Relative entropy representations via tracial joint spectral measures

We show that for positive semidefinite $A$ and $B$ the relative entropy can be written as $D(A\|B)=2\int a\log(a/b)\,\mathrm{d}μ_{A,B}(a,b)$, where $μ_{A,B}$ denotes Heinävaara's tracial joint spectral measure. This allows us to obtain a unified derivation of several known integral representations o