Proposition: A genuine quantum-transfer consequence

Every entanglement-breaking $d$-dimensional channel $\Phi$ satisfies

for every unitary $U$.

Proof. Its Bell-input output is separable. A product pure state has overlap at most $1/d$ with the maximally entangled target, by Cauchy–Schwarz. Convexity gives the same bound for separable states. Testing the ideal Bell projector proves the result.