## 1 Introduction
In our scheme, we also eliminate this serious drawback and extend the reconciliation of Gaussian variables to arbitrary high dimensions. The proposed approach also makes possible to get a closer and more precise approximation of the binary Gaussian channel, in comparison to the multidimensional case. Since the post-processing phase uses the binary form of the continuous variables, in fact, we do not have to decode the Gaussian variables in the multidimensional space. As a corollary, arbitrary high-precision approximation of the logical binary Gaussian channel can be made in the non-spherical space by using considerable dimensions. We exploit it in this work to construct a scalar reconciliation that breaks with the traditions of the previously introduced approaches [9; 10; 11; 42; 46], and uses only the space of scalar variables. The proposed scalar reconciliation is also able to transform the physical Gaussian quantum channel into a logical binary Gaussian channel in two-way CVQKD, and the same benefits can be exploited as in the case of multidimensional reconciliation. However since our scheme is not limited to eight dimensions, an arbitrary precision can be reached in the approximation of the logical binary Gaussian channel. As follows, the accuracy of the conversion between the physical Gaussian quantum channel and the logical Gaussian channel can be improved beyond the current limits. Another issue in the current approaches is the requirement of spherical calculations. To make the existing post-processing approaches more efficient, we have to eliminate the multidimensional operations. The reconciliation of Gaussian variables would be much easier, if we found a solution that would make it possible to extract the final key from the noisy data by simple calculations in the level of scalar space. It immediately follows that this would significantly increase the efficiency of the reconciliation process, and would lead to a negligible complexity and computational power in the error-correction procedure. In this paper, we define low-dimensional (scalar) reconciliation for CVQKD. It brings significantly higher noise-resistance and information-transmission capability, extended transmission distances, and improved key rates. The proposed method does the reconciliation of Gaussian variables without the need of any physical-layer tomography or multidimensional operations. We demonstrate the results for two-way CVQKD. The scheme is backward compatible it also can be applied to one-way CVQKD. The novel contribution of our paper is as follows:  
* The reconciliation process of correlated Gaussian variables is a complex problem that requires either tomography in the physical layer that is intractable in a practical scenario, or high-cost calculations in the multidimensional spherical space with strict dimensional limitations. * To avoid these issues, we propose an efficient logical layer-based reconciliation method for CVQKD to extract binary information from correlated Gaussian variables. * We demonstrate that by operating on the raw-data level, the noise of the quantum channel can be corrected in the low-dimensional scalar space and the reconciliation can be extended to arbitrary dimensions. * We prove that the error probability of scalar reconciliation is zero in any practical CVQKD scenario, and provides unconditional security. * The results allow to significantly improve the currently available key rates and transmission distances of CVQKD. This paper is organized as follows. In Section 2, preliminary findings are summarized. In Section 3, we introduce the reconciliation scheme. Section 4 provides the theorems and proofs. In Section 5, numerical evidence is proposed. Finally, in Section 6, we conclude the paper. Supplementary Materials are also included, notations are summarized in Table S1.