The $f⟷\tilde{f}$ correspondence and its applications in quantum information theory

Let ${\mathcal{F}}_{op}$=\{symmetric, normalized, operator monotone functions\}. If we set $${\mathcal{F}}_{op}^r:=\{f\in {\mathcal{F}}_{op}|f(0)>0\},{\mathcal{F}}_{op}^n:=\{f\in {\mathcal{F}}_{op}|f(0)=0\},$$ trivially it holds $${\mathcal{F}}_{op}={\mathcal{F}}_{op}^r\dot{\cup }{\mathcal{F}}_{op}^n.$$ Define $$\tilde{f}(x):=\frac{1}{2}[(x+1)-(x-1{)}^2\frac{f(0)}{f(x)}]x>0.$$ It is possible to prove that the map $f\to \tilde{f},$ is a bijection from ${\mathcal{F}}_{op}^r$ to ${\mathcal{F}}_{op}^n$, namely a bijection between regular and non-regular functions.

In the last years a number of consequences has been derived from this fact: 1) the dynamical uncertainty principle; 2) its generalization to von Neumann algebras; 3) a new proof of the fact that the Wigner-Yanase-Dyson is an example of a quantum Fisher information; 4) a new proof the monotonicity property for the WYD information; 5) a link between quantum relative entropy and metric adjusted skew information.

The purpose of my talk is to describe the above applications.

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