Upper bounds on the quantum capacity for a general attenuator and amplifier

There have been several upper bounds on the quantum capacity of the single-mode Gaussian channels with thermal noise, such as thermal attenuator and amplifier. We consider a class of attenuator and amplifier with more general noises, including squeezing or even non-Gaussian noises. We derive upper bounds the energy-constrained quantum capacity of those channels by using the quantum conditional entropy power inequality. Also, we obtain lower bounds for the same channels by means of Gaussian optimizer with fixed input entropy. They give narrow bounds when the transmissivity is near unity and the energy of input state is low.

Figure 1

A schematic diagram for complementary and weak complementary Gaussian channels. For a mixed state of environment for a given channel, the purifying system is represented as $R$.

Figure 2

Comparison between our two upper bounds with known upper bound $Q_{\text{SWAT}}$ [19] for (a) the thermal attenuator with $\tau =0.99$ and (b) the amplifier with $\kappa =1.02$. Average photon number of the thermal environment is $N_E=N_{\text{th}}=1$.

Figure 3

Two upper bounds and a lower bound on the quantum capacity of Gaussian (a) attenuator with $\tau =0.98$ and (b) amplifier with $\kappa =1.02$ on the squeezed thermal environment, when $N_{\text{th}}=0.01$ and squeezing parameter $r=0.1$, and thus $N_E\sim 0.02$. $N$ is the mean photon number of input state, and $\mathcal{Q}$ is the quantum capacity (bits). Note that the two upper bounds are very close; thus, they are overlapped in both cases.

Figure 4

Upper and lower bounds on the quantum capacity of general (a) attenuator with $\tau =0.98$ and (b) amplifier with $\kappa =1.02$. The mean photon number of the environment is $N_E=0.2$.

Figure 5

Upper and lower bounds on the quantum capacity of general (a) attenuator with $\tau =0.98$ and (b) amplifier with $\kappa =1.02$. Entropy of the environment is $S\left({\rho }_E\right)\sim 2.75$, thus $N_{\text{th}}=2,$ and we set $N_E=3$.