## 1 Introduction  
Quantum estimation theory as introduced by [PERSON] [1] in the 1960s sets fundamental bounds on the extraction of classical parameters encoded in quantum systems. Since then, a large body of work has been devoted to the estimation of parameters, especially phases encoded in quantum optical states (see, e.g., Refs. [2; 3; 4; 5; 6] for recent works on multiparameter phase estimation and [7; 8; 9; 10; 11; 12] for the problem of estimation of parameters induced by non-commuting generators). We recommend to the interested reader the review by [PERSON] et al. [13]. In most of these studies, it is usually assumed that we have no control on the value of the parameters being estimated. In this paper, we consider a communication protocol between [PERSON] and [PERSON] where [PERSON] can _choose_ the values of the parameters she wants to send and _how_ she encodes these parameters in a collection of coherent states she has at her disposal. More precisely, we consider the problem of encoding and estimating classical parameters in canonically conjugate quantum variables, such as the quadrature components of coherent states of light. We derive the quantum Cramer-Rao bounds (QCRB) for an arbitrary linear encoding of two classical parameters into the quadratures of two coherent states. Furthermore, we present an encoding and estimation protocol that achieves the QCRB for the simultaneous estimation of the two parameters. Finally, we generalize our protocol to encode a set of \(n\) classical parameters into the tensor product of \(n\) coherent states so that one can always simultaneously estimate all parameters optimally, using a measurement technique involving linear optics components followed by homodyne measurements. A corollary of this work is the proof of optimality of the scheme based on phase-conjugate coherent states proposed by [PERSON] and [PERSON] [14] in 2001, which was left as an open problem in [15].  
This article is organized as follows. In Section 2, we introduce the parameter communication problem that we address and explicitly write the encoding of two classical parameters in the quadratures of two coherent states. In Section 3, we review the quantum Cramer-Rao bound, which provides a lower bound on the variance of any (unbiased) estimator of the classical parameters encoded into the quantum states. This bound, which is valid for any measurement, relies on the quantum Fisher information. We thus calculate the quantum Fisher information with respect to the parameters encoded into the coherent states in the considered schemes, and write conditions on the attainability of the corresponding quantum Cramer-Rao bound. In Section 4, we show that the variances obtained in the Cerf-Iblisdir scheme based on phase-conjugate coherent states [14] saturate the quantum Cramer-Rao bound, hence proving the optimality of the scheme. More generally, we characterize a family of encoding schemes of two variables into a pair of coherent states which all saturate the quantum Cramer-Rao bound. Within this family, the Cerf-Iblisdir scheme provides the highest precision enhancement achieved by a joint measurement in comparison with local measurements. In Section 5, we further generalize the scheme to \(n\) variables encoded into \(n\) states, while we conclude in Section 6.