Asymmetric quantum multicast network coding: asymmetric optimal cloning over quantum networks

In this study, we consider a quantum version of multicast network coding as a multicast protocol for sending universal quantum clones (UQCs) from a source node to the target nodes on a quantum network. By extending Owari et al.'s previous results for symmetric UQCs, we derive a protocol for multicasting $1\to 2$ ($1\to 3$) {\it asymmetric} UQCs of a $q^r$-dimensional state to two (three) target nodes.Our protocol works under the condition that each edge on a quantum network represented by an undirected graph $G$ transmits a $q$-dimensional state. There exists a classical solvable linear multicast network code with a source rate of $r$ on a classical network $G^{\mathrm{\prime }}$, where $G$ is an undirected underlying graph of an acyclic directed graph $G^{\mathrm{\prime }}$. We also assume free classical communication over a quantum network.