We propose an alternative measure of quantum uncertainty for pairs of
arbitrary observables in the 2-dimensional case, in terms of collision
entropies. We derive the optimal lower bound for this entropic uncertainty
relation, which results in an analytic function of the overlap of the
corresponding eigenbases. Besides, we obtain the minimum uncertainty states. We
compare our relation with other formulations of the uncertainty principle.Comment: The manuscript has been accepted for publication as a Regular Article
in Physical Review
We present a quantum version of the generalized $(h,\phi)$-entropies,
introduced by Salicr\'u \textit{et al.} for the study of classical probability
distributions. We establish their basic properties, and show that already known
quantum entropies such as von Neumann, and quantum versions of R\'enyi,
Tsallis, and unified entropies, constitute particular classes of the present
general quantum Salicr\'u form. We exhibit that majorization plays a key role
in explaining most of their common features. We give a characterization of the
quantum $(h,\phi)$-entropies under the action of quantum operations, and study
their properties for composite systems. We apply these generalized entropies to
the problem of detection of quantum entanglement, and introduce a discussion on
possible generalized conditional entropies as well.Comment: 26 pages, 1 figure. Close to published versio
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