Quantum and Classical Data Transmission through Completely Depolarizing Channels in a Superposition of Cyclic Orders

Completely depolarizing channels are often regarded as the prototype of physical processes that are useless for communication: any message that passes through them along a well-defined trajectory is completely erased. When two such channels are used in a quantum superposition of two alternative orders, they become able to transmit some amount of classical information, but still no quantum information can pass through them. Here, we show that the ability to place $N$ completely depolarizing channels in a superposition of $N$ alternative causal orders enables a high-fidelity heralded transmission of quantum information with error vanishing as $1/N$. This phenomenon highlights a fundamental difference with the $N=2$ case, where completely depolarizing channels are unable to transmit quantum data, even when placed in a superposition of causal orders. The ability to place quantum channels in a superposition of orders also leads to an increase of the classical communication capacity with $N$, which we rigorously prove by deriving an exact single-letter expression. Our results highlight the more complex patterns of correlations arising from multiple causal orders, which are similar to the more complex patterns of entanglement arising in multipartite quantum systems.

Figure 1

Communication through $N$ channels in a superposition of $N$ cyclic orders. A sender, located at node 1 of a quantum communication network, sends messages to a receiver, located at node $N+1$, through a sequence of intermediate nodes, labeled as $2,\dots ,N$. The intermediate nodes are connected by $N$ quantum channels ${\mathcal{C}}^{(1)},\dots ,{\mathcal{C}}^{(N)}$, which have been placed in one of $N$ configurations related by cyclic permutations, as shown in the graphic. The choice of configuration is controlled by a quantum system in a coherent superposition.

Figure 2

Classical capacity of the effective channel ${\mathcal{C}}_{\mathrm{eff}}$, plotted with respect to $N$ for message systems of dimension $d=2$, 3, 4, and 5.