Quantum cost of dense coding and teleportation

Quantum cost is a key ingredient in evaluating the quality of quantum protocols from a practical viewpoint. We show that the quantum cost of a d-dimensional dense coding protocol is equal to $d+3$ when transmitting the classical message (0,0) and $d+4$ when transmitting another classical message. It appears as linear growth with the dimension and thus makes sense for implementation. In contrast, the quantum cost of high-dimensional teleportation protocols is equal to 13, which is the maximum value of the cost for the two-dimensional case. As an application, we establish the relation between the quantum cost and fidelity of dense coding protocols in terms of four typical noise scenarios.