From Rényi Entropy Power to Information Scan of Quantum States

Abstract

In this paper, we generalize the notion of Shannon’s entropy power to the Rényi-entropy setting. With this, we propose generalizations of the de Bruijn identity, isoperimetric inequality, or Stam inequality. This framework not only allows for finding new estimation inequalities, but it also provides a convenient technical framework for the derivation of a one-parameter family of Rényi-entropy-power-based quantum-mechanical uncertainty relations. To illustrate the usefulness of the Rényi entropy power obtained, we 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.

Coherent states for generalized uncertainty relations and their cosmological implications

“…By using Beckner-Babenko's inequality for the Fourier transform duals (basically inequality between their norms) [34,35], one can derive the Shannon entropy-based UR [37]…”

The Statistical Foundations of Entropy

“…The main result of the sixth contribution [6] was to introduce a generalization the concept of Shannon's entropy power based on Rényi entropy. Consequently, the authors could generalize several popular identities, including the de Bruijn identity, isoperimetric inequality, or Stam inequality.…”