Completeness of the symmetric measurement ansatz beyond two settings

Establish whether restricting all parties to identical coplanar qubit observables in the xz plane is without loss of generality for optimizing permutation-invariant Bell inequalities with more than two measurement settings per party.

Background

The paper evaluates permutation-invariant Bell expressions using a symmetric measurement strategy in which every party measures the same observables, parameterized by angles in the xz plane. The authors state that this restriction is known to incur no loss of generality when there are two measurement settings per party.

For scenarios with more than two settings, the authors use the same coplanar symmetric strategy only as a variational ansatz. Determining whether arbitrary Bloch-vector measurements can yield better optima is therefore an unresolved methodological question relevant to the design and optimization of permutation-invariant Bell inequalities beyond the two-setting case.

References

For m=2 this choice involves no loss of generality. Beyond this case the reductions are not known to be complete, and Eq.~eq:sym_meas_supp serves as a variational ansatz.

eq:sym_meas_supp:

Ax(i)=cos⁡θx σz(i)+sin⁡θx σx(i),x=0,…,m−1,A^{(i)}_x=\cos\theta_x\,\sigma_z^{(i)}+\sin\theta_x\,\sigma_x^{(i)}, \qquad x=0,\dots,m-1,

— Experimental certification of multipartite Bell correlations using only few-body symmetric correlations  (2609.37442 - Sun et al., 29 Sep 2026) in Supplemental Material, Section SIV.B, subsection “Quantum value on the target Dicke state”