Bidirectional and cyclic quantum dense coding in a high-dimension system

Abstract

In light of quantum dense coding in the case of high-dimension quantum states between two parties, by introducing additional local operations for encoding, we propose a brand-new bidirectional quantum dense coding scheme, in which two legitimate agents can simultaneously transmit their different encoded messages to each other. In addition, we present a bidirectional quantum dense coding scheme with a control by virtue of the generalized Hadamard transformation. We show how to implement the cyclic quantum dense coding in an arbitrary high dimension where Alice can transfer her encoded messages $$n_1m_1$$ to Bob; meanwhile, Bob can transfer his encoded messages $$n_2m_2$$ to Charlie and Charlie can also transfer his encoded messages $$n_3m_3$$ to Alice in both clockwise and counterclockwise directions. We can also generalize the cyclic scheme to system having $$n\ge 3$$ agents. Thereby, our scheme can realize dense coding in quantum information networks with n agents in arbitrary directions.