## 4 Two-Mode Encoding and Estimation Schemes
### Optimal Scheme  
As we show now, an optimal encoding and estimation scheme attaining the QCRB can be obtained by selecting two appropriate input coherent states \(|\alpha_{1}\rangle\) and \(|\alpha_{2}\rangle\) and realizing a joint measurement, processing them through a beam splitter of transmittance \(T\) followed by two homodyne measurements, as shown in Figure 1.  
To realize this optimal measurement, we choose a particular solution of Equation (23) of the form  
\[ \begin{array}{ll}\epsilon_{x1}=\sqrt{2T},&\epsilon_{p1}=0,\\ \epsilon_{x2}=\sqrt{2(1-T)},&\epsilon_{p2}=0,\\ \eta_{x1}=0,&\eta_{p1}=\sqrt{2(1-T)},\\ \eta_{x2}=0,&\eta_{p2}=-\sqrt{2T},\end{array}  \tag{28}\]  
which corresponds to the following encoding of the classical parameters \(a\) and \(b\) into the quadratures of the two input coherent states  
\[ \begin{array}{ll}x_{1}^{in}=\sqrt{2T}\ a,\\ x_{2}^{in}=\sqrt{2(1-T)}\ a,\end{array} \qquad \begin{array}{ll}p_{1}^{in}= \sqrt{2(1-T)}\ b,\\ p_{2}^{in}=-\sqrt{2T}\ b,\end{array}  \tag{29}\]  
that is, [PERSON] sends the state \(|\psi\rangle=|\alpha_{1}\rangle\otimes|\alpha_{2}\rangle=\left|\sqrt{T}\ a+i\, \sqrt{1-T}\ b\right\rangle\otimes\left|\sqrt{1-T}\ a-i\,\sqrt{T}\ b\right\rangle\). At the output of [PERSON]'s beam splitter, the quadratures are  
\[ \begin{array}{ll}x_{1}^{out}=\sqrt{2}\ a,&p_{1}^{out}=0,\\ x_{2}^{out}=0,&p_{2}^{out}=\sqrt{2}\ b,\end{array}  \tag{30}\]  
regardless of the value of the transmittance \(T\). It is then easy to see that the homodyne detection of the two output modes (measuring \(\hat{x}\) on mode 1 and \(\hat{p}\) on mode 2) provides estimates of the parameters \(a\) and \(b\) with variances \(\Delta^{2}\tilde{a}=\Delta^{2}\tilde{b}=1/4\) saturating the QCRB in Equation (18). The optimal encoding and decoding protocol is thus physically implementable with linear optics. To shed light on the feature that makes this protocol interesting, let us compare it with individual measurements of the quadratures of the two modes (eliminating the beam splitter) for the whole range of values of \(T\in[0,1]\). Individual homodyne measurements of two coherent states encoded according to Equation (29) achieve smaller relative errors when measuring \(x_{1}^{in}\) and \(p_{2}^{in}\) for \(T\geq 1/2\), or \(x_{2}^{in}\) and \(p_{1}^{in}\) for \(T\leq 1/2\). Thus, when comparing with the joint measurement protocol (including the beam splitter), we need to consider the individual measurements of \(x_{1}^{in}\) and \(p_{2}^{in}\) for \(T\geq 1/2\) and \(x_{2}^{in}\) and \(p_{1}^{in}\) for \(T\leq 1/2\). Interestingly, we note that, for \(T=1/2\), the encoding \(\mathbf{r}_{1/2}^{in}=(a,b,a,-b)\) given by Equation (29) reduces to a scheme based on phase-conjugate coherent states introduced by [PERSON] and [PERSON] in 2001 [14]. In their paper, [PERSON] and [PERSON] noted that this particular encoding (\(\alpha_{2}=\alpha_{1}^{*}\)) provides an enhancement of the measurement precision by reducing the error variances of the quadratures by a factor 2 compared to individual measurements. Nevertheless, no proof of optimality of the [PERSON] scheme had been found in a further work investigating the superiority of joint measurements over local strategies for the estimation of product coherent states [15]. As we show below, this particular optimal scheme is the one which shows the best improvement when comparing it with individual measurements (see Figure 2). A natural question is indeed to compare this improvement with the one exhibited by the other optimal protocols in this family (with other values of the beam splitter transmissivity \(T\)). Observing first the limiting trivial cases \(T=1\) and \(T=0\), we see that the corresponding encodings\(\mathbf{r}_{1}^{in}=(\sqrt{2}a,0,0,-\sqrt{2}b)\) and \(\mathbf{r}_{0}^{in}=(0,\sqrt{2}a,\sqrt{2}b,0)\) are already in optimal configurations, so that the variances of the estimators obtained by individual homodyne measurement of the input quadratures already saturate QCRB.