Directed percolation effects emerging from superadditivity of quantum networks

Entanglement-induced nonadditivity of classical communication capacity in networks consisting of quantum channels is considered. Communication lattices consisting of butterfly-type entanglement-breaking channels augmented, with some probability, by identity channels are analyzed. The capacity superadditivity in the network is manifested in directed correlated bond percolation which we consider in two flavors: simply directed and randomly oriented. The obtained percolation properties show that high-capacity information transfer sets in much faster in the regime of superadditive communication capacity than otherwise possible. As a by-product, this sheds light on a type of entanglement-based quantum capacity percolation phenomenon.

Figure 1

Model A: layered communication network. (a) Passive network; users are located at nodes of a square lattice, the lattice is filled with butterfly shaped primitives, information flow is directed from layer $t\to t+1$. (b) Butterfly primitive consists of two MACs (solid and dashed wedges) from Ref. [15]; node (user) $N_{i,t}$ can communicate with $N_{i-1,t+1},N_{i,t+1},N_{i+1,t+1}$ with maximal capacity $C^{\prime }\ll C_0$. (c) Active network filled randomly by triples of ideal channels allowing HCC (shown is a configuration not allowing HCC from A to B). (d) Entanglement activates the slanted lines of the corresponding passive channels (a) for HCC using dense coding, and thus changes the geometry of the initial HCC network (c) leading to directed communication paths (black arrows) with capacities $\ge C_0$ from A to B.

Figure 2

Model B: multidirectional communication networks. (a) Passive network, allowing communication in each direction; (b) this is due to the structure of the butterfly primitive depicted here—each quartet of nodes forming an elementary square is connected by 4 passive channels, of the type shown in Fig. 1b. (c) Active channels can be placed on any bond and tagged for use in any direction. (d) Emergent network geometry from superimposing (a) and (c), due to dense coding allowing HCC along slanted lines, facilitated by entanglement between active and passive channels.

Figure 3

Comparison of the percolation critical lines $[p_{+}^c,p_{-}^c\left(p_{+}^c\right)]$ for percolation process on randomly oriented square lattice (classical scheme) and butterfly network (EA scheme). Dots are numerically obtained points.

Figure 4

Comparison of critical exponents $\tau$, $\beta$ for classical and EA schemes of multidirectionally oriented percolation.