  High-dimensional quantum information processing has become a mature field of
research with several different approaches being adopted for the encoding of
$D$-dimensional quantum systems. Such progress has fueled the search of
reliable quantum tomographic methods aiming for the characterization of these
systems, being most of these methods specifically designed for a given
scenario. Here, we report on a new tomographic method based on multiply
symmetric states and on experimental investigations to study its performance in
higher dimensions. Unlike other methods, it is guaranteed to exist in any
dimension and provides a significant reduction in the number of measurement
outcomes when compared to standard quantum tomography. Furthermore, in the case
of odd dimensions, the method requires the least possible number of measurement
outcomes. In our experiment we adopt the technique where high-dimensional
quantum states are encoded using the linear transverse momentum of single
photons and are controlled by spatial light modulators. Our results show that
fidelities of $0.984\pm0.009$ with ensemble sizes of only $1.5\times10^5$
photons in dimension $D=15$ can be obtained in typical laboratory conditions,
thus showing its practicability in higher dimensions.
