  Following [OW16], we continue our analysis of: (1) "Quantum tomography",
i.e., learning a quantum state, i.e., the quantum generalization of learning a
discrete probability distribution; (2) The distribution of Young diagrams
output by the RSK algorithm on random words. Regarding (2), we introduce two
powerful new tools: (i) A precise upper bound on the expected length of the
longest union of $k$ disjoint increasing subsequences in a random length-$n$
word with letter distribution $\alpha_1 \geq \alpha_2 \geq \cdots \geq
\alpha_d$; (ii) A new majorization property of the RSK algorithm that allows
one to analyze the Young diagram formed by the lower rows $\lambda_k,
\lambda_{k+1}, \dots$ of its output.
  These tools allow us to prove several new theorems concerning the
distribution of random Young diagrams in the nonasymptotic regime, giving
concrete error bounds that are optimal, or nearly so, in all parameters. As one
example, we give a fundamentally new proof of the fact that the expected length
of the longest increasing sequence in a random length-$n$ permutation is
bounded by $2\sqrt{n}$. This is the $k = 1$, $\alpha_i \equiv \frac1d$, $d \to
\infty$ special case of a much more general result we prove: the expected
length of the $k$th Young diagram row produced by an $\alpha$-random word is
$\alpha_k n \pm 2\sqrt{\alpha_kd n}$.
  From our new analyses of random Young diagrams we derive several new results
in quantum tomography, including: (i) Learning the eigenvalues of an unknown
state to $\epsilon$-accuracy in Hellinger-squared, chi-squared, or KL distance,
using $n = O(d^2/\epsilon)$ copies; (ii) Learning the optimal rank-$k$
approximation of an unknown state to $\epsilon$-fidelity (Hellinger-squared
distance) using $n = \widetilde{O}(kd/\epsilon)$ copies.
