Robustness certificates for complex-composition sectors in real Hilbert-space dynamics

Abstract

Independent realification of finite-dimensional complex subsystems produces an ambient real tensor product that is twice as large as the real form of their ordinary complex composite. The established balancing relation is selected by specified local complex structures J_A, J_B: the symmetric involution M = J_A ⊗ J_B has a −1 eigenspace representing the complex-composition sector and a +1 complement removed by the balanced tensor product. We develop quantitative robustness certificates for departures from this sector. For states, the leakage population L(ρ) = Tr(P_leak ρ), P_leak = (I + M)/2, is directly readable from M and tightly calibrates the trace distance to the zero-leakage state set as L ≤ d_bal ≤ √L. For real completely positive trace-preserving maps, the worst-case balanced-input leakage is the largest eigenvalue of the balanced compression of Φ†(P_leak); its zero set is characterized Kraus by Kraus, and the quantity is Lipschitz under diamond distance. For real skew-symmetric generators, ‖P_leak G P_bal‖ is both the sharp initial leaked-amplitude coefficient and the exact operator-norm distance to the sector-preserving generator subspace. We also give a multipartite extension, a finite-sample trusted-model test, and reproducible finite-dimensional checks. These certificates concern sector support; full recovery of complex quantum mechanics additionally requires compatibility with the induced complex structure.