The $f \longleftrightarrow \tilde f$ correspondence and its applications in quantum information theory

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