Coherent-state constellations and polar codes for thermal Gaussian channels

Optical communication channels are ultimately quantum mechanical in nature, and we must therefore look beyond classical information theory to determine their communication capacity as well as to find efficient encoding and decoding schemes of the highest rates. Thermal channels, which arise from linear coupling of the field to a thermal environment, are of particular practical relevance; their classical capacity has been recently established, but their quantum capacity remains unknown. While the capacity sets the ultimate limit on reliable communication rates, it does not promise that such rates are achievable by practical means. Here we construct efficiently encodable codes for thermal channels which achieve the classical capacity and the so-called Gaussian coherent information for transmission of classical and quantum information, respectively. Our codes are based on combining polar codes with a discretization of the channel input into a finite “constellation” of coherent states. Encoding of classical information can be done using linear optics.

Figure 1

(a) The thermal channel $\mathcal{E}$ mixes the input state $\rho$ with the thermal state of mean photon number $N_0$ on a beamsplitter of transmittivity $k$. (b) The modulator taking classical input $j$ to the $j\mathrm{th}$ coherent state $|z_j\rangle$ in the constellation. (c) The effective channel resulting from combining the modulator with the thermal channel. (d) The classical coding scheme using the effective channel. In the quantum scheme the encoder accepts a quantum state and the modulator must be able to output superpositions of constellation coherent states.

Figure 2

Four constellations of $m=7$ points which approximate a normally distributed random variable. The random-walk and equilattice constellations consist of equally spaced points, while the quantile and equilattice have equal probabilities. Gauss-Hermite quadrature has neither, but precisely reproduces the largest number of moments of the Gaussian distribution, the first $2m-1$. The others have only the same mean and variance.

Figure 3

Achievable rates for classical and quantum information transmission with increasing constellation size $m$, for the pure loss channel ($N_0=0$) with $k=0.8$ and input state ${\tau }_N$ with $N=7$. Although the Gauss-Hermite constellation (straight line) is optimal as $m\to \infty$, it is inferior for small $m$. For practical purposes, the random-walk constellation (dashed line) is best, rising very quickly to rates quite close to capacity.