On a generalized entropic uncertainty relation in the case of the qubit

TL;DR

This paper derives a generalized, tight entropic uncertainty relation for qubits using Rènyi entropy, overcoming previous constraints and providing explicit bounds and minimizing states.

Contribution

It introduces a generalized entropic uncertainty relation for qubits with explicit bounds for any entropic indices, extending previous results and overcoming H"older conjugacy limitations.

Findings

Derived a general expression for the tight lower bound of Rènyi entropy sums.

Provided analytical and semi-analytical bounds in the a0a0 a0a0 plane and on the line a0a0a0.

Identified the minimizing states for the uncertainty bounds.

Abstract

We revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two-dimensional Hilbert space. R\'enyi entropy is used as uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables. We derive a general expression for the tight lower bound of the sum of R\'enyi entropies for any couple of (positive) entropic indices (\alpha,\beta). Thus, we have overcome the H\"older conjugacy constraint imposed on the entropic indices by Riesz-Thorin theorem. In addition, we present an analytical expression for the tight bound inside the square [0 , 1/2] x [0 , 1/2] in the \alpha-\beta plane, and a semi-analytical expression on the line \beta = \alpha. It is seen that previous results are included as particular cases. Moreover, we present an analytical but suboptimal bound for any couple of indices. In all cases, we provide the minimizing states.