Sample-optimal classical shadows for pure states

Huang, Kueng, Preskill introduced the learning task now known as “classical shadows”: given few copies of an unknown state ρ, construct a classical description of the state from independent measurements that can be used to predict certain properties of the state. Specifically, they show Θ(B/epsilon^2) samples of ρ suffice to approximate the expectation value Tr(Oρ) of any Hermitian observable O to within additive error epsilon provided Tr(O^2) ≤ B and the eigenvalues of O are contained in [-1,1]. We consider the task of constructing classical shadows with joint measurements and pure unknown states. We show Θ(√B/epsilon + 1/epsilon^2) copies are necessary and sufficient to construct classical shadows in this setting.