Theorem: Clean flagged formulas and the complete record

For $q=q_n$,

Immediate cell readout returns histories consisting of either no emission or an emission time $j$ and basis label $i$. For $u_i=⟨i|\rho |i⟩$, their probabilities are

Its recovery error and complete-record entropy are

If $q_{n-1}>0$ and $t_n=q_n/q_{n-1}$, then

If $q_{n-1}=0$, the increment is zero.

Proof. The channels are orthogonal-block channels

The decoder compressing the occupied block and preparing $I_C/d$ on vacancy gives the radiation upper bound $q{a}_d$. On a Bell input, no decoder has overlap greater than $1-q+q/d^2$, proving optimality. Interchanging occupied probabilities gives the residual recovery error. The exact residual leakage is proved in Appendix app:flag; it is $q{a}_d$, not residual recovery. The entropy formulas follow from the orthogonal blocks.

Successive local readouts may be commuted past later rotations on other cells for the purpose of computing the complete outcome law. Their probabilities are therefore $p_j{u}_i$ and $q_n$.

Preparing $|i⟩$ on a reported label and $I_C/d$ on no emission gives decoded channel $(1-q_n)\Delta +q_n{\mathcal{T}}_{I_C/d}$. Its error is at most $(1-q_n)b_d+q_n{a}_d$. The Bell test gives the same lower bound, irrespective of emission time. The history entropy follows by splitting the no-emission branch and then its label. At the next opportunity only the former no-emission outcome is refined, giving the displayed increment.