Beyond Mean Subsystem Distinguishability a Distributional Heterogeneity of Local-Perturbation Accessibility

This work investigates information lost when subsystem distinguishishability after a local perturbation is compressed into a cardinality-resolved mean. We introduce the scale-resolved perturbation-accessibility distribution (SPAD), \(P_V(\chi|k,t)\), which retains the full distribution of normalized trace distinguishabilities across equal-size subsystems. Exact statevector calculations are performed across free and interacting Aubry–André chains, the SSH chain, and random Clifford circuits, covering 2,048 independently generated principal dynamics, additional fixed-parameter realizations, and targeted finite-size controls up to \(N=12\). The results demonstrate that nearly identical mean distinguishability profiles can coexist with substantially different accessibility distributions. An exact one-excitation construction proves that this mean/distribution separation can occur analytically. The effect is nevertheless model-, time-, and protocol-dependent and is not shown to persist in the thermodynamic limit. Importantly, SPAD is not established as a superior information diagnostic: all 723 qualifying pairs in the principal ensembles are also distinguished by centered subsystem information capacity (SIC), while unlabelled SPAD is invariant under site permutations and therefore characterizes heterogeneity rather than spatial geography. The central conclusion is deliberately limited: mean subsystem distinguishability is not a sufficient statistic for perturbation-accessibility heterogeneity. The mean and the full distribution answer different questions, motivating distributional reporting when heterogeneity itself is physically relevant.