# Is the capacity achieving input of a discrete memoryless channel unique?

Consider a classical discrete memoryless channel (DMC). Let $$p$$ be an input probability distribution and $$Q$$ be the channel's transition matrix. $$q = Qp$$ is a valid output probability distribution. The components of $$p$$ and $$q$$ are given as $$p_i$$ and $$q_i$$ respectively.

The capacity of this channel is given by maximizing the mutual information.

$$I = \max_{p}\left[\sum_ip_i\left(\sum_{j}Q_{ij}\log(Q_{ij})\right) - \sum_{j}q_j\log q_j\right]$$

But, we know (Theorem 2.7.4 in Cover and Thomas, 2006) that $$I(p)$$ is concave in $$p$$.

If $$I(p)$$ is not simply a linear function (i.e. $$q = Qp$$ is not a trivial relationship) and is differentiable everywhere (seems like a reasonable assumption), does this not imply that the capacity achieving $$p$$ is unique?

• The capacity is a function of the channel. Given the channel, it has a unique value. Maybe you are asking whether there is a unique maximizer distribution $p$? Jul 4, 2019 at 7:27
• @YuvalFilmus that's right, my question is about the uniqueness of the capacity achieving distribution $p$. Jul 4, 2019 at 9:11

I believe the answer is "yes", when $$Q$$ is square and nonsingular, and "not necessarily", otherwise. See "On the Capacity of a Discrete Channel. I", by Saburo Muroga, Journal of the Physical Society of Japan, Vol. 8, No. 4, July - August 1953, pp 484-494.