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 $C \succeq 0$, $\mathrm{Tr} C = 1$, 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 $ρ=R/(r/N)$ separates the actual retained mass from a rank-only random-orientation reference. Standard Grassmann averaging gives $\mathbb{E}ρ=1$ and $\mathrm{Var}(ρ) \le 2/(r d_{\mathrm{eff}})$, 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.