## 4 Two-Mode Encoding and Estimation Schemes
### Optimal Scheme
Thus, for \(T=1\) or \(T=0\), a joint measurement cannot provide any enhancement of the measurement precision. For other values of \(T\), we have a continuous evolution of the precision enhancement with respect to individual measurements ranging between 2 (maximum enhancement) and 1 (no enhancement at all). To see this, let us compare the error variances of the estimators of the encoded parameters \(a\) and \(b\) obtained by the optimal joint or optimal individual measurement. The variances of the joint measurement saturate the QCRB by construction, namely \(\Delta\tilde{a}_{\text{QCRB}}^{2}=\Delta\tilde{b}_{\text{QCRB}}^{2}=1/4\). As already mentioned, the optimal individual measurement consists in measuring \(\hat{x}_{1}\) and \(\hat{p}_{2}\) for \(T\geq 1/2\), or \(\hat{x}_{2}\) and \(\hat{p}_{1}\) for \(T\leq 1/2\), so we may deduce from Equation (29) the corresponding variances of the estimators  
\[\Delta\tilde{a}_{ind}^{2}(T) = \frac{\Delta\tilde{x}_{i}^{2}}{1+|1-2T|}=\frac{1}{2+|2-4T|}, \tag{31}\] \[\Delta\tilde{b}_{ind}^{2}(T) = \frac{\Delta\hat{p}_{3-i}^{2}}{1+|1-2T|}=\frac{1}{2+|2-4T|},\] (32) \[i = \left\{ \begin{array}{ll}1,&\quad(T>1/2)\\ 2,&\quad(T<1/2),\end{array} \right. \tag{33}\]  
where we have made the dependence on \(T\) explicit. Due to this dependence, the enhancement in the measurement precision can be expressed in terms of the ratio between the error variance on the estimators obtained by individual and joint measurements as a function of \(T\):  
\[\frac{\Delta\tilde{a}_{ind}^{2}(T)}{\Delta\tilde{a}_{\text{QCRB}}^{2}}=\frac{ \Delta\tilde{b}_{ind}^{2}(T)}{\Delta\tilde{b}_{\text{QCRB}}^{2}}=\frac{2}{1+ |1-2T|}. \tag{34}\]  
As shown in Figure 2, the precision enhancement attains its maximum value 2 at \(T=1/2\) and its minimum value 1 at \(T=0\) or 1. Given \(a\) and \(b\), the beam splitter transmissivity \(T\) defines a family of points \((x_{i},p_{i})\) in phase space--hence, a family of pairs of coherent states--which allow an optimal encoding as defined by  
Figure 2: Enhancement of the measurement precision of the joint (compared to individual) measurement as a function of the beam splitter transmissivity \(T\). This is calculated as the reduction of the error variance for the joint measurement attaining the QCRB with respect to the error variance of the optimal individual measurements. Equation (29). From this equation, it is easy to deduce that these points belong to an elliptic curve determined by the classical parameters \(a\) and \(b\)  
\[\frac{x_{i}^{2}}{a^{2}}+\frac{p_{i}^{2}}{b^{2}}=2,\quad i=1,2. \tag{35}\]  
Therefore, the general encoding protocol implies the generation of two coherent states determined by the parameters \(a\) and \(b\) to be transmitted (as shown in Figure 3) and a measurement setting \(T\), which has to be specified before the measurement. We note that, by allowing local phase rotations of the input states and by suitably choosing phase angles \(\theta\) and \(\phi\), one can use any pair of coherent states for encoding two classical parameters \(a\) and \(b\) (satisfying the energy constraint), which can be further optimally extracted by joint measurement including a beam splitter with a suitable transmissivity \(T\) followed by homodyne measurements on two output modes. The corresponding protocol is described in Appendix A.