Independent environments

There are two inaccessible key-purification qubits $C_a,C_b$, initially in $|00⟩$, and a dephasing qubit $Z_{\mathrm{e}}$, initially in $|0⟩$. Their free Hamiltonians are zero.

All initial environments are independent of $RM$. The key purification is not discarded from the complementary channel.