More entanglement, less reliability

Entanglement of quantum states is a very important resource in quantum communication systems. For many special tasks, entanglement is found to be helpful. But it is still a problem how to find the relationship between the amount of entanglement and the effect of doing the task. In this work, we consider a particular task-trying to send a single classical bit as effective as possible through a noisy quantum channel and two transmissions through the channel are available as a resource. For two special channels-amplitude damping channel and two-Pauli channel, we separately establish the quantitative connections, and try to find whether the relationship is always monotonous, i.e., the states with the larger entanglement necessarily result in the better effect. Unfortunately, this is not true. At the same time, we can get the optimal codes for the task using this method.