Multicast quantum network coding as optimal symmetric universal cloning over a quantum network

In classical networks, multicast network coding allows a sender to transmit identical information to multiple receivers efficiently, constrained by the network's minimum cut capacity. Translating this capability to quantum networks has been hindered by the no-cloning theorem, which prevents the perfect copying of unknown quantum states. Prior quantum network coding schemes have largely focused on multiple-unicast settings, or have produced suboptimal outputs like GHZ-type states that fail to serve as effective cloning machines. This paper investigates whether optimal symmetric universal clones of unknown quantum states can be successfully multicast from multiple source nodes to terminal nodes over a quantum network, defining the best possible multicast communication permitted by quantum mechanics.

The study models a quantum network as an undirected graph where each edge represents a noiseless q-dimensional quantum channel, supplemented by free classical communication between any pair of nodes. The authors extend the multi-source quantum network coding protocol of Kobayashi et al. to achieve symmetric universal cloning. By leveraging the interplay between quantum operations, classical communication, and pre-shared entanglement among terminal nodes, the protocol achieves multicast distribution at rates dictated by network cuts. The required entanglement is small and depends only on the number of sources and terminals, remaining negligible for large input state dimensions.

The authors establish that if a classical linear multi-source multicast network code exists over an alphabet of size q for an acyclic directed graph derived from the network, a corresponding quantum protocol can be constructed to output optimal symmetric universal clones at every terminal node. In the single-source setting, when a single copy of a high-dimensional input state is available, perfect multicast of its symmetric universal clone is achievable using a small amount of shared entanglement among the target nodes, provided the minimum cut of the directed network graph meets the required threshold. This demonstrates that optimal cloning over quantum networks can attain the minimum cut rate.