The state of a trit $$\{p_1,p_2,p_3=1-p_1-p_2\}$$ can be represented as a triangular simplex. The centre of the simplex is the maximally mixed state $$m=\{\frac{1}{3},\frac{1}{3},\frac{1}{3}\}$$. And the pure states are the three corners. If I plot the Shannon entropy $$H(s)=-\sum\limits_{s\in\{0,1,2\}}p_s\log\left(p_s\right) $$as a contour map on this, I get something like
Apologies, I can't plot it without the simplex being skewed, but imagine it as an equilateral triangle. States on the same contour have the same entropy.
My question is: Is there an orbiting contour for all states (except the pure states) or is there a subset of states with an orbiting contour (as shown) whereas there are other sets of states whose entropy contour touch the sides? Any help greatly appreciated!
The entropy function is strictly concave, has unique maximum $\log 3$ at the center, and its restriction to the boundary segments is also strictly concave with unique maxima at $(0, 1/2, 1/2)$ and the permutations, the value at each of these 3 points being $\log 2$. Thus the boundary states (where at least one $p_i$ is $0$) have possible entropy values between $0$ and $\log 2$, while the entropy between $\log 2$ and $\log 3$ is only achieved in the interior. The higher values give closed contours, the lower give "3 segments"-type contours.
Here's how it all looks: