Information-Theoretic Confinement in Quantum Correlation Recovery under Rank-2 Defects: A Numerical Study via Stinespring Dilation

We investigate the fundamental limits of quantum correlation recovery under structural rank-2 defects, employing systematic numerical optimization over the full class of completely positive trace-preserving (CPTP) maps. The recovery channel is parameterized via Stine spring dilation, enabling exhaustive exploration of the CPTP class by varying the environ ment dimension E ∈ {1,2,3,4}, corresponding to optimization landscapes of 81 to 1296 real parameters. Our central finding is a striking saturation phenomenon: the optimal recovery channel at E =1is strictly unitary, achieving a conditional mutual information I(A : C|B) = 1.76258 bits. Crucially, we demonstrate that genuine mixed CPTP channels at E = 2 and E = 3— with effective Kraus ranks equal to E—saturate this exact same value within a numerical spread of < 10−5 bits, establishing that the unitary bound is not an artifact of dimensional restriction but a fundamental feature of the recovery landscape. While optimization at E =4 yields a slightly suboptimal result (1.75198 bits) due to high-dimensional landscape complexity, the overall saturation trend is robust. The optimal unitary operator is revealed to be a targeted subspace rotation: it preserves most basis states with probabilities > 94% while performing a precise coherent rotation in the subspace directly affected by the defect’s asymmetry. This optimal value lies strictly below both the analytical upper bound of 2.0 bits (derived from the effective dimension reduction induced by the defect) and the ideal quantum secret sharing limit of 3.1699 bits, revealing a finite information-theoretic confinement gap ∆conf ≈ 1.4073 bits. We interpret this saturation as evidence that environmental degrees of freedom provide no information-theoretic advantage for correlation recovery under structural defects, estab lishing a concrete realization of information-theoretic confinement within the framework of Information-Copying Cosmology (ICC).