A Free Analogue of Shannon's Problem on Monotonicity of Entropy

The royal road to automatic noncommutative real analyticity, monotonicity, and convexity

It was shown classically that matrix monotone and matrix convex functions must be real analytic by Löwner and Kraus respectively. Recently, various analogues have been found in several noncommuting variables. We develop a general framework for lifting automatic analyticity theorems in matrix analysis from one variable to several variables, the so-called "royal road theorem." That is, we establish the principle that the hard part of proving any automatic analyticity theorem lies in proving the one variable theorem. We use our main result to prove the noncommutative Löwner and Kraus theorems ove…

Maximal correlation and monotonicity of free entropy and Stein discrepancy

We introduce the maximal correlation coefficient $R(M_1,M_2)$ between two noncommutative probability subspaces $M_1$ and $M_2$ and show that the maximal correlation coefficient between the sub-algebras generated by $s_n:=x_1+\ldots +x_n$ and $s_m:=x_1+\ldots +x_m$ equals $\sqrt{m/n}$ for $m\le n$, where $(x_i)_{i\in \mathbb{N}}$ is a sequence of free and identically distributed noncommutative random variables. This is the free-probability analogue of a result by Dembo--Kagan--Shepp in classical probability. As an application, we use this estimate to provide another simple proof of the monotoni…

Semicircularity, Gaussianity and Monotonicity of Entropy

S. Artstein, K. Ball, F. Barthe, and A. Naor have shown that if (X_j) are i.i.d. random variables, then the entropy of n^{-1/2}(X_1+....+X_n) increases as n increases. The free analogue was recently proven by D. Shlyakhtenko. That is, if (x_j) are freely independent, identically distributed, self-adjoint elements in a noncommutative probability space, then the free entropy of n^{-1/2}(x_1+....+x_n) increases as n increases. In this paper we prove that if X_1 (x_1, resp.) has finite entropy (free entropy, resp.), and if the entropy (the free entropy, resp.) is not a strictly increasing function…

有限自由信息不等式

Finite Free Information Inequalities

本文提出有限自由信息理论，推导出实根多项式相关的熵和Fisher信息单调性不等式，并解决了相关猜想。