Readout-Rank Laws for Isotropic Quantum Tangents

Deep parameterized quantum circuits may remain sensitive to a parameter change while the observables retained by a learning model barely respond. We study this separation for a fixed computational-basis measurement. For a pure-state tangent, we compare the quantum Fisher information F Q, the Fisher information F full in the complete bitstring distribution, and the largest variance-normalized response I A available to a diagonal readout space A. If the joint state--tangent frame is Haar random, we prove that the two successive information fractions are independent Beta variables whose means are 1/2 and r/(2 n-1), where r is the centered dimension of the readout. Consequently, even the joint span of all computational-basis Pauli strings through any fixed weight k retain only O(n k2-n) of the full-record information. Exact-statevector experiments across six circuit families show increasing finite-size agreement with this hierarchy in five nonconserving ensembles as the circuit depth grows. A number-conserving family departs strongly from the isotropic prediction even after correcting the support and readout rank, showing that rank alone is insufficient without tangent isotropy.