  Numerical range of a Hermitian operator X is defined as the set of all
possible expectation values of this observable among a normalized quantum
state. We analyze a modification of this definition in which the expectation
value is taken among a certain subset of the set of all quantum states. One
considers for instance the set of real states, the set of product states,
separable states, or the set of maximally entangled states. We show exemplary
applications of these algebraic tools in the theory of quantum information:
analysis of k-positive maps and entanglement witnesses, as well as study of the
minimal output entropy of a quantum channel. Product numerical range of a
unitary operator is used to solve the problem of local distinguishability of a
family of two unitary gates.
