Fidelity preserving and decoherence for mixed unitary quantum channels

Distinguishable and non-distinguishable quantum states are fundamental resources in quantum mechanics and quantum technologies. Interactions with the environment often induce decoherence, impacting both the distinguishability and non-distinguishability between quantum states. In this paper, we investigate mixed unitary quantum channels and the conditions under which fidelity, a measure of quantum state closeness, is preserved. More precisely, for quantum channels in the form $\Phi(\rho) = \sum_{i=1}^N p_i U_i \rho U_i^*$, we analyze their effect on quantum state $|\varphi\rangle$ through the associated purification $|\Psi\rangle$, explore the structure of such quantum channels that preserve either distinguishable or non-distinguishable states and then discuss the challenges of maintaining fidelity, particularly under the influence of phase damping.