  We consider the classical shadows task for pure states in the setting of both
joint and independent measurements. The task is to measure few copies of an
unknown pure state $\rho$ in order to learn a classical description which
suffices to later estimate expectation values of observables. Specifically, the
goal is to approximate $\mathrm{Tr}(O \rho)$ for any Hermitian observable $O$
to within additive error $\epsilon$ provided $\mathrm{Tr}(O^2)\leq B$ and
$\lVert O \rVert = 1$. Our main result applies to the joint measurement
setting, where we show $\tilde{\Theta}(\sqrt{B}\epsilon^{-1} + \epsilon^{-2})$
samples of $\rho$ are necessary and sufficient to succeed with high
probability. The upper bound is a quadratic improvement on the previous best
sample complexity known for this problem. For the lower bound, we see that the
bottleneck is not how fast we can learn the state but rather how much any
classical description of $\rho$ can be compressed for observable estimation. In
the independent measurement setting, we show that $\mathcal O(\sqrt{Bd}
\epsilon^{-1} + \epsilon^{-2})$ samples suffice. Notably, this implies that the
random Clifford measurements algorithm of Huang, Kueng, and Preskill, which is
sample-optimal for mixed states, is not optimal for pure states. Interestingly,
our result also uses the same random Clifford measurements but employs a
different estimator.
