On Uncertainty Relations and Entanglement Detection with Mutually Unbiased Measurements

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.

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Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements

Entropy ◽

10.3390/e21030270 ◽

2019 ◽

Vol 21 (3) ◽

pp. 270 ◽

Cited By ~ 4

Author(s):

Kyunghyun Baek ◽

Hyunchul Nha ◽

Wonmin Son

Keyword(s):

Three Dimensional ◽

Uncertainty Relations ◽

Dimensional System ◽

Direct Sum ◽

Entropic Uncertainty Relations ◽

Finite Dimensional ◽

Uncertainty Bound ◽

Three Dimensional System ◽

Unsharp Observables ◽

Positive Operator Valued Measure

We derive an entropic uncertainty relation for generalized positive-operator-valued measure (POVM) measurements via a direct-sum majorization relation using Schur concavity of entropic quantities in a finite-dimensional Hilbert space. Our approach provides a significant improvement of the uncertainty bound compared with previous majorization-based approaches (Friendland, S.; Gheorghiu, V.; Gour, G. Phys. Rev. Lett. 2013, 111, 230401; Rastegin, A.E.; Życzkowski, K. J. Phys. A, 2016, 49, 355301), particularly by extending the direct-sum majorization relation first introduced in (Rudnicki, Ł.; Puchała, Z.; Życzkowski, K. Phys. Rev. A 2014, 89, 052115). We illustrate the usefulness of our uncertainty relations by considering a pair of qubit observables in a two-dimensional system and randomly chosen unsharp observables in a three-dimensional system. We also demonstrate that our bound tends to be stronger than the generalized Maassen–Uffink bound with an increase in the unsharpness effect. Furthermore, we extend our approach to the case of multiple POVM measurements, thus making it possible to establish entropic uncertainty relations involving more than two observables.

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Entanglement detection in quantum many-body systems using entropic uncertainty relations

Physical Review A ◽

10.1103/physreva.103.052412 ◽

2021 ◽

Vol 103 (5) ◽

Author(s):

Bjarne Bergh ◽

Martin Gärttner

Keyword(s):

Uncertainty Relations ◽

Many Body ◽

Entropic Uncertainty Relations ◽

Entanglement Detection ◽

Many Body Systems

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Renyi and Tsallis formulations of noise-disturbance trade-off relations

Quantum Information and Computation ◽

10.26421/qic16.3-4-7 ◽

2016 ◽

Vol 16 (3&4) ◽

pp. 313-331

Author(s):

Alexey E. Rastegin

Keyword(s):

Probability Distributions ◽

Conditional Entropy ◽

Uncertainty Relations ◽

Theoretic Approach ◽

Trade Off ◽

Information Theoretic ◽

Entropic Uncertainty Relations ◽

Related Information ◽

Information Theoretic Measures ◽

Information Theoretic Approach

We address an information-theoretic approach to noise and disturbance in quantum measurements. Properties of corresponding probability distributions are characterized by means of both the R´enyi and Tsallis entropies. Related information-theoretic measures of noise and disturbance are introduced. These definitions are based on the concept of conditional entropy. To motivate introduced measures, some important properties of the conditional R´enyi and Tsallis entropies are discussed. There exist several formulations of entropic uncertainty relations for a pair of observables. Trade-off relations for noise and disturbance are derived on the base of known formulations of such a kind.

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