Proposition: Coherent loading and carrier accounting

For every reference-entangled logical input ${\rho }_{QC}$, the loading operation produces the encoded source input and leaves the carrier in a fixed vacuum state. In particular, it introduces no message-dependent inaccessible carrier output.

Proof. Second quantization sends each carrier creation operator to the corresponding source creation operator and preserves the vacuum. It therefore maps every occupation-basis vector on the carrier to the same source occupation vector, with the fixed Fock ordering convention. By linearity, it maps all matrix units and all superpositions correctly.

The carrier output is vacuum on every input vector. Tensoring with $I_Q$ proves the reference statement.