We study operators acting on a tensor product Hilbert space and investigate
their product numerical range, product numerical radius and separable numerical
range. Concrete bounds for the product numerical range for Hermitian operators
are derived. Product numerical range of a non-Hermitian operator forms a subset
of the standard numerical range containing the barycenter of the spectrum.
While the latter set is convex, the product range needs not to be convex nor
simply connected. The product numerical range of a tensor product is equal to
the Minkowski product of numerical ranges of individual factors.Comment: 17 pages, 4 figures. Original preprint "Local numerical range: a
versatile tool in the theory of quantum information" [arXiv:0905.3646v1] was
broadened and split into two papers: "Restricted numerical range: a versatile
tool in the theory of quantum information", and "Product numerical range in a
space with tensor product structure