Capacities of Lossy Bosonic Memory Channels

We introduce a general model describing correlated noise effects in quantum optical communication via attenuating media. The memory effects account for the environment finite relaxation times, which are unavoidable in any realistic model. The use of a proper set of collective field variables allows us to unravel the memory, showing that the $n$-fold concatenation of the memory channel is unitarily equivalent to the direct product of $n$ single-mode lossy bosonic channels. We then compute the ultimate (classical and quantum) transmission rates, showing their enhancement with respect to the memoryless case and proving that coherent state encoding is optimal.

Figure 1

Left: a single use of the memory channel (see text for details). Right: the $n$-fold concatenation of the memory channel: photons entering in the $k$th input mode $a_k$ can only emerge in the output ports $b_{k^{\prime }}$ with $k^{\prime }\ge k$.

Figure 2

(a) Contour plot of the classical capacity $C_N$ as function of $\eta$ and $ϵ$ for $N=8$. (b) Unconstrained quantum capacity $Q_{\infty }$, in the $XY=EE$, $AE$ setups, as a function of $\eta$ and $ϵ$ (it diverges logarithmically at $\eta =1$ and $ϵ=1$). (c) Distribution of the effective transmissivities ${\tau }_k^{XY}$ for $\eta =0.7$, $ϵ=0.3$: the solid line shows the asymptotic distribution computed from Eq. (2), the dots and the circles, respectively, show the distribution of transmissivities ${\tau }_k^{EE}$ and ${\tau }_k^{AB}$ for $n=20$.