  Quantum information protocols often rely on tomographic techniques to
determine the state of the system. A popular method of encoding information is
on the different paths a photon may take, for example, parallel waveguides in
integrated optics. However, reconstruction of states encoded onto a large
number of paths is often prohibitively resource intensive and requires
complicated experimental setups. Addressing this, we present a simple method
for determining the state of a photon in a superposition of d paths using a
rotating one-dimensional optical Fourier Transform. We establish the theory and
experimentally demonstrate the technique by measuring a wide variety of
six-dimensional density matrices. The average fidelity of these with the
expected state is as high as 0.9852 +/- 0.0008. This performance is comparable
or exceeds established tomographic methods for other types of systems.
