Superadditivity of the Classical Capacity with Limited Entanglement Assistance

Finding the optimal encoding strategies can be challenging for communication using quantum channels, as classical and quantum capacities may be superadditive. Entanglement assistance can often simplify this task, as the entanglement-assisted classical capacity for any channel is additive, making entanglement across channel uses unnecessary. If the entanglement assistance is limited, the picture is much more unclear. Suppose the classical capacity is superadditive, then the classical capacity with limited entanglement assistance could retain superadditivity by continuity arguments. If the classical capacity is additive, it is unknown if superadditivity can still be developed with limited entanglement assistance. We show this is possible, by providing an example. We construct a channel for which the classical capacity is additive, but that with limited entanglement assistance can be superadditive. This shows entanglement plays a weird role in communication, and we still understand very little about it.

Figure 1

Consider a channel $\mathcal{N}$ for classical communication, with additive classical capacity. We have the following three scenarios. (a) Entanglement across channel uses does not help if we do not have any assistance. (c) Entanglement across channel uses also does not help if we have unlimited entanglement assistance (this is always true regardless of the channel). The question addressed is case (b), whether entanglement across channel uses can help if we have some entanglement assistance.

Figure 2

(a) Schematic plot of the superadditivity in $\mathrm{C}\mathrm{E}$ trade-off for our channel. ${\mathcal{C}}_P={\mathcal{C}}_P^{(1)}$ at $P=0$ and $P_{\max }$, but not for all values in between. (b) $\mathrm{Q}\mathrm{E}$ trade-off curve for channels with additive quantum capacity. $P_{\max }$ is the maximum amount of available entanglement assistance.

Figure 3

Diagrammatic representation of $\mathcal{N}$.