Abstract

We introduce a method to estimate unknown pure d-dimensional quantum states using the probability distributions associated with only three measurement bases. The measurement results of 2d projectors are used to generate a set of 2d-1 states, the value of the likelihood function of which is evaluated using the measurement results of the remaining d projectors. The state with the highest value of the likelihood function is the estimate of the unknown state. The method estimates all pure states except for a null-measure set, which can be overcome by adapting two bases. 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 eight-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.