Complete positivity of the general quantum master equations for arbitrary potentials

Establish that the quantum master equations defined by Eqs. (maindiffforces) and (maindiffforcesbis), derived via canonical quantization of the generalized Klein–Kramers operator with symmetric friction and noise, generate a completely positive, trace-preserving evolution (i.e., are of Lindblad type) for arbitrary potential energy functions, not only for the harmonic oscillator.

Background

The paper develops a classical-to-quantum framework starting from a generalized Langevin dynamics with friction and noise acting symmetrically on both Hamilton equations, yielding a generalized Klein–Kramers equation classically and, upon canonical quantization, two quantum master equations depending on whether friction operators are Hermitian or non-Hermitian.

For the harmonic oscillator, the authors analyze these master equations and, under a shared constraint on the friction coefficients, show that the dynamics is completely positive (Lindblad type). However, the extension of this complete positivity result to general potentials remains unresolved, motivating an explicit open problem to verify Lindblad-type structure and complete positivity for the general models beyond the harmonic oscillator.