= Finite-reference balanced homodyne variance
{title2=$\operatorname{Var}I=4|\beta|^2\operatorname{Var}X_\vartheta+\langle a^\dagger a\rangle$}
For the convention of <balanced homodyne detection>, assume independent signal and coherent reference states and ideal detectors. Write $b=\beta+\delta b$, with $\delta b$ in vacuum. The current difference splits into $2|\beta|X_\vartheta$ and $i(a^\dagger\delta b-a\delta b^\dagger)$. Cross terms have zero expectation, and the second term has variance $\langle a^\dagger a\rangle$. Consequently the exact variance is $4|\beta|^2\operatorname{Var}X_\vartheta+\langle a^\dagger a\rangle$. Division by $4|\beta|^2$ approaches the signal <optical quadrature> variance when the photon-number correction is negligible.
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