (ii) minimization of the sum of the minimal entropies, subject to the Landau-Pollak inequality, which links the maximum of the distributions associated to the POVM. As a consequence, our bound depends on the triplet overlap (cA; cB; cAB), where cA and cB are the intrinsic overlaps of each POVM whereas cAB is the overlap between both POVM that quanties the degree of incompatibility between them. In the case of nondegenerate observables [(1; 1; c)], the bound appears to be optimum for given c > 1/sqrt2 and improves all c-dependent bounds that exists in the literature such as Deutsch and Maassen-Unk ones.