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arXiv:quant-ph/0606244 · Physical Review A

Entropic uncertainty relations and locking: Tight bounds for mutually unbiased bases

Manuel A. Ballester, Stephanie Wehner

Published Feb 21, 2007• 113 citations
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Abstract

We prove tight entropic uncertainty relations for a large number of mutually unbiased measurements. In particular, we show that a bound derived from the result by Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)] for two such measurements can in fact be tight for up to $\sqrt{d}$ measurements in mutually unbiased bases. We then show that using more mutually unbiased bases does not always lead to a better locking effect. We prove that the optimal bound for the accessible information using up to $\sqrt{d}$ specific mutually unbiased bases is $\mathrm{log}\phantom{\rule{0.2em}{0ex}}d∕2$, which is the same as can be achieved by using only two bases. Our result indicates that merely using mutually unbiased bases is not sufficient to achieve a strong locking effect and we need to look for additional properties.

DOI
10.1103/physreva.75.022319

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We prove tight entropic uncertainty relations for a large number of mutually unbiased measurements. In particular, we show that a bound derived from the result by Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)] for two such measurements can in fact be tight for up to d\sqrt{d}d​ measurements in mutually unbiased bases. We then show that using more mutually unbiased bases does not always lead to a better locking effect. We prove that the optimal bound for the accessible information using up to d\sqrt{d}d​ specific mutually unbiased bases is logd∕2\mathrm{log}\phantom{\rule{0.2em}{0ex}}d∕2logd∕2, which is the same as can be achieved by using only two bases. Our result indicates that merely using mutually unbiased bases is not sufficient to achieve a strong locking effect and we need to look for additional properties.
We show that using more mutually unbiased bases does not always lead to a better locking effect. In particular, we prove that the optimal bound for the accessible information using three specific mutually unbiased bases is n/2, which is the same as can be achieved by using only two bases. Our result indicates that merely using mutually unbiased bases is not sufficient to achieve a strong locking effect, and we need to look for additional properties. We also prove an entropic uncertainty relation that may be of independent interest.
Entropic uncertainty relations provide an information-theoretic framework for quantifying the fundamental indeterminacy inherent in quantum mechanics. We propose more stringent quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. We present lower and upper bounds of the quantum uncertainties based on the complementarity of the observables, the purity of the measured state, the (conditional) von-Neumann entropies, the Holevo quantities and mutual information. The results are illustrated by several representative cases, showing that our bounds are tighter than and outperform previously existing bounds.
Recently, an entropic uncertainty relation for multiple measurements was proposed by Liu et al. [Phys. Rev. A 91, 042133 (2015)]. However, the lower bound of the relation is not always tight with respect to different measurements. Herein, we improve the lower bound of the entropic uncertainty relation for multiple measurements, referred to as a simply constructed bound (SCB). We verify that the SCB is tighter than Liu et al.'s result for arbitrary mutually unbiased basis measurements, which might play a fundamental and crucial role in practical quantum information processing. Moreover, we optimize the SCB by considering mutual information and the Holevo quantity, and propose an optimized SCB. Notably, the proposed bounds are extrapolations of the behavior of two measurements to a larger collection of measurements. It is believed that our findings would shed light on entropy-based uncertainty relations in the multiple measurement scenario and will be beneficial for security analysis in quantum key distributions.