Local numerical range for a class of $2otimes d$ hermitian operators

A local numerical range is analyzed for a family of circulant observables and states of composite $2 otimes d$ systems. It is shown that for any $2otimes d$ circulant operator $cal O$ there exists a basis giving rise to the matrix representation with real non-negative off-diagonal elements. In this basis the problem of finding extremum of $cal O$ on product vectors $ket{x}otimes ket{y} in mathbb{C}^2otimes mathbb{C}^d$ reduces to the corresponding problem in $mathbb{R}^2otimes mathbb{R}^d$. The final analytical result for $d=2$ is presented.