Product numerical range

We study operators acting on a composite Hilbert space and investigate their local numerical range, local spectral radius and local $C$--spectral radius. Concrete bounds for the local numerical range for Hermitian operators are derived. Local numerical range of a non-Hermitian operator forms a subset of the standard numerical range. While the latter set is convex, the local range needs not to be convex nor simply connected. Local numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors. As an exemplary application of these algebraic tools in the theory of quantum information, we study block positive matrices and entanglement witnesses. Furthermore, we apply local numerical range to solve the problem of local distinguishability of a family of two unitary gates. Local $C$--spectral radius is useful for characterizing quantum entanglement and finding local fidelity between two states of a composite system, while higher order local numerical range can be used to design local dark spaces and local error correction codes.