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> 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.Comment: edited for clarity; 13 pages, 1 figur
