Nonlocal Network Coding in Interference Channels

In a network, a channel introduces correlations to the parties that aim to establish a communication protocol. We present a framework of nonlocal network coding by exploiting a Bell scenario and show the usefulness of nonlocal and quantum resources in network coding. Two-sender and two-receiver interference channels are considered, for which network coding is characterized by two-input and four-outcome Bell scenarios. It is shown that nonsignaling correlations lead to strictly higher channel capacities than quantum correlations in general. This also holds true for quantum and local correlations: network coding with quantum resources shows a strictly higher channel capacity than local ones. It turns out, however, that more nonlocality does not necessarily imply a higher channel capacity. The framework can be generally applied to network communication protocols.

Figure 1

In a network of $M+N$ parties [see Eq. (1)], two senders $A_1$ and $A_2$ choose two inputs $m_1$ and $m_2$ as the messages to deliver to parties $B_1$ and $B_2$. For a channel considered throughout in Eq. (5) that transmits $X_i=(x_{i1},x_{i2})$ to $Y_i$, network coding denoted by $\mathbb{E}$ [see also Eq. (6)] corresponds to a Collins-Gisin-Linden-Massar-Popescu (CGLMP) scenario. The senders exploit a Bell scenario for network coding $\mathbb{E}$.

Figure 2

In the gray area, it holds that (a) $C^{(\mathcal{Q})}(\mathcal{N})\ge C^{({\mathcal{Q}}_{LB})}(\mathcal{N})>C^{(\mathcal{L})}(\mathcal{N})$ from Eqs. (14) and (16), and (b) $C^{(\mathcal{Q})}(\mathcal{N})\le C^{({\mathcal{Q}}_{UB})}(\mathcal{N})<C^{(\mathcal{N}\mathcal{S})}(\mathcal{N})$ from Eqs. (15) and (17). For almost all $p,q\in [0,1]$, nonlocal network coding is more useful than quantum network coding, which is more useful than local network coding.