  Given many copies of an unknown quantum state $\rho$, we consider the task of
learning a classical description of its principal eigenstate. Namely, assuming
that $\rho$ has an eigenstate $|\phi\rangle$ with (unknown) eigenvalue $\lambda
> 1/2$, the goal is to learn a (classical shadows style) classical description
of $|\phi\rangle$ which can later be used to estimate expectation values
$\langle \phi |O| \phi \rangle$ for any $O$ in some class of observables. We
consider the sample-complexity setting in which generating a copy of $\rho$ is
expensive, but joint measurements on many copies of the state are possible. We
present a protocol for this task scaling with the principal eigenvalue
$\lambda$ and show that it is optimal within a space of natural approaches,
e.g., applying quantum state purification followed by a single-copy classical
shadows scheme. Furthermore, when $\lambda$ is sufficiently close to $1$, the
performance of our algorithm is optimal--matching the sample complexity for
pure state classical shadows.
