Gui-Lu Long

Dense coding is known as an attractive quantum information protocol. While the original study considers the noiseless setting, many subsequent studies extended this result to more general settings. However, all of them focused only on the communication speed in various noisy settings. While dense coding with the noiseless setting realizes twice communication speed, it also realizes quantum secure direct communication (QSDC) as follows.In dense coding, the sender, Alice, and the receiver, Bob, share perfect Bell states and Alice encodes her message by application of a unitary operation. Since Alice's local state is a completely mixed state, the eavesdropper, Eve, cannot obtain any information about the message even when Eve intercepts the transmitted quantum state. However, it is not easy to share a perfect Bell state. Hence, we need to consider secure communication under imperfect shared state. Specifically, we study secure direct communication by using a general preshared quantum state and a generalization of dense coding. In this scenario, Alice is allowed to apply a unitary operation on the preshared state to encode her message, and the set of allowed unitary operations forms a group. To decode the message, Bob is allowed to apply a measurement across his own system and the system he receives. In the worst scenario, we guarantee that Eve obtains no information for the message even when Eve access the joint system between the system that she intercepts and her original system of the preshared state. For a practical application, we construct a modular wiretap code by concatenating inverse universal hashing and an arbitrary error correcting code. Combining the wiretap code with error verification, we propose a concrete protocol for the private dense coding model and derive an upper bound of information leakage in the finite-length setting. We also discuss how to apply our scenario to the case with discrete Weyl-Heisenberg representation when the preshared state is unknown. In this case, Pauli encoding operation and Pauli channel are considered. Hence, our protocol can be applied many similar tasks.