Full characterisation of polarisation states of light via direct measurement

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.