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

Question

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?

Answer 1

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.

You might also be interested in my note on Muroga's paper, in which I attempt to provide an alternate proof of the feasibility of the optimal input probability distribution, in the special case of a binary channel, and to correct what I think is a trivial error in Muroga's paper (although this might simply have been Muroga being too polite to point out a trivial mistake by Shannon), and I also try to simplify a part of Muroga's argument:

"On the Computation of the Shannon Capacity of a Discrete Channel with Noise", S. Cowell, arXiv:1701.08731v2 [cs.IT] [here]1