  The impossibility of deterministic and error-free discrimination among
nonorthogonal quantum states lies at the core of quantum theory and constitutes
a primitive for secure quantum communication. Demanding determinism leads to
errors, while demanding certainty leads to some inconclusiveness. One of the
most fundamental strategies developed for this task is the optimal unambiguous
measurement. It encompasses conclusive results, which allow for error-free
state retrodictions with the maximum success probability, and inconclusive
results, which are discarded for not allowing perfect identifications.
Interestingly, in high-dimensional Hilbert spaces the inconclusive results may
contain valuable information about the input states. Here, we theoretically
describe and experimentally demonstrate the discrimination of nonorthogonal
states from both conclusive and inconclusive results in the optimal unambiguous
strategy, by concatenating a minimum-error measurement at its inconclusive
space. Our implementation comprises 4- and 9-dimensional spatially encoded
photonic states. By accessing the inconclusive space to retrieve the
information that is wasted in the conventional protocol, we achieve significant
increases of up to a factor of 2.07 and 3.73, respectively, in the overall
probabilities of correct retrodictions. The concept of concatenated optimal
measurements demonstrated here can be extended to other strategies and will
enable one to explore the full potential of high-dimensional nonorthogonal
states for quantum communication with larger alphabets.
