Characterization of all quadratic Bell constraints

Prove or refute the conjecture that every quadratic Bell constraint reduces, up to affine transformations, to stabilizer constraints and subsystem-purity constraints.

Background

The paper establishes a transformation from quadratic symmetries of Bell distributions to quadratic symmetries of Pauli distributions and observes that the image of this transformation is a proper subset of the latter. Consequently, some nonlinear properties visible in the Pauli distribution may not be directly accessible through Bell measurements.

The authors conjecture that the quadratic constraints accessible at the Bell level have no additional forms beyond stabilizer and subsystem-purity constraints, modulo affine transformations. Proving this would simplify the paper’s analysis and identify the intrinsic information content of Bell sampling; disproving it by finding genuinely different symmetry families could reveal new efficiently learnable classes of structured quantum states.

References

We conjecture that all quadratic Bell constraints necessarily reduce to stabilizer and subsystem purities (up to affine transformations). Proving this, would both simplify the analysis developed here, and clarify the intrinsic information content of the Bell distribution. Conversely, finding genuinely different families of symmetries, would then point to new classes of structured states admitting similar efficient learning algorithms.

— Efficient learning of Clifford disentanglers and typical $t$-doped unitaries with exponentially more $T$ gates  (2609.27565 - Aguilar et al., 23 Sep 2026) in Discussion and future work, paragraph “Open questions”