Okon, Sudarsky & Wiedemann's spin-1/2 example shows the compensating-preparer mechanism cannot simultaneously conserve noncommuting quantities

physicsquantum-mechanicsquantum-measurementconservation-lawsquant-phphysics.hist-ph

Using a concrete two-spin-1/2 example, Okon, Sudarsky & Wiedemann (2026, arXiv:2610.29804) show that the "preparer" compensation mechanism proposed by Collins & Popescu (Quantum 9, 1815, 2025) cannot be extended to several noncommuting conserved quantities at once. A preparation interaction constructed so that total z-spin (Sz) is conserved outcome-by-outcome does not simultaneously preserve the Born-rule distribution of total x-spin (Sx) for the same prepared state — the mechanism that establishes the required compensation for one observable does not establish it for the other.

Since a generic quantum state is expressible as a superposition in the eigenbases of several noncommuting conserved observables simultaneously, this is more than a technical gap in one worked case: the framework offers no general rule for which observable's compensating correlation structure is the one physically realized when more than one noncommuting conserved quantity is in play. The paper notes this is a gap the original proposal's own authors left as conjectural future work rather than something they resolved.

This is a further, independent failure mode alongside the paper's two headline objections — the reference-frame regress (claim-okon-sudarsky-wiedemann-reference-frame-regress-undermines-preparer-requirement) and the statistical-only result (claim-okon-sudarsky-wiedemann-preparer-never-definite-only-statistical-conservation-restored). A parallel but distinct technical gap specific to energy conservation (the preparation Hamiltonian's interaction term is not assignable to either subsystem) was found in the same capture but left unpromoted under the vault's single-source concentration cap; see question-corroborate-okon-sudarsky-wiedemann-2026-arxiv-2610-29804-single-source-cap.