Let $f$ be a differentiable strictly convex function on $(0,\infty)$. The Bregman $f$-divergence of the positive definite operators $A$ and $B$ acting on a finite dimensional Hilbert space is defined by $$ H_f(A,B)= \mathrm{Tr} \left(f(A)-f(B)-f'(B)(A-B)\right). $$ If $f$ can be extended to $0$ by continuity, then the Bregman $f$-divergence can be extended to positive semidefinite operators by continuity, as well. The Jensen $f$-divergence of the positive semidefinite operators $A$ and $B$ is defined by $$ J_{f}(A,B)=\mathrm{Tr} \left(\frac{1}{2} \left(f(A)+f(B)\right)-f\left(\frac{1}{2} \left(A + B\right)\right)\right) $$ for a strictly convex continuous function $f$ defined on $[0, \infty).$

Both Bregman and Jensen divergences are generalized distance measures. This latter notion stands for any function $d: X\times X \to [0,\infty )$ on any set $X$ with the mere property that for $x,y \in X$ we have $d(x,y)=0$ if and only if $x=y$. Transformations which preserve generalized distance measures are called generalized isometries.

The aim of the talk is to describe the structure of the generalized isometries of the set of density operators – which used to represent the state space of a finite quantum system – with respect to Bregman and Jensen divergences. It turns out that every bijective transformation which leaves the Bregman or Jensen divergence invariant is implemented by either a unitary or an antiunitary operator on the underlying Hilbert space. Such a result may be considered as a Wigner type result. At several points of our argument we use ideas and techniques of the work of Lajos Molnár which determines the structure of the relative entropy preserving transformations of the state space [1].

The results presented in the talk are collected in [2].