of what makes quantum theory nonclassical resides in the fact that $\Phi$ cannot equal $I$. How close, then, can we make $\Phi$ to $I$ by wisely choosing our MIC and post-measurement states? It turns out that $\Phi_{\rm SIC}$ is closest to the identity with respect to the distance measure induced by any member of a large family of operator norms called unitarily invariant norms (see section 3.5 in [@Horn:1994]). A unitarily invariant norm is one such that ${\left\lVertA\right\rVert}={\left\lVertUAV\right\rVert}$ for all unitary matrices $U$ and $V$. These norms include the Schatten $p$-norms (among which are the trace norm, the Frobenius norm, and the operator norm when $p=1,2,$ and $\infty$ respectively) and the Ky Fan $k$-norms. This result codifies the intuition that Eq.  represents the “simplest modification one can imagine to the LTP” [@Fuchs:2014 p. 1971].

To prove this, we will make use of the theory of majorization [@Horn:1994; @Marshall:2011]. Suppose $x$ and $y$ are vectors of $N$ real numbers and that $x^\downarrow$ and $y^\downarrow$ are $x$ and $y$ sorted in nonincreasing order. Then we say that $x$ weakly majorizes $y$ from below, denoted $x\succ_w y$, if $$\label{MajBaker}
    \sum_{i=1}^k x_i^\downarrow\geq\sum_{i=1}^k y_i^\downarrow\;,\quad \text{for } k=1,\ldots,N\;.$$ If the last inequality is an equality, we 