We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of entanglement detection in bipartite finite-dimensional systems may be realized. A certain trade-off between a sensitivity of the scheme and its experimental complexity is discussed.We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of entanglement detection in bipartite finite-dimensional systems may be realized. A certain trade-off between a sensitivity of the scheme and its experimental complexity is discussed.

arXiv:1407.7333Open Systems & Information DynamicsMar 1, 201545 citations

Generalized uncertainty relations

Burcu Elif Akten

Dec 1, 19990 citations

Uncertainty relations in the presence of quantum memory for mutually unbiased measurements

Kun Wang, Nan Wu, Fangmin Song

In a work by Berta et al. [Phys. Rev. A 90, 062127 (2014)], uncertainty relations in the presence of quantum memory were formulated for mutually unbiased bases using conditional collision entropy. In this paper, we generalize their results to the mutually unbiased measurements. Our primary result is an equality between the amount of uncertainty for a set of measurements and the amount of entanglement of the measured state, both of which are quantified by the conditional collision entropy. Implications of this equality relation are discussed. We further show that similar equality relations can be obtained for generalized symmetric informationally complete measurements. We also derive an interesting equality for arbitrary orthogonal basis of the space of Hermitian, traceless operators.In a work by Berta et al. [Phys. Rev. A 90, 062127 (2014)], uncertainty relations in the presence of quantum memory were formulated for mutually unbiased bases using conditional collision entropy. In this paper, we generalize their results to the mutually unbiased measurements. Our primary result is an equality between the amount of uncertainty for a set of measurements and the amount of entanglement of the measured state, both of which are quantified by the conditional collision entropy. Implications of this equality relation are discussed. We further show that similar equality relations can be obtained for generalized symmetric informationally complete measurements. We also derive an interesting equality for arbitrary orthogonal basis of the space of Hermitian, traceless operators.

arXiv:1807.01047Physical Review ASep 26, 201811 citations

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

Manuel A. Ballester, Stephanie Wehner

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.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.

arXiv:quant-ph/0606244Physical Review AFeb 21, 2007113 citations

\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.