Quantum water-filling solution for the capacity of Gaussian information channels

We study the transmission of classical information via optical Gaussian channels with a classical additive noise under the physical assumption of a finite input energy including the energy of classical signal (modulation) and the energy spent on squeezing the quantum states carrying information. Multiple uses of a certain class of memory channels with correlated noise is equivalent to one use of parallel independent channels generally with a phase-dependent noise. The calculation of the channels capacity implies finding the optimal distribution of the input energy between the channels. Above a certain input energy threshold, the optimal energy distribution is given by a solution known in the case of classical channels as water-filling. Below the threshold, the optimal distribution of the input energy depends on the noise spectrum and on the input energy level, so that the channels fall into three different classes: the first class corresponds to very noisy channels excluded from information transmission, the second class is composed of channels in which only one quadrature (q or p) is modulated and the third class corresponds to the water-filling solution. Although the non-modulated quadrature in the channels of the second class is not used for information transmission, a part of the input energy is used for the squeezing the quantum state which is a purely quantum effect. We present a complete solution to this problem for one mode and analyze the influence of the noise phase dependence on the capacity. Contrary to our intuition, in the highly phase-dependent noise limit, there exists a universal value of the capacity which neither depends on the input energy nor on the value of noise temperature. In addition, similarly to the case of lossy channels for weak thermal contribution of the noise, there exists an optimal squeezing of the noise, which maximizes the capacity. © 2010 SPIE.SCOPUS: cp.p