We investigate the single qubit transformations under several typical coherence-free operations, such as, incoherent operation (IO), strictly incoherent operation (SIO), physically incoherent operation (PIO), and coherence-preserving operation (CPO). Quantitative connection has been built between IO and SIO in single qubit systems. Moreover, these coherence-free operations have a clear hierarchical relationship in single qubit systems: CPO ⊂ PIO ⊂ SIO=IO. A new and explicit proof for the necessary and sufficient condition of single qubit transformation via IO or SIO has been provided, which indicates that SIO with only two Kraus operators are enough to realize this transformation. The transformation regions of single qubits via CPO and PIO are also given. Our method provides a geometric illustration to analyze single qubit coherence transformations by introducing the Bloch sphere depiction of the transformation regions, and tells us how to construct the corresponding coherence-free operations.

We study the non-Markovian dynamics of an open quantum system with machine learning. The observable physical quantities and their evolutions are generated by using the neural network. After the pre-training is completed, we fix the weights in the subsequent processes thus do not need the further gradient feedback. We find that the dynamical properties of physical quantities obtained by the dynamical learning are better than those obtained by the learning of Hamiltonian and time evolution operator. The dynamical learning can be applied to other quantum many-body systems, non-equilibrium statistics and random processes.