## 2 Two-Mode Coherent-State Parameter Communication Scheme with Linear Encoding  
Usually, a parameter estimation problem can be divided into three stages: the first stage consists in preparing a probe state, which, in the second stage, acquires some unknown parameters (often phases) by interacting with the probed media. The probe state is then measured in the third stage and the parameters are inferred from the measurement outcomes. The goal is of course to estimate the unknown parameters as precisely as possible, that is, to minimize the variance of each parameter estimator.  
In this paper, we adopt a variant of this procedure, more suitable for parameter communication purposes. We consider the first two stages of the above scheme to be performed by the sender, [PERSON], who prepares the states and encodes some parameters into them. The third stage is then performed by the receiver, [PERSON], who is allowed to use linear optics and homodyne measurements. Moreover, we authorize [PERSON] and [PERSON] to exchange prior messages on the measurement settings with the restriction that no relevant information about the unknown parameters can be inferred from these messages.  
Specifically, we consider the following parameter communication scheme. To send [PERSON] a complex number \(z=(a+ib)/\sqrt{2}\) through an optical channel, [PERSON] generates two coherent states \(|\alpha_{1}\rangle\otimes|\alpha_{2}\rangle\) at her choice, encoding the real numbers \(a\) and \(b\). Coherent states can be regarded as resulting from the action of the displacement operator on the vacuum state \(|0\rangle\), namely  
\[|\alpha\rangle=\exp(ip\hat{x}-ix\hat{p})|0\rangle, \tag{1}\]  
where \(x\) and \(p\) are the displacement parameters and \(\hat{x}\) and \(\hat{p}\) are the quadrature operators. In this way, the state \(|\alpha_{1}\rangle\otimes|\alpha_{2}\rangle\) is characterized by the quadrature vector \(\mathbf{r}=(x_{1},p_{1},x_{2},p_{2})\). The commutation relations between the quadrature operators are \(\begin{bmatrix}\hat{x}_{i},\hat{p}_{j}\end{bmatrix}=i\delta_{ij}\) in natural units (\(\hbar=1\)) for \(i,j=1,2\). A natural condition, which we impose on these coherent states, is some total energy constraint on the input state:  
\[x_{1}^{2}+p_{1}^{2}+x_{2}^{2}+p_{2}^{2}=2(a^{2}+b^{2}). \tag{2}\]  
This constraint means that average energy per mode is equal to some \"signal energy\" \(a^{2}+b^{2}\). [PERSON], on his side, has only passive linear optics (beam splitters and phase shifters) and homodyne detectors in order to build a measurement apparatus for estimating parameters \(a\) and \(b\). The goal for [PERSON] is to estimate optimally both \(a\) and \(b\) knowing that they are encoded linearly in the quadratures of the coherent states \(|\alpha_{1}\rangle\otimes|\alpha_{2}\rangle\).  
The most general linear encoding of the real parameters \(a\) and \(b\) in the quadratures of a two-mode coherent state system is  
\[\begin{split} x_{1}=&\epsilon_{x1}a+\eta_{x1}b,\\ p_{1}=&\epsilon_{p1}a+\eta_{p1}b,\\ x_{2}=&\epsilon_{x2}a+\eta_{x2}b,\\ p_{2}=&\epsilon_{p2}a+\eta_{p2}b.\end{split} \tag{3}\]Thus, [PERSON] has to choose eight real constants \((\epsilon_{k,l},\eta_{k,l})\), where \(k=x,p\) and \(l=1,2\), in a way that allows [PERSON] to optimally retrieve the encoded parameters \(a\) and \(b\). In the next section, optimality is defined as the attainability of the quantum Cramer-Rao bound on the variance of both estimators of classical parameters \(a\) and \(b\). As we will show, attaining this bound imposes several constraints on the eight constants.