General entropy-like uncertainty relations in finite dimensions

Abstract

We revisit entropic formulations of the uncertainty principle for an arbitrary pair of positive operator-valued measures (POVM) $A$ and $B$, acting on finite dimensional Hilbert space. Salicrú generalized $(h,\varphi )$-entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet $(c_A,c_B,c_{A,B})$ with $c_A$ (resp. $c_B$) being the overlap between the elements of the POVM $A$ (resp. $B$) and $c_{A,B}$ the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz [Phys.\ Rev.\ A \textbf{77}, 042110 (2008)] and consists in a minimization of the entropy sum subject to the Landau-Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the uncertainty principle, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz-Thorin theorem. In the case of nondegenerate observables, we show that for given $c_{A,B}>\frac{1}{\sqrt{2}}$, the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen-Uffink bound prevails when $c_{A,B}\le \frac{1}{2}$. Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.

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