Proposition: Enter-compatible realization with two fresh ancillary qubits

For each $\eta$, let

and set

A two-qubit ancillary preparation

followed by controlled $U_a$ on the carrier and destructive readout of the ancillary register, has resolved selective maps

If the label is not available to later decoding, the carrier channel is ${\mathcal{P}}_{\eta }$.

Repeated steps can be represented with peak occupancy three qubits and one carrier retained between steps.

Proof. A unitary on the two fresh ancillary qubits prepares $|{\chi }_{\eta }⟩$ from $|00⟩$, by completing the specified unit vector to an orthonormal basis. The controlled unitary sends an input vector to

Enter consumes both ancillary names. Its branch map on the carrier is $\sqrt{w_a}{U}_a$, which proves the selective-map formula.

The Pauli identity

gives

The two classical label bits are either consumed by $\mathsf{dropBit}$ without feedback or exported to an experimenter-side archive with no return path. No ancillary quantum name is silently dropped. A fresh pair can be allocated at the next step.