From Rényi Entropy Power to Information Scan of Quantum States

Abstract

In the estimation theory context, we generalize the notion of Shannon's entropy power to the Rényi-entropy setting. This not only allows to find new estimation inequalities, such as the Rényi-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 Rényi-entropy-power-based quantum-mechanical uncertainty relations. To put more flesh on the bones, we use the Rényi 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.

28 pages, 2 figures

References in corpus (6)

• Experimental Test of Quantum Jarzynski Equality with a Trapped Ion System
• Systematic Analysis of the Non-extensive Statistical Approach in High Energy Particle Collisions - Experiment vs. Theory
• Experimental quantum key distribution with finite-key security analysis for noisy channels
• On the Role of Information Theoretic Uncertainty Relations in Quantum Theory
• Generalized statistics: yet another generalization
• The multi-mode quantum Entropy Power Inequality

Cited by in corpus (1)

• Coherent states for generalized uncertainty relations as Tsallis probability amplitudes: new route to non-extensive thermostatistics