Local numerical range for a class of $2\otimes d$ 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 $2\otimes d$ 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 $\ket{x}\otimes \ket{y} \in \mathbb{C}^2\otimes \mathbb{C}^d$ reduces to the corresponding problem in $\mathbb{R}^2\otimes \mathbb{R}^d$. The final analytical result for $d=2$ is presented.