\equiv \{P_i\}$ and $q \equiv \{Q_i\}$ satisfy the set of relations $$\sum_{i=1}^{k} P_i \geq \sum_{i=1}^{k} Q_i \;\; , \;\;\; \forall k,
\label{majdef}$$ where it is understood that the elements of both sets have been placed in decreasing order (e.g., $P_i > P_{i+1}, \forall i$), then $p$ is said to majorize $q$, and this is written $q \prec p$. While at first Eq.(\[majdef\]) looks a little complicated, a few moments consideration reveals that it captures precisely what one means by uncertainty - if $p$ majorizes $q$, then $p$ is more sharply peaked than $q$, and consequently describes a state of knowledge containing less uncertainty. What is more, majorization implies an ordering with Shannon entropy $H[\cdot]$. That is, if $p$ majorizes $q$, then $H[p] \leq H[q]$. [@MO; @Bhatia]

In a sense, majorization is a more basic notion of uncertainty than entropy in that it captures that concept alone – the Shannon entropy on the other hand characterizes the more specific notion of information. To characterize the uncertainty of a density matrix, we can apply majorization to the vector consisting of its eigenvalues. If $\rho$ and $\sigma$ are density matrices, then we will write $\sigma \prec \rho$ if $\rho$’s eigenvalues majorize $\sigma$’s. Various 