Spin-coherent quantum designs tie exact operator reconstruction to spherical (2J+S)-designs

The paper resolves the rank-resolved tomography problem for spin-coherent states by showing that the canonical contravariant-symbol formula succeeds precisely when the sampling set satisfies the design condition of degree 2J+S. A positive-weight Gauss-Legendre construction is supplied that bypasses the equal-weight requirement, yielding explicit measurement protocols for polarimetry and magnetometry. These results unify earlier discrete coherent-state bases under a single design-theoretic criterion and directly support moment estimation from few samples.

Contextually, the work sits at the intersection of quantum information and spherical design theory, extending results on finite coherent-state tomography that previously lacked rank specificity. It addresses a gap between overcomplete continuous representations and minimal discrete sets needed for exact reconstruction in finite-dimensional spin spaces, where non-orthogonality has long complicated readout.

Practical uptake will hinge on whether experimental groups can realize the prescribed point sets with current control hardware. If the Gauss-Legendre weights prove robust to noise, the framework could reduce sample counts in quantum state tomography by 30-50 percent for low-rank spin states, a threshold that would matter for near-term sensors.

Next steps include numerical benchmarking against existing spherical-design libraries and integration into adaptive measurement loops for real-time magnetometry.