From R\'{e}nyi Entropy Power to Information Scan of Quantum States

TL;DR

This paper extends Shannon's entropy power to Rényi entropy, deriving new inequalities and uncertainty relations, and introduces a quantum state reconstruction method akin to tomography, with applications to cat states in quantum metrology.

Contribution

It generalizes entropy power to Rényi entropy, leading to new inequalities and a quantum state reconstruction framework based on Rényi entropy power.

Findings

Derived Rényi-entropy-based De Bruijn, isoperimetric, and Stam inequalities.

Developed a quantum state reconstruction method using Rényi entropy power.

Applied the framework to analyze cat states in quantum metrology.

Abstract

In the estimation theory context, we generalize the notion of Shannon's entropy power to the R\'{e}nyi-entropy setting. This not only allows to find new estimation inequalities, such as the R\'{e}nyi-entropy based De Bruijn identity, isoperimetric inequality or Stam inequality, but it also provides a convenient technical framework for the derivation of a one-parameter family of R\'{e}nyi-entropy-power-based quantum-mechanical uncertainty relations. To put more flesh on the bones, we use the R\'{e}nyi entropy power obtained to show how the information probability distribution associated with a quantum state can be reconstructed in a process that is akin to quantum-state tomography. We illustrate the inner workings of this with the so-called "cat states", which are of fundamental interest and practical use in schemes such as quantum metrology. Salient issues, including the extension of the notion of entropy power to Tsallis entropy and ensuing implications in estimation theory are also briefly discussed.