Layered quantum key distribution

We introduce a family of quantum key distribution protocols for distributing shared random keys within a network of $n$ users. The advantage of these protocols is that any possible key structure needed within the network, including broadcast keys shared among subsets of users, can be implemented by using a particular multipartite high-dimensionally entangled quantum state. This approach is more efficient in the number of quantum channel uses than conventional quantum key distribution using bipartite links. Additionally, multipartite high-dimensional quantum states are becoming readily available in quantum photonic labs, making the proposed protocols implementable using current technology.

Figure 1

Examples of layered key structures. Left: Each of the two security agencies requires a secure communication channel with their respective agents, as well as an interagency channel secret even from their agents. Additionally, the agents require a secure channel shared only among themselves. Right: The central bank shares a key with each of its branches, while each branch shares a key with each of its automated teller machines (ATMs) and, additionally, all parties share a common secret key.

Figure 2

Entanglement distribution model. Each user $u_j$ is connected to the source of entanglement via a quantum channel. Additionally each pair of users shares a classical channel.

Figure 3

Simplest layered QKD (LQKD) example. (a) In this example, three users want to share keys in two layers $k_1=\{1,2,3\}$ and $k_2=\{1,2\}$. $p$ is the probability that users 1 and 2 share the state $|{\psi }_4^{+}\rangle$. (b) An EPR implementation results in the idealized rates $\left[\right\{1,2,3\};(1-p\left)\right],\left[\right\{1,2\};3p-1]$. (c) A GHZ implementation results in the idealized rates $\left[\right\{1,2,3\};(1-p\left)\right],\left[\right\{1,2\};2p]$. (d) An implementation with the state $|{\mathrm{\Psi }}_{442}\rangle =\frac{1}{2}\left(\right|000\rangle +|111\rangle +|220\rangle +|331\rangle )$ results in the idealized rates $\left[\right\{1,2,3\};1],\left[\right\{1,2\};1]$.

Figure 4

A key structure with idealized rate 1 with EPR implementation. There are six two-user layers, which can be grouped into three partitions $\{\{u_1,u_2\},\{u_3,u_4\}\},\{\{u_1,u_4\},\{u_2,u_3\}\},\{\{u_1,u_3\},\{u_2,u_4\}\}$. Eight-dimensional EPR pairs can be distributed to each partition in parallel. If each of these distribution rounds happens with probability $\frac{1}{3}$, the average rate for every layer is 1.

Figure 5

The number of channel uses needed in order to share a multipartite key with EPR pairs. Users $u_1$ and $u_m$ need to share only a single EPR pair with their neighbors. The rest of the users need to share two EPR pairs each. To share the multipartite key, user $u_1$ generates a random string locally and sends it to the user $u_2$ secretly via one-time-pad encryption. Then each user $u_i$, after receiving the key from the user $u_{i-1}$, sends it secretly to the user $u_{i+1}$, until all the users share the new secret key.

Figure 6

Classification of different states for key structure $\mathcal{K}=P\left(\right\{1,2,3,4\left\}\right)\setminus \{\varnothing ,\{1\},\{2\},\{3\},\{4\left\}\right\}$. (a) Key structure $\mathcal{K}$ ordered with respect to inclusion. (b) Decomposition of ordered $\mathcal{K}$ into binary trees. Note that union of two children vertices is always equal to their parent. Each tree can now be implemented using the trade-off recursive step multiple times. Joining the trees together can be done by the recursive step 1. This is illustrated by the dotted circle that joins three states together. (c)–(f) Different decompositions of ordered $\mathcal{K}$ into a nonbinary tree. Using reductions to qudit protocols allows us to use the trade-off recursive step for all layers. Each decomposition leads to a different state.