is the simplex $\Phi^{-1}(\Delta)$, the blue circle is the image of $\mathcal{Q}_d$ under the Born Rule, denoted $\mathcal{P}$, and the red circle is $\Phi(\mathcal{P})$. $\mathcal{P}$ and $\Phi(\mathcal{P})$ are portrayed with circles to capture convexity and inclusion relationships only; they need not bear any resemblance to spheres.[]{data-label="figure2"}](simplexes_and_convex_sets.png){width="\linewidth"}

Let $\mathcal{N}$ be the normalized hyperplane of $d^2$-element quasiprobability vectors. Within this is the $(d^2-1)$-simplex of probability vectors, $\Delta$. For any MIC, $d$-dimensional quantum state space $\mathcal{Q}_d$ is mapped by the Born Rule to a convex subset of $\Delta$, denoted $\mathcal{P}$. Note that $\Phi^{-1}(\Delta)$ is equal to the convex hull of the $d^2$ probability vectors $\tr H_j\sigma_i$, that is, the probabilities for the MIC measurement for each post-measurement state. Consequently, $\Phi^{-1}(\Delta)\subset \mathcal{P}$, which implies $\Delta\subset \Phi(\mathcal{P})$. These inclusions must be strict, i.e., $\Phi\neq I$: When the MIC and post-measurement states are rank-$1$, the vertices of the simplex will be among the pure-state probability vectors, but $\mathcal{P}$ contains more pure states than there are vertices of $\Phi^{-1}(\Delta)$. Since the image of some probability vectors consistent with quantum theory must leave the probability simplex under the application of $\Phi$, we have demonstrated that the appearance of negativity is unavoidable in our framework and is in fact 