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 four- and nine-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.

非正交量子态之间的确定性和无差错区分是不可能的，这是量子理论的核心，并且构成了安全量子通信的原语。要求确定性会导致错误，而要求确定性会导致一些不确定性。为此任务开发的最基本的策略之一是最佳的明确测量。它包括结论性结果，允许以最大成功概率进行无错误状态追溯，以及由于不允许完美识别而被丢弃的不确定结果。有趣的是，在高维希尔伯特空间中，不确定的结果可能包含有关输入状态的有价值信息。这里，我们通过在其不确定空间连接最小误差测量，从理论上描述并通过实验证明最佳明确策略中非正交状态与确定性和不确定性结果的区别。我们的实现包括四维和九维空间编码光子态。通过访问不确定的空间来检索传统协议中浪费的信息，我们在正确追溯的整体概率上分别显着增加了 2.07 和 3.73 倍。这里展示的级联最优测量的概念可以扩展到其他策略，并使人们能够探索高维非正交状态的全部潜力，用于与更大字母的量子通信。