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Ascertaining the physical state of a system is vital in order to understandand predict its behaviour. Tomography is a standard approach used to determinethe form of an unknown state. Here we show that an alternative approach, basedon sequential weak and strong measurements, can be used to determine thedensity matrix in a simple, fast, and general style. We directly measure theprobability amplitudes of a variety of pure polarisation states of light. Wethen generalise this experiment to directly measure the Dirac distribution andconsequently determine the density matrix. Our work is the first to demonstratethe direct measurement of the full description of the state of atwo-dimensional system, and it has applications to measurements in foundationalquantum mechanics, quantum information, and physical chemistry.

Full characterisation of polarisation states of light via direct measurement