Nonlinear majorization monotone for blockwise bistochastic quantum measurements

Let $\bm{P}\in\Delta_{n,d}$ and $\bm{Q}\in\Delta_{n,d}$ be blockwise probability vectors. Let $\mathcal{B}_{n,d}$ denote the set of blockwise bistochastic matrices, and let $\*$ denote the blockwise product. For a matrix $A$, define its 2-norm by

$$\lVert A\rVert_2=\sqrt{\operatorname{tr}[A^\dagger A]}.$$

Nonlinear majorization monotone conjecture. If there exists a blockwise bistochastic matrix $\bm{B}\in\mathcal{B}_{n,d}$ such that $\bm{Q}=\bm{B}\*\bm{P}$, then for any ordering of $\{Q_i\}$ there exists an ordering of $\{P_i\}$ such that

$$\left\lVert\sum_{i=1}^k\left(P_i-\frac{1}{n}\mathds{1}\right)\right\rVert_2\geq\left\lVert\sum_{i=1}^k\left(Q_i-\frac{1}{n}\mathds{1}\right)\right\rVert_2$$

for all $1\leq k\leq n$.

This conjecture proposes a nonlinear monotone extending majorization beyond the linear monotones available for sortable quantum measurements. It asserts that blockwise bistochastic dynamics cannot increase the cumulative 2-norm after a suitable ordering of the input blocks; the paper provides the surrounding motivation and numerical evidence but does not establish the claim.