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.
Highlights
The notion of entropy is undoubtedly one of the most important concepts in modern science
The notion of entropy stemmed from thermodynamics, where it was developed to quantify the annoying inefficiency of steam engines
We could rephrase the generalized Stam inequality (21) and generalized isoperimetric inequality (14) in terms of Tsallis entropy power (EP). Though such inequalities are quite interesting from a mathematical point of view, it is not yet clear how they could be practically utilized in the estimation theory as there is no obvious operational meaning associated with Tsallis entropy
Summary
The notion of entropy is undoubtedly one of the most important concepts in modern science. The use of ITEs has been stimulated by new high-precision instrumentation [22,23] and by, e.g., recent advances in stochastic thermodynamics [24,25] or observed violations of Heisenberg’s error-disturbance uncertainty relations [26,27,28,29,30] In his seminal 1948 paper, Shannon laid down the foundations of modern information theory [5]. We further illuminate the role of Rényi’s EP by deriving (through the Stam inequality) Rényi’s EP-based quantum uncertainty relations for conjugate observables To flesh this out, the second part of the paper is devoted to the development of the use of Rényi EPs to extract the quantum state from incomplete data. For the reader’s convenience, we relegate some technical issues concerning the generalized De Bruijn identity and associated isoperimetric and Stam inequalities to three appendices
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