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 $\mathrm{\Phi }(\rho )={\sum }_{i=1}^N{p}_i{U}_i\rho U_i^{\ast }$, we analyze their effect on quantum state $\mathrm{\mid }\phi ⟩$ through the associated purification $\mathrm{\mid }\mathrm{\Psi }⟩$, 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.