Figure 2

(a) A quantum Internet scenario with a set of incoming entangled connections \({{\mathscr {S}}}_{{\mathscr {P}}}^{{\mathrm{L}}_{l} \left( R_{i} ,R_{j} \right) } \left( R_{i} \right) \bigcap {{\mathscr {S}}}_{{\mathscr {P}}}^{{\mathrm{L}}_{l} \left( R_{i} ,R_{j} \right) } \left( R_{j} \right) \) that traverse the entangled connection \({\mathrm{L}}_{l} \left( R_{i} ,R_{j} \right) \) between quantum repeaters \(R_{i} \) and \(R_{j} \). The entangled states in the set \({{\mathscr {S}}}_{{\mathscr {P}}}^{{\mathrm{L}}_{l} \left( R_{i} ,R_{j} \right) } \left( R_{i} \right) \) of \(R_{i} \) and in the set \({{\mathscr {S}}}_{{\mathscr {P}}}^{{\mathrm{L}}_{l} \left( R_{i} ,R_{j} \right) } \left( R_{j} \right) \) of \(R_{j} \) (depicted by gray circles) are to be swapped with the entangled state that forms \({\mathrm{L}}_{l} \left( R_{i} ,R_{j} \right) \). The entanglement swapping is performed by the entanglement swapping operator \(U_{S} \). The other incoming entangled states in the quantum repeaters that do not traverse \({\mathrm{L}}_{l} \left( R_{i} ,R_{j} \right) \) are not elements of \({{\mathscr {S}}}_{{\mathscr {P}}}^{{\mathrm{L}}_{l} \left( R_{i} ,R_{j} \right) } \left( R_{i} \right) \bigcap {{\mathscr {S}}}_{{\mathscr {P}}}^{{\mathrm{L}}_{l} \left( R_{i} ,R_{j} \right) } \left( R_{j} \right) \). (b) A deadlock situation in the entanglement swapping procedure in a quantum Internet setting. The aim of quantum node A is to share an entangled connection with the distant quantum repeater \(R_{k} \). The source node A generates an entangled pair and transmits one half, \(\rho _{A} \), to \(R_{i} \) and keeps the other half, \(R_{A} \left( \beta \left( \rho _{A} \right) \right) \). In \(R_{i} \), the set \({{\mathscr {A}}}\left( \rho _{A} \right) \) (depicted by a yellow circle) does not contain the target entangled system \(\sigma _{B} \) from the target node \(R_{k} \) for the swapping; therefore, \(R_{i} \) generates an entangled pair (depicted by black dots) and shares an entangled connection \({\mathrm{L}}_{l} \left( R_{i} ,R_{j} \right) \) with \(R_{j} \). Quantum repeater \(R_{j} \) also generates an entangled pair (depicted by blue dots) and shares the entangled connection \({\mathrm{L}}_{l} \left( R_{j} ,R_{k} \right) \) with \(R_{k} \). Then, the target quantum node \(R_{k} \) generates an entangled connection (depicted by red dots) and sends one half, \(\sigma _{B} \), to \(R_{i} \) to form the entangled connection \({\mathrm{L}}_{l} \left( R_{k} ,R_{i} \right) \), while it keeps the other half, \(R_{k} \left( \beta \left( \sigma _{B} \right) \right) \). (c) Quantum repeater \(R_{i} \) receives \(\sigma _{B} \) and swaps it with \(\rho _{A} \) to form the distant entangled connection \({\mathrm{L}}_{l} \left( A,R_{k} \right) \). The deadlock in the entanglement swapping is caused by the fact that set \({{\mathscr {A}}}\left( \rho _{A} \right) \) in \(R_{i} \) does not contain \(\sigma _{B} \), so \(R_{i} \) does not establish the entangled connection \({\mathrm{L}}_{l} \left( R_{i} ,R_{j} \right) \) with \(R_{j} \), and \(R_{j} \) does not establish the entangled connection \({\mathrm{L}}_{l} \left( R_{j} ,R_{k} \right) \) with \(R_{k} \).