  We introduce a new method to estimate unknown pure $d$-dimensional quantum
states using the probability distributions associated with only three
measurement bases. Measurement results of $2d$ projectors are employed to
generate a set of $2^{d-1}$ possible states, the likelihood of which is
evaluated using the measurement results of the $d$ remaining projectors. The
state with the highest likelihood is the estimate of the unknown state. The
method estimates all pure states but a null-measure set. The viability of the
protocol is experimentally demonstrated using two different and complementary
high-dimensional quantum information platforms. First, by exploring the
photonic path-encoding strategy, we validate the method on a single
8-dimensional quantum system. Then, we resort to the five superconducting qubit
IBM quantum processor to demonstrate the high performance of the method in the
multipartite scenario.
