Example: Adaptive peak need not obey a prefix Choi bound

A qubit depolarizing channel is the uniform mixture of the four Pauli conjugations. Its four Pauli Kraus vectors are orthogonal, so its Choi rank is four. Its one-shot environment ledger requires two environment qubits and peak three including the input.

An adaptive implementation needs only one coin qubit at a time. Prepare and destructively read a Hadamard coin, apply $X$ or $I$ to the input according to that bit, reuse the freed slot for a second coin, and apply $Z$ or $I$. Classically forgetting the two bits gives the depolarizing channel. The peak is two live qubits.

Thus the Choi rank of the accumulated channel bounds a coherent environment that retains its whole dilation, not the instantaneous quantum memory of every adaptive implementation.