Two-time correlations functions and quantumness of an open two-level system

Joint probability of obtaining outcomes in projective measurement performed sequentially on a two-level system at two different times is derived to investigate quantumness of the system, where the two-level system is influenced by a thermal reservoir with finite correlation time. Two-time correlation functions of the Pauli operators are also obtained. The joint probability and the two-time correlation functions are calculated by making use of the formulas developed recently by Ban et al. [Phys. Rev. A 97, 052101 (2018)] since the quantum regression theorem cannot be applied due to the finiteness of the reservoir correlation time. It is shown that the finiteness of the reservoir correlation time enhances the quantum witnesses of the two-level system. Furthermore the violation of the Leggett-Garg inequality is examined.