say $x$ majorizes $y$, denoted $x\succ y$.

Another variant of majorization, called log majorization or multiplicative majorization, is also studied [@Marshall:2011]. We say that $x$ weakly log majorizes $y$ from below, denoted $x\succ_{w\log}y$, if $$\label{logmaj}
    \prod_{i=1}^k x_i^\downarrow\geq\prod_{i=1}^ky_i^\downarrow\;, \quad \text{for } k=1,\ldots,N\;.$$ If the last inequality is an equality, we say $x$ log majorizes $y$, denoted $x\succ_{\rm log} y$. Taking the log of both sides of Eq.  demonstrates that log majorization is majorization between the vectors after an element-wise application of the log map. Log majorization is strictly stronger than regular majorization; $x\succ_{w\log}y\implies x\succ_wy$, but the reverse implication is not true. Majorization is a partial order on vectors of real numbers sorted in nonincreasing order.

Throughout this paper we will make use of the standard inequalities between the arithmetic, geometric, and harmonic means for vectors of $n$ positive numbers $x_i$: $$\label{meanordering}
    \frac{1}{n}\sum_{i=1}^nx_i\ge\left(\prod_{i=1}^nx_i\right)^{\!1/n}\ge\left(\frac{1}{n}\sum_{i=1}^n\frac{1}{x_i}\right)^{\!-1}\;.$$ with equality in all cases if and only if $x_i=c$ for all $i$. We now turn to two lemmas.

\[det\] Let $\Phi_{\rm p}$ denote the column-quasistochastic matrix associated with a MIC and a proportional post-measurement set. Then $\det\Phi_{\rm p}\geq\det\Phi_{\rm SIC}$ with equality iff the MIC is a SIC.

We may write $\Phi^{-1}_{\rm p}=GA^{-1}$ where $G_{ij}:=\tr H_iH_j$ is the Gram 