Preservers of matrix pairs with a fixed inner product value
The authors characterize bijective maps \(\phi:V\rightarrow V\), where \(V\) is a set of matrices, such that \(\text{tr}(AB)=c \Leftrightarrow \text{tr}\phi(A)\phi(B)=c\) for a fixed \(c\).NEWLINENEWLINEIn Section 1, they give an overall view of the motivation of the article, the connection with quantum physics, and related open questions.NEWLINENEWLINEIn Section 2, they completely characterize the bijections for \(V\) to the set of \(n\times n\) Hermitian matrices or the set of \(n\times n\) symmetric matrices and for arbitrary \(c\).NEWLINENEWLINEIn Section 3, they completely characterize the bijections for \(V\) to be the set of all real effects (i.e matrices \(A\) satisfying \(0\leq A\leq I\)) or the set of all complex effects, and for arbitrary \(c\in [0,1]\).NEWLINENEWLINEIn Section 4, they characterize the bijections for \(V\) to be the set of all projections of rank one and \(c\) lies within \([\frac12, 1)\). It turns out that they are all of the form \(\phi(P)=OPO^t\) for some orthogonal transformation \(O\).
scientific article; zbMATH DE number 6084687