Fundamental limit on angular measurements and rotations from quantum mechanics and general relativity

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? (C) 2021 The Author(s). Published by Elsevier B.V.