On uncertainty relations and entanglement detection with mutually unbiased measurements

TL;DR

This paper explores properties of mutually unbiased measurements to derive entropic uncertainty relations and revisits their application in entanglement detection, providing new bounds and methods for bipartite systems.

Contribution

It introduces new properties of mutually unbiased measurements and derives entropic uncertainty relations in terms of Rényi and Tsallis entropies, enhancing entanglement detection techniques.

Findings

Estimated upper bounds for the sum of indices of coincidence.

Derived entropic uncertainty relations in Rényi and Tsallis entropies.

Proposed a bipartite entanglement detection method using mutually unbiased measurements.

Abstract

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 R\'{e}nyi 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.