 Solution

Use the unsquared convention for quantum fidelity:Choose a purification of a density operator $|\Psi {⟩}_{RA}$ of $\rho$, and put ${\tau }_{RA}=({\mathrm{id}}_R\otimes \Lambda )(|\Psi ⟩⟨\Psi |)$. By definition,The monotonicity of quantum fidelity under partial trace follows from Uhlmann's theorem: maximizing purifications of two joint states are also candidate purifications of their marginals, with the discarded subsystem included in the purifying system. The optimization for the marginals therefore cannot give a smaller overlap. Here those marginals are $\rho$ and $\Lambda (\rho )$. ConsequentlyThis entanglement fidelity bound by state fidelity distinguishes preserving a reference entanglement from merely reproducing the input marginal.