$\Phi$ be the column-quasistochastic matrix associated with an arbitrary reference process. Then $$s(\Phi)\succ_{w\log}s(\Phi_{\rm SIC})\;,$$ with equality iff the MIC and post-measurement states are SICs.

We are now poised to prove:

\[unitarilyinvariant\] Let $\Phi$ be the column-quasistochastic matrix associated with an arbitrary reference process. Then for any unitarily invariant norm ${\left\lVert\cdot\right\rVert}$, $${\left\lVertI-\Phi\right\rVert}\geq{\left\lVertI-\Phi_{\rm SIC}\right\rVert}\;,$$ with equality iff the MIC and post-measurement states are SICs.

By Corollary 3.5.9 in [@Horn:1994], every unitarily invariant norm is monotone with respect to the partial order on matrices induced by weak majorization of the vector of singular values. $I-\Phi$ is singular with exactly one eigenvalue equal to zero, so one of its singular values is zero as well. Then $$\begin{split}
    s(I-\Phi)&\succ\left\{\frac{\sum_is_i(I-\Phi)}{d^2-1},\ldots,\frac{\sum_is_i(I-\Phi)}{d^2-1}\right\}\\
    &\succ_w\{d,\ldots,d\}=s(I-\Phi_{\rm SIC})
\end{split}$$ if $$\sum_is_i(I-\Phi)\geq d(d^2-1)\;.$$ We have $$\begin{aligned}
    \sum_is_i(I-\Phi)&\geq&\sum_i|\lambda_i(I-\Phi)|=\sum_i|\lambda_i(\Phi)-1|\nonumber\\
    &\geq&\sum_i(|\lambda_i(\Phi)|-1)\geq\sum_i\lambda_i(\Phi_{\rm SIC})-d^2\nonumber\\
    &=&d(d^2-1)\;,\end{aligned}$$ where the first inequality follows Eq. 3.3.13a in [@Horn:1994], the second follows from the triangle inequality, and the last follows from Lemma \[maj\].

It is known that no quasiprobability representation of quantum theory can be entirely nonnegative [@Ferrie:2009]. What does this mean in our formalism?

![$\mathcal{N}$ is the normalized hyperplane of $d^2$-element quasiprobability vectors and the outer, black triangle represents the $(d^2-1)$-simplex $\Delta$ of probabilities. For a given MIC, the inner, green triangle 