Quantitative error bounds for the positivity-restored Krylov dissipative generator

Derive a quantitative error bound for the approximate dissipative generator obtained by projecting the Krylov-truncated coefficient matrix onto the positive semidefinite cone, without imposing additional assumptions.

Background

For interacting systems, Krylov–Lanczos truncation of the modular transformation does not generally preserve conditional complete positivity, so the resulting object need not be an exact Lindbladian. The paper proposes replacing the truncated coefficient matrix with its closest positive semidefinite approximation, denoted by $\Xi_*$, thereby restoring the Lindbladian condition. However, the authors explicitly state that no quantitative estimate is available for the error introduced by this projection. Establishing such an estimate would relate the positivity-restoration procedure to the accuracy of the original dissipative dynamics.