Proposition: A priced coherent one-shot realization

The frozen channel has Choi rank four. In a pure-environment isometric realization, its minimum environment dimension is four, encodable in two qubits. The coherent output-plus-environment interface therefore occupies three qubits before any environmental readout.

This is a one-shot dilation ledger, not a lower bound on every adaptive implementation or on a collective decoder's memory.

Proof. The middle diagonal entries of (eq:choi) are positive. The determinant of its remaining two-by-two block is

Thus its rank is four.

A Kraus decomposition gives $J(\mathcal{A})$ as a sum of rank-one Kraus-vector projectors, so at least four Kraus matrices are needed. Conversely, its spectral decomposition gives four. The minimal pure dilation environment has that dimension. A two-qubit environment and one output qubit attain the stated coherent interface count.