From Rényi Entropy Power to Information Scan of Quantum States

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.