Distance-Independent Entanglement Generation in a Quantum Network using Space-Time Multiplexed Greenberger–Horne–Zeilinger (GHZ) Measurements

In a quantum network that successfully creates links—shared Bell states between neighboring repeater nodes—with probability p in each time slot, and performs Bell State Measurements at nodes with success probability q < 1, the end-to-end entanglement generation rate drops exponentially with the distance between consumers, despite multi-path routing. If repeaters can perform multi-qubit projective measurements in the GHZ basis that succeed with probability q, the rate does not change with distance in a certain (p,q) region, but decays exponentially outside. This region where the distance-independent rate occurs is the super-critical region of a new percolation problem. We extend this GHZ protocol to incorporate a time-multiplexing blocklength k, the number of time slots over which a repeater can mix-and-match successful links to perform fusion on. As k increases, the super-critical region expands. For a given (p,q), the entanglement rate initially increases with k, and once inside the super-critical region for a high enough k, it decays as 1/k GHZ states per time slot. When memory coherence time exponentially distributed with mean μ is incorporated, it is seen that increasing k does not indefinitely increase the super-critical region; it has a hard μ-dependent limit. Finally, we find that incorporating space-division multiplexing, i.e., running the above protocol independently in up to d disconnected network regions, where d is the network’s node degree, one can go beyond the 1 GHZ state per time slot rate that the above randomized local-link-state protocol cannot surpass. As (p,q) increases, one can approach the ultimate min-cut entanglement-generation capacity of d GHZ states per slot.