Theorem: Flagged-transfer calibration

After $j$ pulses, the complete map has the form

where the occupied ranges of $A_j,L$ are orthogonal to their vacuum flags. Its optimal recovery and leakage are

An optimal decoder compresses the occupied radiation code and prepares $I_C/d$ on its orthogonal complement.

If

each pulse conserves $H_0$. For one initially occupied cell,

For $K$ independent cells with the same $q_j$, this expectation is $K\omega (1-q_j)$, even for correlated logical labels.

Proof. Induction gives the pulse amplitudes above, and

This proves (eq:flagged) and its isometry property.

The accessible channel is

The stated decoder produces $(1-q_j){\mathrm{id}}_C+q_j{\mathcal{T}}_{I/d}$, with error $q_j{a}_d$. Any decoder's Bell overlap is at most $(1-q_j)+q_j/d^2$, proving optimality.

The complement is

Replacement $q_{j}L{I}_C{L}^{†}/d+(1-q_j)|\mathrm{vac}⟩⟨\mathrm{vac}|$ gives leakage at most $q_j{a}_d$. For an arbitrary replacement state, apply a CPTP pinching and retraction onto the occupied code block and vacuum flag. Let $u$ be its occupied weight. On a Bell input, the occupied Bell projector has expectation difference $q_j-u/d^2$. The effect equal to that projector on the occupied block and identity on vacuum has difference $u{a}_d$. If $u\le q_j$, the first is at least $q_j{a}_d$; if $u\ge q_j$, the second is at least $q_j{a}_d$. This proves the leakage lower bound.

Each pulse mixes equal-energy one-excitation states and is the identity on all other sectors. It commutes with $H_0$. The transfer probability is $1-q_j$, proving the energy formula. Linearity and addition of the independent number observables prove the $K$-cell statement.