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

TL;DR

This paper introduces map-dependent quantum characteristic functions to analyze non-Markovian quantum dynamics, linking positivity conditions to CP-divisibility and information backflow.

Contribution

It establishes a new framework connecting characteristic functions with the structural properties of quantum dynamical maps, especially CP-divisibility.

Findings

Negativity of the Gram matrix indicates breakdown of CP-divisibility.

The framework applies to amplitude damping and dephasing models.

Positivity conditions correspond to information backflow phenomena.

Abstract

We introduce map-dependent quantum characteristic functions constructed from the normalized Choi operator of quantum dynamical maps. We prove a Bochner--Choi positivity theorem establishing that the positive-type condition of the associated Gram matrix is equivalent to complete positivity of the underlying quantum channel. Applying the construction to intermediate dynamical maps, we obtain a characterization of CP-divisibility in terms of positivity of two-time characteristic functions. Numerical examples for amplitude damping and pure dephasing models demonstrate that negativity of the Gram matrix coincides with the breakdown of CP-divisibility and the emergence of information backflow. The proposed framework provides a new bridge between characteristic-function methods in quantum statistics and structural properties of quantum dynamical maps.