  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.
