Map-Dependent Quantum Characteristic Functions and CP-Divisibility in Non-Markovian Quantum Dynamics

Referee: [§3] §3 (Bochner-Choi positivity theorem and definition of map-dependent QCF): The equivalence between the positive-type condition of the Gram matrix and complete positivity is stated for the normalized Choi operator. For intermediate maps Φ(t,s) that are not CP, the normalization step (division by the trace of the Choi operator) can encounter division by zero or yield non-positive or complex quantities when the un-normalized Choi operator has negative eigenvalues or non-positive trace. The proof does not explicitly demonstrate that the Gram matrix remains real and the equivalence continues to hold in this regime, which is load-bearing for the CP-divisibility claim.

Authors: We agree that the Bochner-Choi theorem is rigorously proven under the assumption that the map is CP, so that the normalized Choi operator is a valid density operator. For trace-preserving intermediate maps (standard in the dynamical-map setting), the trace of the Choi operator equals the Hilbert-space dimension and is therefore strictly positive, precluding division by zero. The resulting normalized operator is Hermitian with unit trace even when it possesses negative eigenvalues. The associated Gram matrix is constructed from the (real) matrix elements of this Hermitian operator and remains real-valued. While the full equivalence holds only for CP maps, the contrapositive implies that negativity of the Gram matrix is a reliable indicator of non-CP. We will add an explicit remark in §3 clarifying the domain of the theorem and noting that the construction extends formally to Hermitian operators for the purpose of detecting CP breakdown. revision: partial

Referee: [§4] §4 (Characterization of CP-divisibility): The application to two-time characteristic functions for intermediate maps assumes the normalized construction is valid even when the maps are not CP by definition during non-Markovian intervals. No separate argument or regularization is provided to ensure the positive-type condition remains a faithful indicator of CP-divisibility when the underlying Choi operator is not positive semidefinite, risking that the claimed characterization depends on an unexamined extension of the theorem.

Authors: The characterization of CP-divisibility is based on the observation that an intermediate map is CP if and only if the associated two-time Gram matrix is positive semidefinite. When the map fails to be CP, the Gram matrix can (and does) become indefinite; this negativity is used as a diagnostic of non-divisibility. Because the underlying Choi operator is always Hermitian for Hermiticity-preserving maps, the characteristic-function construction and the Gram matrix remain well-defined without additional regularization. We will insert a short dedicated paragraph in §4 that explicitly invokes the contrapositive of the Bochner-Choi theorem to justify the indicator for non-CP intervals, thereby removing any ambiguity about the extension. revision: partial

Referee: [§5] §5 (Numerical examples for amplitude damping and pure dephasing): The reported coincidence between Gram-matrix negativity and the breakdown of CP-divisibility is shown for specific time intervals, but the manuscript provides no details on discretization step size, cutoff criteria, or sensitivity analysis. Without such checks it is unclear whether the observed agreement is robust or could be affected by post-hoc parameter choices.

Authors: We thank the referee for highlighting this omission. In the revised manuscript we will add a new subsection (or appendix) that specifies the time-discretization step size employed in the numerical integration, the numerical threshold used to declare an eigenvalue negative, and the results of a sensitivity analysis performed by varying the step size by factors of two and altering the negativity cutoff. These additions will confirm that the reported coincidence is robust and not an artifact of particular parameter choices. revision: yes