Negative marginal densities in mixed quantum–classical Liouville dynamics

The mixed quantum-classical Liouville equation (QCLE) provides an approximate perturbative framework for describing the dynamics of systems with coupled quantum and classical degrees of freedom of disparate thermal wavelengths. The evolution governed by the Liouville operator preserves many properties of full quantum dynamics, including the conservation of total population, energy, and purity, and has shown quantitative agreement with exact quantum results for the expectation values of many observables where direct comparisons are feasible. However, since the QCLE density matrix operator is obtained from the partial Wigner transform of the full quantum density matrix, its matrix elements can have negative values, implying that the diagonal matrix elements behave as pseudo-densities rather than densities of classical phase space. Here, we compare phase-space distributions generated by exact quantum dynamics with those produced by QCLE evolution from pure quantum initial states. We show that resonance effects in the off-diagonal matrix elements differ qualitatively, particularly for low-energy states. Furthermore, numerical and analytical results for low-dimensional models reveal that the QCLE can violate the positivity of marginal phase-space densities, a property that should hold at all times for any physical system. A perturbative analysis of a model system confirms that such violations arise generically. We also show that the violations of positivity of the marginal densities vanish as the initial energy of the system increases relative to the energy gap between subsystem states. These findings suggest that a negativity index, quantifying deviations from positivity, may provide a useful metric for assessing the validity of mixed quantum-classical descriptions.