Purity and state fidelity of quantum channels

We associate with every quantum channel $T$ acting on a Hilbert space $\mathcal{H}$ a pair of Hermitian operators, referred to as “Hamiltonians,” 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.