Quantum network coding based on remote state preparation over the k-pair network

We first propose a quantum network coding protocol that enables the deterministic cross remote preparation of arbitrary single-qudit states over the butterfly network. Only two pairs of maximally entangled states are required to be pre-shared between two senders. Each sender takes a positive operator-valued measurement and a projective measurement sequentially. The two receivers can synchronously reconstruct the target states by performing the appropriate unitary operations which are derived by general formulas. Notably, two classical dits of communication redundancy are eliminated on the bottleneck channel, making the transmission efficiency twice as high as that without network coding. Subsequently, the proposed protocol is extended to a more general k-pair quantum network. Furthermore, we explore the efficient realization of probabilistic remote state preparation over the aforementioned network architecture under the condition that non-maximally entangled states are utilized as the quantum resource.