Proposition: Sequential preservation and scheduled readout

Every $V_j$ in (eq:sequential-isometry) is an isometry. If subsequent emissions act only on the residual and are not controlled by earlier measurement outcomes, destructive readout of an already emitted $S_j$ commutes, as a channel, with those subsequent emissions. Immediate and deferred readout therefore give the same final hybrid channel, including every reference.

Proof. Induction gives

Readout on an already emitted factor and emission on the residual act on disjoint tensor factors. Their Kraus operators commute after the fixed output reorderings. Composing the two orders therefore gives identical unnormalized branch maps and hence the same hybrid channel. Tensoring with ${\mathrm{id}}_Q$ preserves the identity.