  Ascertaining the physical state of a system is vital in order to understand
and predict its behaviour. Tomography is a standard approach used to determine
the form of an unknown state. Here we show that an alternative approach, based
on sequential weak and strong measurements, can be used to determine the
density matrix in a simple, fast, and general style. We directly measure the
probability amplitudes of a variety of pure polarisation states of light. We
then generalise this experiment to directly measure the Dirac distribution and
consequently determine the density matrix. Our work is the first to demonstrate
the direct measurement of the full description of the state of a
two-dimensional system, and it has applications to measurements in foundational
quantum mechanics, quantum information, and physical chemistry.
