Residual instruments

A sufficient accessible state need not be the carrier marginal alone. Hidden memory can invalidate a purported carrier-only interval-channel description.

For example, start $S$ in $\rho$ and a bath qubit in $I/2$. Two successive SWAPs have carrier channels: depolarizing after the first and identity after the second. A second model stores a uniformly random Pauli label $a$ classically and applies the same Pauli twice. It has the same two unintervened single-time channels. Replace the carrier by $|0⟩⟨0|$ between steps. The SWAP model returns $\rho$, whereas the Pauli-label model returns $I/2$. Thus agreement without that intervention does not extend to the replacement protocol.

Likewise, after the outputs in eq:obstruction-outputs, an $X$-basis readout gives probability $1$ versus $1/2$ for $+$. The residual difference becomes a later record difference.

If approximations obey

then the same input gives

Improved implementation does not remove incompatible ideal targets.