Readout Orientation Controls Measurement-Accessible Quantum Tangent Geometry

Abstract

A fixed quantum measurement can expose substantially more tangent information than a restricted observable readout retains. We study this second restriction. For a normalized covariance $C⪰0$, $\mathrm{Tr}C=1$, of measurement-induced tangent scores in an $N$-dimensional centered score space, and a rank-$r$ readout projector $P$, we quantify retained tangent mass by $R=\mathrm{Tr}(PC)$. The ratio $\rho =R/(r/N)$ separates the actual retained mass from a rank-only random-orientation reference. Standard Grassmann averaging gives $\mathbb{E}\rho =1$ and $\mathrm{Var}(\rho )\le 2/(r{d}_{\mathrm{eff}})$, where $d_{\mathrm{eff}}=1/\mathrm{Tr}(C^2)$. We use this identity as a null model rather than as a new random-projection theorem. Numerically, family-balanced one- and two-body readouts remain close to the rank reference through $n=16$ even as the tangent covariance becomes strongly anisotropic. The decisive equal-rank comparison holds the circuit, measurement record, readout rank, and evaluation shot budget fixed. For Haar-$U(4)$ at $n=12$, cross-fitted alignment increases the mean directional gradient-energy proxy by a factor 9.584 and the finite-shot signal-to-noise ratio by a factor 3.111 relative to the physical one-body readout, while a random rank-matched subspace remains near the rank baseline. A half-filled $U(1)$-conserving family provides a structured counterexample to generic orientation: physical low-weight $Z$ readouts are already strongly aligned with leading tangent directions over the tested finite-size range. We treat this symmetry result as a case study, not as a claim that $U(1)$ symmetry generically prevents barren plateaus or that hydrodynamics is the established mechanism. The results isolate readout orientation as a degree of freedom invisible to rank alone that directly controls how much measured tangent information remains usable after readout restriction.

15 pages, 14 figures. Source code and reproducibility materials are available at: https://github.com/AHDMarwan/Spectral-Geometry-of-Accessible-Quantum-Tangents-Beyond-Isotropic-Readout-Rank-Laws