Finite-reference balanced homodyne variance ($\mathrm{Var}I=4|\beta {|}^2\mathrm{Var}{X}_{\vartheta }+⟨a^{†}a⟩$)

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 $i(a^{†}\delta b-a\delta b^{†})$. Cross terms have zero expectation, and the second term has variance $⟨a^{†}a⟩$. Consequently the exact variance is $4|\beta {|}^2\mathrm{Var}{X}_{\vartheta }+⟨a^{†}a⟩$. Division by $4|\beta {|}^2$ approaches the signal optical quadrature variance when the photon-number correction is negligible.