Coordinate-free Nagaoka–Hayashi bound for constrained parameters

Determine whether a coordinate-free formulation of the Nagaoka–Hayashi bound, analogous to the coordinate-free formulations of the quantum Fisher information matrix and Helstrom bound, exists for constrained parameters on the probability simplex.

Background

The paper estimates relative emitter brightnesses represented by a probability vector on a simplex, so the parameters satisfy a normalization constraint and the associated Fisher information matrices are rank-deficient in the ambient coordinate space. The authors handle this constraint by selecting orthonormal coordinates on the simplex tangent space and projecting the relevant information matrices onto that tangent space.

Although the Helstrom bound and quantum Fisher information matrix admit formulations that are invariant under orthogonal changes of tangent-space coordinates, the Nagaoka–Hayashi bound is evaluated numerically after choosing an explicit coordinate system. The authors explicitly identify whether an analogous coordinate-free formulation exists as unresolved.

References

It remains an interesting open problem whether a coordinate-free formulation of the NH bound, similar to that of $Q_{\bm{\theta}}$ and $c_{\bm{\theta}{\rm SLD}$, exists in the context of constrained parameters (see Supplementary Materials for more discussion).

— Attaining Fundamental Limits of Multiparameter Incoherent Optical Imaging Using Joint-Detection Quantum Measurements  (2608.19524 - Deshler et al., 20 Aug 2026) in Footnote following Eq. (relative-brightness reparameterization), Section “Separable v. Joint Measurements”