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arXiv:1509.00398 · International Journal of Quantum Information

Optimality of entropic uncertainty relations

Kais Abdelkhalek, René Schwonnek, Hans Maassen, Fabian Furrer, Jörg Duhme, Philippe Raynal, Berthold‐Georg Englert, Reinhard F. Werner

Published Sep 1, 2015• 3 citations
arXivHTMLPDF

Abstract

The entropic uncertainty relation proven by Maassen and Uffink for arbitrary pairs of two observables is known to be nonoptimal. Here, we call an uncertainty relation optimal, if the lower bound can be attained for any value of either of the corresponding uncertainties. In this work, we establish optimal uncertainty relations by characterizing the optimal lower bound in scenarios similar to the Maassen–Uffink type. We disprove a conjecture by Englert et al. and generalize various previous results. However, we are still far from a complete understanding and, based on numerical investigation and analytical results in small dimension, we present a number of conjectures.

DOI
10.1142/s0219749915500458

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The entropic uncertainty relation proven by Maassen and Uffink for arbitrary pairs of two observables is known to be nonoptimal. Here, we call an uncertainty relation optimal, if the lower bound can be attained for any value of either of the corresponding uncertainties. In this work, we establish optimal uncertainty relations by characterizing the optimal lower bound in scenarios similar to the Maassen–Uffink type. We disprove a conjecture by Englert et al. and generalize various previous results. However, we are still far from a complete understanding and, based on numerical investigation and analytical results in small dimension, we present a number of conjectures.
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Entropic uncertainty relations H(A)+H(B)⩾γH(A)+H(B)\geqslant γH(A)+H(B)⩾γ give a nonzero lower bound γγγ to the sum of the Shannon entropies HHH of the outcome probabilities of incompatible observables AAA and BBB. They are better than the variance-based uncertainty relations because they only depend on the Born statistics of the outcomes and not on the outcomes themselves, and because bounds γγγ typically are state independent. Here we provide a state-independent lower bound γsγ_sγs​ that is better than the textbook Maassen-Uffink bound and, in the limit of the parameter s→2s\to 2s→2, becomes asymptotically tight for all A,BA,BA,B. The bound can be extended to Renyi entropies. After the submission of this paper, we found out that the main results presented here were already on page 6 of https://quantum-journal.org/papers/q-2018-03-30-59/pdf/
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