Figure 4

The strongly-entangled structure \({{\mathscr {S}}}_{{\mathscr {N}}} \) as formed by \(\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| \) quantum repeaters and \(\left| E\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) \right| \) entangled connections with heterogeneous entanglement levels, where \(\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| =5\), \(\left| E\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) \right| ={\textstyle \frac{\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| \cdot \left( \left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| -1\right) }{2}} \) , and \({{{\mathscr {F}}}}\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) =\left( 1-{\textstyle \frac{1}{\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| }} \right) \). (a) The low-priority node \(R_{i} \) is associated with the entanglement throughput request \(B\left( R_{i} \right) \). The \(\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| \) quantum repeaters of \({{\mathscr {S}}}_{{\mathscr {N}}} \) establish \(\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| \) entangled connections with \(R_{i} \) (depicted by the outgoing dashed black lines), with each connection having entanglement throughput \(B\left( R_{q}^{\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) } ,R_{i} \right) ={\textstyle \frac{1}{\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| }} B\left( R_{i} \right) \), where \(\sum _{q=1}^{\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| }B\left( R_{q}^{\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) } ,R_{i} \right) =B\left( R_{i} \right) \). A given quantum repeater \(R_{q}^{\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) } \) of \({{\mathscr {S}}}_{{\mathscr {N}}} \) establishes \(\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| -1\) entangled connections within \({{\mathscr {S}}}_{{\mathscr {N}}} \), each with entanglement throughput \(Q\left( R_{q}^{\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) } ,R_{z}^{\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) } \right) ={\textstyle \frac{1}{\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| }} B\left( R_{q}^{\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) } ,R_{i} \right) \), where \(R_{z}^{\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) } \) is a neighbor of \(R_{q}^{\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) } \). (b) Each of the \(\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| -1\) quantum repeaters of \({{\mathscr {S}}}_{{\mathscr {N}}} \) applies entanglement swapping \(U_{S} \) to establish the entangled connection between \(R_{i} \) and the egress quantum repeater \(R_{E}^{\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) } \) of \({{\mathscr {S}}}_{{\mathscr {N}}} \). Then, an arbitrary routing is applied to establish the entangled connection between \(R_{i} \) and the destination node \(D\left( R_{i} \right) \) of \(R_{i} \). The request from \(R_{i} \) to the strongly-entangled structure \({{\mathscr {S}}}_{{\mathscr {N}}} \) is served via \(n_{{\mathscr {P}}} =\left| {{\mathscr {S}}}_{{\mathscr {N}}} \right| \) parallel entangled paths \({{\mathscr {P}}}\left( R_{q}^{\left( {{\mathscr {S}}}_{{\mathscr {N}}} \right) } ,R_{i} \right) \) between the quantum repeaters of \({{\mathscr {S}}}_{{\mathscr {N}}} \) and \(R_{i} \). The dashed entangled connections are rebuilt within \({{\mathscr {S}}}_{{\mathscr {N}}} \) after the entanglement swapping operations.