Classical Communications with Indefinite Causal Order for $N$ completely depolarizing channels

In the presence of indefinite causal order, two identical copies of a completely depolarizing channel can transmit non-zero information. This arises due to quantum superposition of two alternative orders of quantum channels. Here, we address the question: Can we transmit perfect classical information with superposition of $N$ depolarizing channels with multiple number of causal orders? We find that for $N$ completely depolarizing channels with superposition of causal orders, there always is an additional classical communication advantages. This communication advantage increases with the number $N$ but asymptotically tends to zero for large dimension of the input state. We also find that there is almost $3$ fold gain in communication if we go from $2$ channels to $3$ channels instead of $1.9$ fold. We also show that the gain in classical communication rate decreases exponentially with the dimension of the target state and increases rapidly with the increase in number of causal orders. However, for qubit state, it saturates at $0.31$ bits and can never reach the perfect scenario. We also derive an analytical expression for the Holevo quantity for $N$ completely depolarizing channels with superposition of $M\in [2,N!]$ orders.