The Quantum Repeater Network Saturates the Entanglement Distribution Asymptotically

Efficient and reliable entanglement distribution forms the core of quantum network architecture. The Quantum Max-Flow concept defines the upper boundary for efficiently transmitting entanglement within quantum tensor networks (Calegari, Freedman, and Walker, Journal of the American Mathematical Society, 2010). Despite the general invalidity of the Quantum analog of the Max-Flow Min-Cut theorem, this paper introduces a precise Quantum Max-Flow Min-Cut Theorem tailored to quantum tensor networks enhanced by ancilla entanglement. Specifically, our research establishes infinite 'n' values for any quantum tensor network, where the Quantum Max-Flow matches the Quantum Min-Cut by attaching a maximally entangled state with dimension 'n' to each edge. Our protocol exclusively relies on quantum teleportation and guarantees unambiguous success. This work highlights the potential of quantum repeaters, enabled by ancilla entanglement, to efficiently distribute entanglement throughout quantum networks.