We introduce and study the entanglement breaking rank of an entanglement breaking channel. We show that the entanglement breaking rank of the channel $mathfrak Z: M_d to M_d$ defined by begin{align*} mathfrak Z(X) = frac{1}{d+1}(X+text{Tr}(X)mathbb I
_d) end{align*} is $d^2$ if and only if there exists a symmetric informationally-complete POVM in dimension $d$.
We introduce a game related to the $I_{3322}$ game and analyze a constrained value function for this game over various families of synchronous quantum probability densities.
