Mutually unbiased bases (MUBs) are highly symmetric bases on complex Hilbert spaces, and the corresponding rank-1 projective measurements are ubiquitous in quantum information theory. In this work, we study a recently introduced generalization of MUBs called mutually unbiased measurements (MUMs). These measurements inherit the essential property of complementarity from MUBs, but the Hilbert space dimension is no longer required to match the number of outcomes. This operational complementarity property renders MUMs highly useful for device-independent quantum information processing. It has been shown that MUMs are strictly more general than MUBs. In this work we provide a complete proof of the characterization of MUMs that are direct sums of MUBs. We then proceed to construct new examples of MUMs that are not direct sums of MUBs. A crucial technical tool for these construction is a correspondence with quaternionic Hadamard matrices, which allows us to map known examples of such matrices to MUMs that are not direct sums of MUBs. Furthermore, we show that -- in stark contrast with MUBs -- the number of MUMs for a fixed outcome number is unbounded. Next, we focus on the use of MUMs in quantum communication. We demonstrate how any pair of MUMs with d outcomes defines a d-dimensional superdense coding protocol. Using MUMs that are not direct sums of MUBs, we disprove a recent conjecture due to Nayak and Yuen on the rigidity of superdense coding for infinitely many dimensions. The superdense coding protocols arising in the refutation reveal how shared entanglement may be used in a manner heretofore unknown.

翻译：互不偏基（MUBs）是复希尔伯特空间上高度对称的基，相应的秩-1投影测量在量子信息理论中无处不在。本研究探讨了近期引入的一种MUBs泛化形式——互不偏测量（MUMs）。此类测量继承了MUBs的核心互补性，但希尔伯特空间维度不再要求与结果数量匹配。这种操作互补性使得MUMs对设备无关量子信息处理具有重要价值。已有研究表明MUMs严格泛化于MUBs。本文首先完整证明了MUMs可表征为MUBs直和的充要条件，继而构建了非MUBs直和的新MUMs实例。关键的技术工具是建立与四元数哈达玛矩阵的对应关系，从而将已知四元数哈达玛矩阵实例映射为非MUBs直和的MUMs。进一步研究表明，与MUBs形成鲜明对比的是，固定结果数量下MUMs的数目无上限。随后我们聚焦MUMs在量子通信中的应用，证明任意一对具有d个结果的MUMs可定义d维超密编码协议。通过使用非MUBs直和的MUMs，我们否证了Nayak与Yuen近期关于无穷维超密编码刚性的猜想。该否证过程中涌现的超密编码协议揭示了共享纠缠的一种全新利用方式。