Capacity of a bosonic memory channel with Gauss-Markov noise

We address the classical capacity of a quantum bosonic memory channel with additive noise, subject to an input energy constraint. The memory is modeled by correlated noise emerging from a Gauss-Markov process. Under reasonable assumptions, we show that the optimal modulation results from a “quantum water-filling” solution above a certain input energy threshold, similar to the optimal modulation for parallel classical Gaussian channels. We also derive analytically the optimal multimode input state above this threshold, which enables us to compute the capacity of this memory channel in the limit of an infinite number of modes. The method can also be applied to a more general noise environment which is constructed by a stationary Gauss process. The extension of our results to the case of broadband bosonic channels with colored Gaussian noise should also be straightforward.

Figure 1

(Color online) Quantum water-filling solution for the monomodal channel. Optimal eigenvalues for both quadratures.

Figure 2

(Color online) Global quantum water-filling solution. Output spectra ${\gamma }_{\text{out}}^{\left(q\right)}\left(x\right)$ and ${\gamma }_{\text{out}}^{\left(p\right)}\left(x\right)$ shown by the solid decreasing curve and dashed increasing curve vs spectral parameter $x$. The solid bar represents the global quantum water-filling level ${\mu }_{\text{gl}}$; the light gray area shows the energy used for modulation in $q$.

Figure 3

(Color online) Fraction of input energy used for squeezing $\eta$ and capacity $C$ in bits per use vs noise parameter $N$. The increasing curves correspond to the capacities; the decreasing to $\eta$. Gray horizontal lines correspond to the classical limit $C_{\text{cl}}$ of each capacity. For all graphs, the dotted, dashed, and solid line correspond to $\varphi =0.4,0.7,0.9$. For each $\varphi$, we set $\forall N\ge 1:\overline{n}/N={\overline{n}}_{\text{thr}}\left(\varphi ,N^{\prime }\right)/N^{\prime }$, with $N^{\prime }\equiv 1$.

Figure 4

Function $R^{\left(n\right)}$ and capacity $C$ in bits per use vs number of channel uses $n$. The dotted, dashed, dashed-dotted, and solid curve correspond to $R^{\left(n\right)}$ with $\varphi =0,0.4,0.55,0.7$. For each $\varphi$, the corresponding capacity is shown by a gray horizontal line. For all plots, we took $N=1$ and $\overline{n}=7.5>{\overline{n}}_{\text{thr}}\left(\varphi =0.7,N=1\right)$.