Quantum state restoration and single-copy tomography

Given a single copy of an n qubit quantum state |psi>, the no-cloning theorem greatly limits the amount of information which can be extracted from it. Moreover, given only a procedure which verifies the state, for example a procedure which measures the operator |psi><psi|, we cannot prepare |psi> in time polynomial in n . In this paper, we consider the scenario in which we are given both a single copy of |psi> and the ability to verify it. We show that in this setting, we can do several novel things efficiently. We present a new algorithm that we call quantum state restoration which allows us to extend a large subsystem of |psi> to the full state, and in turn this allows us to copy small subsystems of |psi>. In addition, we present algorithms that can perform tomography on small subsystems of |psi>, and we show how to use these algorithms to estimate the statistics of any efficiently implementable POVM acting on |psi> in time polynomial in the number of outcomes of the POVM.