## 3 Parameter Estimation
### Attainability of the Quantum Cramer-Rao Bound  
A necessary and sufficient condition for the attainability of the QCRB was introduced by [PERSON] (1976) for pure states \(\rho=|\psi\rangle\langle\psi|\). This condition, which was referred to as the _commutation condition_ in [PERSON] (1977), is expressed as  
\[\text{Tr}(\rho[L_{\theta_{i}},L_{\theta_{j}}])=0, \tag{19}\]  
where \(\rho\) is a pure state. It translates into a condition on our encoding constants \((\epsilon_{ij},\eta_{ij})\), as we show below. The SLDs for a two-mode coherent state \(\rho_{12}=|\alpha_{1}\rangle\langle\alpha_{1}|\otimes|\alpha_{2}\rangle \langle\alpha_{2}|\) in which the classical parameters \(a\) and \(b\) are encoded with the linear encoding strategy in Equation (3) are given by  
\[ \begin{array}{llll}L_{a}=&i[\epsilon_{p1}\,(\hat{x}_{1}\rho_{12}-\rho_{12} \hat{x}_{1})&+&\epsilon_{p2}\,(\hat{x}_{2}\rho_{12}-\rho_{12}\hat{x}_{2})\\ &-\epsilon_{x1}\,(\hat{p}_{1}\rho_{12}-\rho_{12}\hat{p}_{1})&-&\epsilon_{x2}\, (\hat{p}_{2}\rho_{12}-\rho_{12}\hat{p}_{2})],\\ L_{b}=&i[\eta_{p1}\,(\hat{x}_{1}\rho_{12}-\rho_{12}\hat{x}_{1})&+&\eta_{p2}\, (\hat{x}_{2}\rho_{12}-\rho_{12}\hat{x}_{2})\\ &-\eta_{x1}\,(\hat{p}_{1}\rho_{12}-\rho_{12}\hat{p}_{1})&-&\eta_{x2}\,(\hat{p }_{2}\rho_{12}-\rho_{12}\hat{p}_{2})],\end{array}  \tag{20}\]  
where we have plugged Equation (9) into the formulas of the SLDs for pure states in Equation (6) where \(\theta_{i}=a,b\). Inserting Equation (20) into Equation (19), we observe that the only surviving terms are those involving the commutator \([\hat{x}_{i},\hat{p}_{i}]\), which leads to  
\[ \begin{array}{llll}\text{Tr}(\rho_{12}[L_{a},L_{b}])&=&i\langle\alpha_{1}|( \epsilon_{p1}\eta_{x1}-\epsilon_{x1}\eta_{p1})[\hat{x}_{1},\hat{p}_{1}]|\alpha _{1}\rangle+i\langle\alpha_{2}|(\epsilon_{p2}\eta_{x2}-\epsilon_{x2}\eta_{p2} )[\hat{x}_{2},\hat{p}_{2}]|\alpha_{2}\rangle\\ &=&-\epsilon_{p1}\eta_{x1}+\epsilon_{x1}\eta_{p1}-\epsilon_{p2}\eta_{x2}+ \epsilon_{x2}\eta_{p2}.\end{array}  \tag{21}\]  
As a result, we obtain the _attainability condition_ on the encoding constants  
\[\epsilon_{p1}\eta_{x1}-\epsilon_{x1}\eta_{p1}+\epsilon_{p2}\eta_{x2}-\epsilon_ {x2}\eta_{p2}=0. \tag{22}\]  
Together with the energy constraint in Equation (2) and equal precision conditions in Equation (15), the attainability condition in Equation (22) gives rise to a system of four equations that an optimal encoding strategy must satisfy, namely  
\[ \begin{array}{l}\epsilon_{x1}^{2}+\epsilon_{p1}^{2}+\epsilon_{x2}^{2}+ \epsilon_{p2}^{2}=2,\\ \eta_{x1}^{2}+\eta_{p1}^{2}+\eta_{x2}^{2}+\eta_{p2}^{2}=2,\\ \epsilon_{p1}\eta_{p1}+\epsilon_{x1}\eta_{x1}+\epsilon_{p2}\eta_{p2}+ \epsilon_{x2}\eta_{x2}=0,\\ \epsilon_{p1}\eta_{x1}-\epsilon_{x1}\eta_{p1}+\epsilon_{p2}\eta_{x2}-\epsilon_ {x2}\eta_{p2}=0.\end{array}  \tag{23}\]As we only have four constraints for eight real constants \((\epsilon_{i,j},\eta_{i,j})\), there is no unique solution to this system. The most general encoding corresponds to the one that can be optimally decoded by using the most general two-mode passive Gaussian state composed of three local phases and one beam splitter. Therefore, the most general two-mode encoding can be written in the following way:  
\[ \begin{array}{ll}\epsilon_{x1}=\sqrt{2\overline{T}}\cos(\theta),&\eta_{x1}= \sqrt{2(1-T)}\left(\sin(\theta)\sin(\psi)-\cos(\theta)\cos(\psi)\right),\\ \epsilon_{p1}=\sqrt{2\overline{T}}\sin(\theta),&\eta_{p1}=-\sqrt{2(1-T)}\left( \sin(\theta)\cos(\psi)+\cos(\theta)\sin(\psi)\right),\\ \epsilon_{x2}=\sqrt{2(1-T)}\cos(\phi),&\eta_{x2}=\sqrt{2T}\left(\cos(\phi) \cos(\psi)-\sin(\phi)\sin(\psi)\right),\\ \epsilon_{p2}=\sqrt{2(1-T)}\sin(\phi),&\eta_{p2}=\sqrt{2T}\left(\cos(\phi) \sin(\psi)+\cos(\phi)\sin(\psi)\right).\end{array}  \tag{24}\]  
On the decoding side (see Figure 1), [PERSON] should first apply two local phase rotations of angles \(\theta\) and \(\phi\), followed by a beam splitter of transmittance \(T\), then a local rotation on the second mode of angle \(\psi\) and finally measure the \(x\)-quadrature on both modes.