Percolation of secret correlations in a network

In this work, we explore the analogy between entanglement and secret classical correlations in the context of large networks—more precisely, the question of percolation of secret correlations in a network. It is known that entanglement percolation in quantum networks can display a highly nontrivial behavior depending on the topology of the network and on the presence of entanglement between the nodes. Here we show that this behavior, thought to be of a genuine quantum nature, also occurs in a classical context.

Figure 1

Secret-key networks: (a) A general secret-key network is composed of a set of nodes (vertices of the graph) distributed with a given geometry, sharing secrecy correlations when connected by a link (edges of the graph). (b) One-time pad: In order to establish an unbiased secret bit between previously unconnected nodes $A$ and $C$, $B$ and $C$ apply one round of one-time pad; (i) $B$ publicly announces the value of $z=b_1\oplus b_2$; (ii) $C$ calculates $c^{\prime }=c\oplus z$, which gives $c^{\prime }=a$.

Figure 2

1D chain (4 nodes in the figure): The protocol giving the best probability of distilling a secret bit between nodes $A_0$ and $A_n$ works as follows: (i) Each intermediate node $A_i$ ($1\le i\le n-1$) makes public the value of the sum (modulo 2) of its two bits: $z_i=a_i\oplus a_{i+1}$; (ii) node $A_n$ calculates privately $a_n\oplus {\sum }_{i=1}^{n-1}{z}_i$; (iii) nodes $A_0$ and $A_n$ apply the optimal probabilistic conversion protocol for the given set ($z_1,\cdots ,z_{n-1}$).

Figure 3

The initial configuration is a honeycomb (hexagonal) lattice, each node being represented by a circle. Each two adjacent nodes are connected by two links, each one representing a biased secret-key bit. Using local operations and public communication, we can transform the topology to a triangular lattice improving the percolation threshold. To do so, every yellow node performs three parallel one-time pad operations, one for each pair of connected links (labeled by a star in the interior of the brown node). This transforms every pair of connected links into a link of the new triangular lattice (see black stars for those corresponding to the brown node).