Classical capacity of Gaussian thermal memory channels

The classical capacity of phase-invariant Gaussian channels has been recently determined under the assumption that such channels are memoryless. In this work we generalize this result by deriving the classical capacity of a model of quantum memory channel, in which the output states depend on the previous input states. In particular we extend the analysis of Lupo et al. [Phys. Rev. Lett. 104, 030501 (2010) and Phys. Rev. A 82, 032312 (2010)] from quantum limited channels to thermal attenuators and thermal amplifiers. Our result applies in many situations in which the physical communication channel is affected by nonzero memory and by thermal noise.

Figure 1

Schematic description of a Gaussian memory channel ${\Phi }_n$ which is iterated $n$ times. The application of the memory channel to $n$ successive input modes $a_1,\cdots ,a_n$ is described by $n$ phase-insensitive channels ${\mathcal{E}}_{\kappa }$ (thermal attenuators or amplifiers) where each of them is coupled to a Gaussian thermal environment and to a memory mode. The initial memory mode $a_1^M$ travels horizontally and correlates the output signals with the previous input signals. A beam splitter of transmissivity $\mu$ is used to tune the memory effect of the channel. For $\mu =1$ the memory mode is perfectly preserved while for $\mu =0$ the channel becomes memoryless. A reasonable choice for the initial state of the memory mode is a Gaussian thermal state in equilibrium with the environment, i.e., we make the identification $a_1^M=a_0^E$. The final state of the memory is assumed to be inaccessible and is traced out.

Figure 2

Asymptotic spectrum $\eta \left(z\right)$ of Eq. (22) for the case where ${\mathcal{E}}_{\kappa }$ of Fig. 1 represents an attenuator channel (i.e., $\kappa \in [0,1])$. In this case the system is operated below the threshold limit $\mu \kappa \le 1$ (no divergency in the spectrum occurs) and the values of $\eta \left(z\right)$ are always bounded below 1 (i.e., the channels ${\mathcal{E}}_{{\eta }_j^{\left(n\right)}}^N$ entering the decomposition (16) are attenuators). In each plot the plane represents the value of $\mu$. Notice that for $\kappa =0$ one has $\eta \left(z\right)=\mu$, while for $\kappa =1,\eta \left(z\right)=1$ independently from $\eta$.

Figure 3

Logarithm of the asymptotic spectrum $\eta \left(z\right)$ of Eq. (22) for the case where ${\mathcal{E}}_{\kappa }$ of Fig. 1 represents an amplifier channel (i.e., $\kappa \ge 1)$. In this case the values of $\eta \left(z\right)$ are always larger than 1 meaning that the channels ${\mathcal{E}}_{{\eta }_j^{\left(n\right)}}^N$ entering the decomposition (16) describe amplifiers. Above threshold (i.e., $\mu \kappa \ge 1)$ the system acquires also a divergent, singular eigenvalue represented in the picture by the cyan vertical region.

Figure 4

Capacity (in nats/channel use) as a function of the thermal photon number $N$ for $\mu =0.8$ and mean input energy $E=8$ for various values of the transmissivity $\kappa$. In particular the upper panel refers to the case where the map ${\mathcal{E}}_{\kappa }$ of Fig. 1 is an attenuator (i.e., $\kappa \in [0,1])$, while the lower panel refers to the case where ${\mathcal{E}}_{\kappa }$ is an amplifier $\left(\kappa \ge 1\right)$. As expected, the capacity is degraded by the temperature and enhanced if the transmissivity is close to unity.

Figure 5

Behavior of the energy density $N\left(z\right)$ for $\kappa =0.9,\mu =0.8,E=8$ and $N$ ranging in steps of 0.1 from top to bottom from 0.5 to 1.2, near to the critical temperature $N_{\text{crit}}\sim 0.8$. As expected, $N\left(z\right)$ is always increasing. If we exclude the region near $z=0$, the functions are almost identical and approach nearly the same constant value for $z\gtrsim 1$. Inset: Zoom on the region $z\to 0$. We can see that above the critical temperature $N\left(z\right)$ is zero on a finite interval, while below it $N\left(z\right)$ approaches a positive value which strongly depends on the temperature.

Figure 6

Behavior of the energy density $N\left(z\right)$ for $\kappa =1.1,\mu =0.8,E=8$ and $N$ ranging in steps of 0.1 from top to bottom from 9.4 to 10.1, near to the critical temperature $N_{\text{crit}}\sim 9.8$. As expected, $N\left(z\right)$ is always increasing. If we exclude the region near $z=0$, the functions are almost identical and approach nearly the same constant value for $z\gtrsim 1$. Inset: Zoom on the region $z\to 0$. We can see that above the critical temperature $N\left(z\right)$ is zero on a finite interval, while below it $N\left(z\right)$ approaches a positive value which strongly depends on the temperature.

Figure 7

Behavior of the fraction $\frac{z_0}{2\pi }$ $(z_0$ ranges from 0 to $2\pi )$ of unused beam splitters as a function of the temperature $N$ for $E=8,\mu =0.8$, and various values of $\kappa$. At zero temperature $\left(N=0\right)$ all the beam splitters are exploited and $z_0=0$; then $z_0$ remains zero up to the critical temperature $N_{\text{crit}}$, and grows for $N>N_{\text{crit}}$. We notice that for typical values of the parameters $N_{\text{crit}}$ is much greater for $\kappa >1$ than for $0<\kappa <1$. In the infinite temperature limit $N\to \infty$ only an infinitesimal fraction of the beam splitters is used and $z_0$ tends to $2\pi$, even if this is not evident from the plots due to the limited range of $N$.

Figure 8

Behavior of the minimal energy for which all the beam splitters are exploited as a function of $\kappa$ for $\mu =0.8$ and various values of the temperature $N$. As expected, $E$ grows with the temperature, and $E=0$ for $\kappa =0,1$ and $\kappa \to \infty$. In the attenuator case we notice the divergence of $E$ for $\kappa =\mu$ $\left(=0.8\right)$, due to the fact that $\eta \left(0\right)=0$ and for any positive temperature the optimal $N\left(z\right)$ must vanish on a finite interval.