We show that the precision of an angular measurement or rotation (e.g., on the orientation of a qubit or spin state) is limited by fundamental constraints arising from quantum mechanics and general relativity (gravitational collapse). The limiting precision is $r^{-1}$ in Planck units, where r is the physical extent of the (possibly macroscopic) device used to manipulate the spin state. This fundamental limitation means that spin states $S_1$ and $S_2$ cannot be experimentally distinguished from each other if they differ by a sufficiently small rotation. Experiments cannot exclude the possibility that the space of quantum state vectors (i.e., Hilbert space) is fundamentally discrete, rather than continuous. We discuss the implications for finitism: does physics require infinity or a continuum?

我们表明，角度测量或旋转（例如，在量子位或自旋状态的方向上）的精度受到量子力学和广义相对论（引力坍缩）产生的基本约束的限制。极限精度为$r^{-1}$以普朗克单位表示，其中r是用于操纵自旋状态的（可能是宏观的）设备的物理范围。这个基本限制意味着自旋状态${秒}_1$ 和 ${秒}_2$如果它们的旋转足够小，则无法通过实验将它们区分开来。实验不能排除量子态向量空间（即希尔伯特空间）从根本上是离散的，而不是连续的的可能性。我们讨论有限论的含义：物理学需要无穷大还是连续统？