Consistency of Hybrid Quantum-Classical Ensembles

Michael J. W. Hall, Marcel Reginatto

José Luis Alonso, Carlos Bouthelier-Madre, Alberto Castro +2 more

In this work we generalize and combine Gibbs and von Neumann approaches to build, for the first time, a rigorous definition of entropy for hybrid quantum-classical systems. The resulting function coincides with the two cases above when the suitable limits are considered. Then, we apply the MaxEnt principle for this hybrid entropy function and obtain the natural candidate for the hybrid canonical ensemble (HCE). We prove that the suitable classical and quantum limits of the HCE coincide with the usual classical and quantum canonical ensembles since the whole scheme admits both limits, thus showing that the MaxEnt principle is applicable and consistent for hybrid systems.

Xuanmin Zhu, Shengshi Pang, Shengjun Wu +1 more

Nikola Burić, Istok P. Mendaš, Duska B Popovic +2 more

General statistical ensembles in the Hamiltonian formulation of hybrid quantum-classical systems are analyzed. It is argued that arbitrary probability densities on the hybrid phase space must be considered as the class of possible physically distinguishable statistical ensembles of hybrid systems. Nevertheless, statistical operators associated with the hybrid system and with the quantum subsystem can be consistently defined. Dynamical equations for the statistical operators representing the mixed states of the hybrid system and its quantum subsystem are derived and analyzed. In particular, these equations irreducibly depend on the total probability density on the hybrid phase space.

L. L. Salcedo

After formulating a no-go theorem for perfect quantum-classical hybrid systems, a consistency requirement based on standard statistical considerations is noted. It is shown that such requirement is not fulfilled by the mean-field approach or by the statistical ensemble in configuration space approach. Further unusual features of the latter scheme are pointed out.

L. E. Ballentine