Aggregating quantum repeaters for the quantum internet

The quantum internet holds promise for accomplishing quantum teleportation and unconditionally secure communication freely between arbitrary clients all over the globe, as well as the simulation of quantum many-body systems. For such a quantum internet protocol, a general fundamental upper bound on the obtainable entanglement or secret key has been derived [K. Azuma, A. Mizutani, and H.-K. Lo, Nat. Commun. 7, 13523 (2016)]. Here we consider its converse problem. In particular, we present a universal protocol constructible from any given quantum network, which is based on running quantum repeater schemes in parallel over the network. For arbitrary lossy optical channel networks, our protocol has no scaling gap with the upper bound, even based on existing quantum repeater schemes. In an asymptotic limit, our protocol works as an optimal entanglement or secret-key distribution over any quantum network composed of practical channels such as erasure channels, dephasing channels, bosonic quantum amplifier channels, and lossy optical channels.

Figure 1

Quantum network. The network is associated with a directed graph $G=(V,E)$ with set $V$ of vertices and set $E$ of edges, where $V$ is composed of Alice's node $A$, Bob's node $B$, and intermediate nodes $C^1,C^2,...$, and $C^n$ ($n=5$ here) and a directed edge $e=X\to Y$ in $E$ for $X,Y\in V$ specifies a quantum channel ${\mathcal{N}}^e$ to send a subsystem in node $X$ to node $Y$. The goal here is to give Alice and Bob pbits or ebits by using quantum channels ${\left\{{\mathcal{N}}^e\right\}}_{e\in E}$ and LOCC.

Figure 2

Bell-pair network ${⨂}_{e\in E}|{\mathrm{\Phi }}^{+}\rangle \langle {\mathrm{\Phi }}^{+}{|}_e^{\otimes \lfloor m^e\rfloor R_{\delta }^e}$ associated with $G^{\prime }=(V,E^{\prime })$. This is an example of the Bell-pair network for Fig. 1. Each undirected edge $e^{\prime }\in E^{\prime }$ represents a Bell pair $|{\mathrm{\Phi }}^{+}{\rangle }_e$. The denominator and the numerator of a fraction describe $\lfloor m^e\rfloor R_{\delta }^e$ and how many Bell pairs are used and consumed in the aggregated quantum repeater protocol, respectively. Dashed edges are unused Bell pairs. Here the choice of $V_A=\{A,C^1,C^3\}$ in Eq. (2) gives $M_{\delta }=8$.