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The impossibility of deterministic and error-free discrimination amongnonorthogonal quantum states lies at the core of quantum theory and constitutesa primitive for secure quantum communication. Demanding determinism leads toerrors, while demanding certainty leads to some inconclusiveness. One of themost fundamental strategies developed for this task is the optimal unambiguousmeasurement. It encompasses conclusive results, which allow for error-freestate retrodictions with the maximum success probability, and inconclusiveresults, which are discarded for not allowing perfect identifications.Interestingly, in high-dimensional Hilbert spaces the inconclusive results maycontain valuable information about the input states. Here, we theoreticallydescribe and experimentally demonstrate the discrimination of nonorthogonalstates from both conclusive and inconclusive results in the optimal unambiguousstrategy, by concatenating a minimum-error measurement at its inconclusivespace. Our implementation comprises 4- and 9-dimensional spatially encodedphotonic states. By accessing the inconclusive space to retrieve theinformation that is wasted in the conventional protocol, we achieve significantincreases of up to a factor of 2.07 and 3.73, respectively, in the overallprobabilities of correct retrodictions. The concept of concatenated optimalmeasurements demonstrated here can be extended to other strategies and willenable one to explore the full potential of high-dimensional nonorthogonalstates for quantum communication with larger alphabets.

Enhanced discrimination of high-dimensional quantum states by concatenated optimal measurement strategies