Complementarity is a phenomenon explaining several core features of quantum theory, such as the well-known uncertainty principle. Roughly speaking, two objects are said to be complementary if being certain about one of them necessarily forbids useful knowledge about the other. Two quantum measurements that do not commute form an example of complementary measurements, and this phenomenon can also be defined for ensembles of states. Although a key quantum feature, it is unclear whether complementarity can be understood more operationally, as a necessary resource in some quantum information task. Here we show this is the case, and relates to a novel task which we term $\eta$-unambiguous exclusion. As well as giving complementarity a clear operational definition, this also uncovers the foundational underpinning of unambiguous exclusion tasks for the first time. We further show that a special type of measurement complementarity is equivalent to advantages in certain encryption tasks. Finally, our analysis suggest that complementarity of measurement and state ensemble can be interpreted as strong forms of measurement incompatibility and quantum steering, respectively.

翻译：互补性是一种解释量子理论若干核心特征（如著名的测不准原理）的物理现象。粗略而言，若对某个对象的确知必定会阻止对另一个对象的有用认知，则称这两个对象具有互补性。两个不对易的量子测量构成互补测量的范例，而该现象也可针对态系综进行定义。尽管这是量子理论的关键特性，但互补性能否被更操作化地理解为某些量子信息任务中的必要资源，尚不明确。本文证明这一观点成立，且关联至一种被命名为$\eta$-无歧义排除的新任务。该研究不仅赋予互补性清晰的操作定义，还首次揭示了无歧义排除任务的基础理论根基。我们进一步证明，特定类型的测量互补性等价于某些加密任务中的优势。最后，我们的分析表明，测量互补性与态系综互补性可分别被诠释为测量不兼容性与量子导引的强形式。