Entanglement fidelity bound by state fidelity ($F_e(\rho ,\Lambda )\le F(\rho ,\Lambda (\rho ){)}^2$)

In the unsquared quantum fidelity convention, the fidelity between a purification and its channel output is the square root of entanglement fidelity. Tracing out the purifying reference leaves the input and output marginals. Monotonicity of quantum fidelity under partial trace therefore gives the bound. Preserving an input marginal is a weaker requirement than preserving its correlations with a reference.