  We associate to every quantum channel $T$ acting on a Hilbert space
$\mathcal{H}$ a pair of Hamiltonian operators over the symmetric subspace of
$\mathcal{H}^{\otimes 2}$. The expectation values of these Hamiltonians over
symmetric product states give either the purity or the pure state fidelity of
$T$. This allows us to analytically compute these measures for a wide class of
channels, and to identify states that are optimal with respect to these
measures.
