Quantum state restoration and single-copy tomography

Given a single copy of a quantum state |ψ〉, the no cloning theorem greatly limits the amount of information which can be extracted from it. On the other hand, given only a procedure which verifies the state, for example access to a measurementM = {I−|ψ〉〈ψ|, |ψ〉〈ψ|}, even distinguishing I−|ψ〉〈ψ| from the identity takes exponential time. In this paper, we consider the scenario in which we are given both a single copy of |ψ〉and the ability to verify it. We show that taken together, these primitives enable us to efficiently learn about |ψ〉. In particular, for any POVM (even with non commuting operators) we give an algorithm which estimates its statistics on |ψ〉, in time polynomial in the number of operators. We show how this algorithm puts severe limitations on possible quantum money shcemes.