Optimal superdense coding over memory channels

We study the superdense coding capacity in the presence of quantum channels with correlated noise. We investigate both the cases of unitary and nonunitary encoding. Pauli channels for arbitrary dimensions are treated explicitly. The superdense coding capacity for some special channels and resource states is derived for unitary encoding. We also provide an example of a memory channel where nonunitary encoding leads to an improvement in the superdense coding capacity.

Figure 1

Superdense coding capacity for a correlated quasiclassical channel and a Bell state ($\eta =1$), as a function of the noise parameter $p$ and the correlation degree $\mu$.

Figure 2

Superdense coding capacity for a correlated quasiclassical channel and a Werner state, as a function of the correlation degree $\mu$ and the parameter $\eta$. The noise parameter is $p=0.05$.

Figure 3

The dashed red curve is the correlation degree $\tilde{\mu }\left(p\right)$ as a function of the noise parameter $p$. The superdense coding capacity $C_{\mathrm{un}}^{\mathrm{Q},\mathrm{B}}$ and the transferred information $C_{\Gamma }^{\mathrm{Q},\mathrm{B}}$ coincide for $\mu =\tilde{\mu }\left(p\right)$ (see main text). For $\mu <\tilde{\mu }\left(p\right)$, the blue (light gray) area, the nonunitary preprocessing $\Gamma$ increases the superdense coding capacity of a quasiclassical channel and a Bell state, in comparison to just unitary encoding.

Figure 4

Comparison between the superdense coding capacity (28) and the transferred information (38) as functions of the noise parameter $p$, with the correlation degree $\mu =0.2$. The dashed red curve corresponds to the capacity $C_{\mathrm{un}}^{Q,B}$ given by Eq. (28), while the solid blue curve represents the transferred information $C_{\Gamma }^{Q,B}$ given by Eq. (38). As we can see, for $\mu =0.2$, in the range of the noise parameter $0.007<p<0.293$, we reach a higher capacity by applying the nonunitary preprocessing $\Gamma$, the solid blue curve.

Figure 5

Another comparison between the superdense coding capacity (28) and the transferred information (38) as functions of the correlation degree $\mu$, with the noise parameter $p=0.05$. The dashed red curve corresponds to the capacity $C_{\mathrm{un}}^{Q,B}$ given by Eq. (28). The solid blue line represents the transferred information $C_{\Gamma }^{Q,B}$ given by Eq. (38). As we can see, for $p=0.05$ and $\mu <0.294$, the nonunitary preprocessing $\Gamma$ is useful to enhancing the capacity, compared to only unitary encoding.