Figure 3

Entanglement distribution with resource balancing in the quantum Internet. (a) Low-priority quantum repeaters. Users \(U_{3} \) and \(U_{6} \) would like to share an entangled connection with B through \(R_{l} \). Quantum repeater \(R_{l} \) has only a single density matrix from B available for the entanglement swapping via entangled connection \(\mathrm{L}_{l} \left( R_{k} ,B\right) \), and as a corollary, \(R_{l} \) can serve only \(U_{3} \) or \(U_{6} \). Users \(U_{1} \), \(U_{2} \), and \(U_{5} \) are served directly, since these users have no common resource requirements. The quantum repeater node \(R_{l} \) serves \(U_{6} \), thus establishing the entangled connection \({\mathrm{L}}_{l} \left( R_{k} ,R_{l} \right) \) between \(R_{k} \) and \(R_{l} \). (b) Resource balancing via random routing. User \(U_{6} \) establishes the distant entangled connection \({\mathrm{L}}_{l} \left( R_{k} ,B\right) \) with B through \(R_{l} \) (depicted by the green line). For a seamless transition of resource saving, a random quantum repeater is selected for user \(U_{3} \) from the set \({{{\mathscr {R}}}}_{{{\mathscr {S}}}} \) of random quantum repeaters (\({{{\mathscr {R}}}}_{{{\mathscr {S}}}} \) is realized by the strongly-entangled structure \({{\mathscr {S}}}_{{\mathscr {N}}} \)) to establish the entangled connection \({\mathrm{L}}_{l} \left( R_{i} ,R^{*} \right) \) (depicted by the red line), where \(R^{*} \in {{{\mathscr {R}}}}_{{{\mathscr {S}}}} \) is a quantum repeater from \({{{\mathscr {R}}}}_{{{\mathscr {S}}}} \).