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Shannon’s monotonicity problem for free and classical entropy. With an appendix by Hanne Schultz. (English) Zbl 1191.46052

Summary: We give a short unified proof of the following theorem, valid in the context of both classical probability theory and Voiculescu’s free probability theory: let \((X_j(1), \dots, X_j(n))\) be independent (resp., freely independent) \(n\)-tuples of random variables. Let \(Z_N^{(p)} = N^{-1/2}(X_1^{(p)} + \dots + X_N^{(p)})\) be their central limit sums. Then the entropy (resp., free entropy) of the \(n\)-tuple \((Z_N^{(1)}, \dots, Z_N^{(n)})\) is a monotone function of \(N\). The classical case (for \(n = 1\)) is a celebrated result of S.Artstein, K.Ball, F.Barthe and A.Naor [J. Am.Math.Soc.17, No.4, 975–982 (2004; Zbl 1062.94006)], and our proof is an adaptation and simplification of their argument.

MSC:

46L54 Free probability and free operator algebras
81S25 Quantum stochastic calculus
94A17 Measures of information, entropy

Citations:

Zbl 1062.94006
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References:

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