Estimating the Shannon entropy and (un)certainty relations for design-structured POVMs

TL;DR

This paper introduces new polynomial-based methods to estimate Shannon entropy and uncertainty relations for quantum measurements, enhancing accuracy and applicability in quantum information science.

Contribution

It presents a novel family of polynomial estimators for Shannon entropy, improving bounds and uniformity, and applies these to quantum design measurements in quantum tomography.

Findings

New polynomial estimators provide tighter entropy bounds.

Estimates are more uniform, reducing large errors at specific points.

Applications include quantum tomography and steerability detection.

Abstract

Complementarity relations between various characterizations of a probability distribution are at the core of information theory. In particular, lower and upper bounds for the entropic function are of great importance. In applied topics, we often deal with situations, where the sums of certain powers of probabilities are known. The main question is how to convert the imposed restrictions into two-sided estimates on the Shannon entropy. It is addressed in two different ways. The more intuitive of them is based on truncated expansions of the Taylor type. Another method is based on the use of coefficients of the shifted Chebyshev polynomials. We propose here a family of polynomials for estimating the Shannon entropy from below. As a result, estimates are more uniform in the sense that errors do not become too large in particular points. The presented method is used for deriving uncertainty and certainty relations for positive operator-valued measures assigned to a quantum design. Quantum designs are currently the subject of active researches due to potential use in quantum information science. It is shown that the derived estimates are applicable in quantum tomography and detecting steerability of quantum states.