## 3 Parameter Estimation
### Optimal Lower Bound  
Here, the parameters \(a,b\) are encoded into the \(x\) and \(p\) quadratures of the input product coherent state \(|\psi\rangle=|\alpha_{1}\rangle\otimes|\alpha_{2}\rangle\), which justifies the use of homodyne detection in the optimal measurement scheme in Section 4. To find the quantum Cramer-Rao bound on the variances \(\Delta^{2}\tilde{a}\) and \(\Delta^{2}\tilde{b}\) of the estimators of \(a\) and \(b\), we use the additivity of the QFIM. It implies that the QFIM for the product state \(|\psi\rangle=|\alpha_{1}\rangle\otimes|\alpha_{2}\rangle\) is the sum of the QFIM of the individual states \(|\alpha_{1}\rangle\) and \(|\alpha_{2}\rangle\), namely  
\[F^{(\alpha_{1}\alpha_{2})}=F^{(\alpha_{1})}+F^{(\alpha_{2})}. \tag{8}\]Thus, we simply need to express the individual QFIM for each of the two coherent states. First, we calculate the derivatives with respect to \(a\) and \(b\) of the individual coherent states  
\[ \begin{split}\partial_{a}|\alpha_{j}\rangle&=i\left( \epsilon_{pj}\hat{x}_{j}-\epsilon_{xj}\hat{p}_{j}+2\epsilon_{xj}\epsilon_{pj}+ \frac{3\epsilon_{xj}\eta_{pj}+\epsilon_{pj}\eta_{xj}}{2}b\right)|\alpha_{j} \rangle,\\ \partial_{b}|\alpha_{j}\rangle&=i\left(\eta_{pj} \hat{x}_{j}-\eta_{xj}\hat{p}_{j}+2\eta_{xj}\eta_{pj}+\frac{3\eta_{xj}\epsilon_ {pj}+\eta_{pj}\epsilon_{xj}}{2}b\right)|\alpha_{j}\rangle,\end{split}  \tag{9}\]  
where \(j=1,2\) indicates the mode number. By using Equation (9) and the definition in Equation (7) with \(\theta_{i}=a,b\), we obtain the QFIM for each mode  
\[F^{(a_{j})}=2\left( \begin{array}{cc}\epsilon_{xj}^{2}+\epsilon_{pj}^{2}& \epsilon_{pj}\eta_{pj}+\epsilon_{xj}\eta_{xj}\\ \epsilon_{pj}\eta_{pj}+\epsilon_{xj}\eta_{xj}&\eta_{xj}^{2}+\eta_{pj}^{2} \end{array} \right). \tag{10}\]  
It is worth noting that the single-mode QFIM in Equation (10) is not in general an invertible matrix, which originates from the non-commutativity of the quadrature operators \(\hat{x}\) and \(\hat{p}\) and is a manifestation of the Heisenberg uncertainty principle. It means that it is in general not possible to extract the two parameters optimally from a single mode, that is, one cannot reach the variance \(\left(F(\alpha_{j})_{ii}\right)^{-1}/N\) simultaneously for both parameter estimators. As a consequence of this non-invertibility, one cannot use the QFIM in Equation (10) to calculate the QCRB on the estimator of \(a\) and \(b\). However, the QFIM becomes invertible when applied to two coherent states \(|\alpha_{1}\rangle\otimes|\alpha_{2}\rangle\), namely  
\[F^{(\alpha_{1}\alpha_{2})}=2\left( \begin{array}{cc}A&C\\ C&B\end{array} \right). \tag{11}\]  
where  
\[ \begin{split} A&=\epsilon_{x1}^{2}+\epsilon_{p1}^{2}+\epsilon _{x2}^{2}+\epsilon_{p2}^{2},\\ B&=\eta_{x1}^{2}+\eta_{p1}^{2}+\eta_{x2}^{2}+\eta_{p2}^{2},\\ C&=\epsilon_{p1}\eta_{p1}+\epsilon_{x1}\eta_{x1}+\epsilon_{p2}\eta_{p2}+ \epsilon_{x2}\eta_{x2}.\end{split}  \tag{12}\]  
Its eigenvalues are given by  
\[\lambda_{\pm}=\frac{A+B\pm\sqrt{(A+B)^{2}+4(C^{2}-AB)}}{2}. \tag{13}\]  
We want to estimate both classical parameters \(a\) and \(b\) with the same precision since they are equally important in order to estimate the complex number \(z=(a+ib)/\sqrt{2}\). This _equal precision condition_ means that both eigenvalues should be equal \(\lambda_{+}=\lambda_{-}\), which implies the condition  
\[(A-B)^{2}=-4C^{2}. \tag{14}\]  
According to Equation (7), \(A\), \(B\) and \(C\) should be real numbers, hence the only solution of Equation (14) is expressed by two conditions  
\[A=B,\qquad C=0. \tag{15}\]  
Furthermore, we can rewrite the energy constraint in terms of the QFIM parameters as  
\[(A-2)(a^{2}+b^{2})=0. \tag{16}\]This equation should be satisfied for all real \(a\) and \(b\), since they can be chosen arbitrarily by [PERSON], so that we only have the condition \(A=2\).