Derivation — C073 (confusion-matrix inversion)

Forward (confusion) model

Let the true populations be $$T_1=P(\uparrow\downarrow)+P(\downarrow\uparrow)\ \ (\text{1 bright ion}),\qquad T_2=P(\uparrow\uparrow)\ \ (\text{2 bright ions}).$$ The supp. defines $p$ = probability a 1-bright event is recorded as 2-bright, and $q$ = probability a 2-bright event is recorded as 1-bright. (0-bright vs 1-bright misclassification is negligible.) The recorded probabilities are then $$P(2)=(1-q)\,T_2+p\,T_1,\qquad P(1)=(1-p)\,T_1+q\,T_2,$$ i.e. $$\begin{pmatrix}P(1)\\P(2)\end{pmatrix} =\underbrace{\begin{pmatrix}1-p & q\\ p & 1-q\end{pmatrix}}_{M} \begin{pmatrix}T_1\\T_2\end{pmatrix}.$$

Inversion

$\det M=(1-p)(1-q)-pq=1-p-q$. Therefore $$M^{-1}=\frac{1}{1-p-q}\begin{pmatrix}1-q & -q\\ -p & 1-p\end{pmatrix},$$ and $$\begin{pmatrix}T_1\\T_2\end{pmatrix}=M^{-1}\begin{pmatrix}P(1)\\P(2)\end{pmatrix} =\frac{1}{1-p-q}\begin{pmatrix}(1-q)P(1)-q\,P(2)\\ -p\,P(1)+(1-p)P(2)\end{pmatrix}.$$

So $$T_2=P(\uparrow\uparrow)=\frac{(1-p)P(2)-p\,P(1)}{1-p-q},$$ $$T_1=P(\uparrow\downarrow)+P(\downarrow\uparrow)=\frac{(1-q)P(1)-q\,P(2)}{1-p-q}.$$

The first line matches the paper's Eq. (S7) exactly: $P(\uparrow\uparrow)=\frac{(1-p)P(2)-pP(1)}{1-p-q}$.

Discrepancy in Eq. (S8)

The paper's Eq. (S8) reads $$P(\uparrow\downarrow)+P(\downarrow\uparrow)=\frac{(1-q)P(2)-q\,P(1)}{1-p-q}.$$ My inversion gives $\frac{(1-q)P(1)-q\,P(2)}{1-p-q}$ — i.e. with $P(1)$ and $P(2)$ swapped in the numerator relative to the paper. The paper's S8 numerator is the same as the S7 numerator with $p\to q$, which is not the correct inverse of the confusion matrix.

Cross-check by consistency: a correct correction must preserve total probability, $T_1+T_2=P(1)+P(2)$ (both equal the total normalization). Summing my expressions: $$T_1+T_2=\frac{(1-q)P(1)-qP(2)+(1-p)P(2)-pP(1)}{1-p-q} =\frac{(1-p-q)P(1)+(1-p-q)P(2)}{1-p-q}=P(1)+P(2).\ \checkmark$$ Summing the paper's S7+S8: $$\frac{(1-p)P(2)-pP(1)+(1-q)P(2)-qP(1)}{1-p-q} =\frac{(2-p-q)P(2)-(p+q)P(1)}{1-p-q}\ne P(1)+P(2)$$ in general. So the paper's S8 as printed does not conserve probability and is inconsistent with S7 + the confusion model; the correct expression has $P(1)$ as the leading term, $\frac{(1-q)P(1)-qP(2)}{1-p-q}$.

Conclusion

The S7 formula for $P(\uparrow\uparrow)$ is verified exactly. The S8 formula for $P(\uparrow\downarrow)+P(\downarrow\uparrow)$ as printed does not match the inverse of the stated confusion matrix and fails the probability-conservation check; the correct form is $\frac{(1-q)P(1)-qP(2)}{1-p-q}$. This is most likely a typo (swapped $P(1)\!\leftrightarrow\!P(2)$ in the numerator) in the supplementary. Reported as a mismatch on the second equation. Verified symbolically in sympy_check.py.