An explicit finite dilation and schedule

A fresh environment qubit $E$, initially in $|0⟩$, realizes the antiqubit relaxation by the isometry

An explicit unitary extension is obtained by applying:

1. a rotation $R_y(2\arcsin \sqrt{\gamma })$ on $E$, controlled by $A=1$;

2. ${\mathrm{CNOT}}_{E\to A}$.

Here $R_y(\omega )=e^{-i\omega Y/2}$. The two formulas in (eq:dilation) follow directly.

Destructive Enter of $E$, with its outcome not exported, gives the two Kraus branches $L_0,L_1$ and consumes the environment's laboratory name. Destructive Enter of $N,A$ then produces the exported word.

A complete one-shot schedule, after the supplier delivers the data registers, is:

1. allocate $E$ in $|0⟩$;

2. apply $V_{\varphi }$ and ${V_{\varphi }}^{†}$ in parallel;

3. apply $\mathrm{CZ}$;

4. apply the controlled environment rotation;

5. apply ${\mathrm{CNOT}}_{E\to A}$;

6. Enter $E$ and forget its unexported outcome;

7. Enter $N,A$ and export both outcome bits.

Forgetting the unexported classical value is an interface operation. No physical cyclic erasure cost is assigned to it.