Environment-Record Leakage for Finite Quantum Error Histories: Balanced Global Minimax, Complete Minimizer Geometry, and Distributional Stability

Overview This preprint develops a finite-history framework for quantifying how coherently stored environment records reveal logical quantum information. Instead of considering only which error occurred, the analysis treats the full complementary environment channel associated with a labeled family of finite quantum-error histories. For two selected history labels, the leakage is expressed exactly through the numerical-range diameter of the compressed relative operator. This yields global finite-history compression bounds, a zero-leakage characterization equivalent to Knill–Laflamme scalarity, and lower and upper bounds for recovery by a single common channel. Main results For a balanced independent two-step dephasing model on one logical qubit encoded into two physical qubits, the paper solves the worst-case optimization over the full rank-two code Grassmannian: $$\inf_{V\in\mathrm{Gr}(2,4)} \sup_{\phi,\psi} \mathcal L_{\mathrm{hist}}(V) = \frac12.$$ The transverse leakage is determined exactly by $$L_{\mathrm{tr}}(\eta,t) = \max\left\{ \frac12,\, 2\eta,\, \sqrt{\frac14-\eta^2}\sqrt{1+t} \right\}.$$ All global minimizers are classified. Besides the parity-isoclinic orthogonal-axis family, the minimizer set contains a genuine non-isoclinic region characterized by $$0 < \eta \le \frac14, \qquad e_z = o_z = 0, \qquad \vert{}\mathbf e \cdot \mathbf o\vert{} \le \frac{4\eta^2}{1-4\eta^2}.$$ A universal quartic inequality needed for the all-code theorem is reduced analytically to an exact polynomial-positivity problem and closed using rational Bernstein-basis verification. Distributional stability On a fixed labeled history-operator family, the complementary channel, worst-parameter leakage, and all-code minimax value are Lipschitz-stable under changes of the history probability distribution. With $$d_{\mathrm A}(p,q) = \left[ \sum_h \left(\sqrt{p_h}-\sqrt{q_h}\right)^2 \right]^{1/2},$$ the minimax value satisfies $$\vert{}\Lambda(p)-\Lambda(q)\vert{} \le 2d_{\mathrm A}(p,q).$$ Consequently, key quantitative implications of the balanced complete solution—including the minimax value, principal near-minimizer coordinate constraints, and a universal operational recovery lower bound—remain stable under nearby biased or correlated perturbations.