We introduce the notion of compatibility dimension for a set of quantum measurements: it is the largest dimension of a Hilbert space on which the given measurements are compatible. In the Schrödinger picture, this notion corresponds to testing compatibility with ensembles of quantum states supported on a subspace, using the incompatibility witnesses of Carmeli, Heinosaari, and Toigo [Phys. Rev. A 98,012126 (2018) and Phys. Rev. Lett. 122, 130402 (2019).]. We provide several bounds for the compatibility dimension using approximate quantum cloning or algebraic techniques inspired by quantum error correction. We analyze in detail the case of two orthonormal bases and, in particular, that of mutually unbiased bases.

中文翻译：

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量子测量的相容性维度

我们为一组量子测量引入了相容维的概念：这是给定测量在其上相容的希尔伯特空间的最大维。在Schrödinger图片中，此概念对应于使用Carmeli，Heinosaari和Toigo [Phys。修订版A 98，012126（2018）和phys。莱特牧师 122，130402（2019）。]。我们使用受量子误差校正启发的近似量子克隆或代数技术为兼容性维提供了几个界限。我们详细分析了两个正交基的情况，尤其是相互无偏的基的情况。