Joint numerical ranges, quantum maps, and joint numerical shadows

We associate with a k-tuple of hermitian N × N matrices a probability measure on Rk supported on their joint numerical range: The joint numerical shadow of these matrices. When k = 2 we recover the numerical range and the numerical shadow of the complex matrix corresponding to a pair of hermitian matrices. We apply this material to the theory of quantum information. Thus, we show that quantum maps on the set of quantum states defined by Kraus operators satisfying the identity resolution assumption shrink joint numerical ranges.