Maps on positive definite operators preserving the quantum $$\chi _\alpha ^2$$ -divergence.

We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum $$\chi _\alpha ^2$$ -divergence for some $$\alpha \in [0,1]$$ . We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived. [ABSTRACT FROM AUTHOR]