Uncertainty Relations in the Presence of Quantum Memory for Mutually Unbiased Measurements

TL;DR

This paper extends uncertainty relations with quantum memory from mutually unbiased bases to measurements and generalized measurements, establishing equalities linking measurement uncertainty and entanglement.

Contribution

It generalizes Berta's uncertainty relations to mutually unbiased measurements and symmetric informationally complete measurements, revealing new equality relations.

Findings

Equality between measurement uncertainty and entanglement quantified by conditional collision entropy.

Extension of uncertainty-entanglement relation to symmetric informationally complete measurements.

Derivation of an equality for orthogonal basis of Hermitian, traceless operators.

Abstract

In [Berta 2014 Entanglement], uncertainty relations in the presence of quantum memory was 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 relation 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.