  Single-qubit measurements are typically insufficient for inferring arbitrary
quantum states of a multi-qubit system. We show that if the system can be fully
controlled by driving a single qubit, then utilizing a local random pulse is
almost always sufficient for complete quantum-state tomography. Experimental
demonstrations of this principle are presented using a nitrogen-vacancy (NV)
center in diamond coupled to a nuclear spin, which is not directly accessible.
We report the reconstruction of a highly entangled state between the electron
and nuclear spin with fidelity above 95%, by randomly driving and measuring the
NV-center electron spin only. Beyond quantum-state tomography, we outline how
this principle can be leveraged to characterize and control quantum processes
in cases where the system model is not known.
