Abstract:
Chentsov created Information Geometry by showing that classical Fisher Information is the only metric on the spaces of probability vectors which ‘`contracts under noise". Chentsov himself posed the same question about the quantum realm and was Petz to find the answer using means. Still using means Petz clarified the notion of quantum variance. I’ll discuss how means enable us to understand the relation between Quantum Variance and Quantum Fisher Information and how this sheds new light on the Uncertainty Principle. [G22b] The Jensen inequality for means Recently the Jensen inequality for 2-means has been proved. Does a general version for $n$-means exist? [G22a] Stam inequality as a mean inequality The Stam inequality for the Gamma distribution appears as a mean inequality. Can this be generalized?[G14]
[G22b] P. Gibilisco, “Uncertainty and Quantum Variance at the light of Quantum Information Geometry”, Information Geometry, 2022
[G22a] P. Gibilisco, “About the Jensen inequality for numerical $n$–means”, International Journal of Modern Physics A, 2022, 2243010
[G14] P. Gibilisco, “Fisher information and means: some questions in the classical and quantum settings”, International Journal of Software and Informatics, 8:3-4 (2014), 265–276
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