Theorem: Referenced quantum-transfer obstruction

For every reference-source density,

is separable across reference and radiation. Every radiation-only deterministic CPTP decoder to a qubit has Bell-input fidelity at most $1/2$ and recovery error at least $1/2$:

Proof. The displayed positive product decomposition proves separability. Local decoding preserves it. A normalized product vector has overlap at most $1/2$ with

by Cauchy–Schwarz. Convexity extends this bound to separable states. Testing the Bell projector gives trace-distance error at least $1-F_e\ge 1/2$, hence the diamond lower bound.