Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies

Abstract

We formulate uncertainty relations for mutually unbiased bases and symmetric informationally complete measurements in terms of the Rényi and Tsallis entropies. For arbitrary number of mutually unbiased bases in a finite-dimensional Hilbert space, we give a family of Tsallis $\alpha$-entropic bounds for $\alpha \in (0;2]$. Relations in a model of detection inefficiences are obtained. In terms of Rényi's entropies, lower bounds are given for $\alpha \in [2;\infty )$. State-dependent and state-independent forms of such bounds are both given. Uncertainty relations in terms of the min-entropy are separately considered. We also obtain lower bounds in term of the so-called symmetrized entropies. The presented results for mutually unbiased bases are extensions of some bounds previously derived in the literature. We further formulate new properties of symmetric informationally complete measurements in a finite-dimensional Hilbert space. For a given state and any SIC-POVM, the index of coincidence of generated probability distribution is exactly calculated. Short notes are made on potential use of this result in entanglement detection. Further, we obtain state-dependent entropic uncertainty relations for a single SIC-POVM. Entropic bounds are derived in terms of the Rényi $\alpha$-entropies for $\alpha \in [2;\infty )$ and the Tsallis $\alpha$-entropies for $\alpha \in (0;2]$. In the Tsallis formulation, a case of detection inefficiences is briefly mentioned. For a pair of symmetric informationally complete measurements, we also obtain an entropic bound of Maassen-Uffink type.

19 pages, no figures. The version 4 includes uncertainty relations with detection inefficiencies and brief notes on entanglement detection. New references are added. Except for the style, this version matches the journal version

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