  In this paper, we investigate the interference engineering of the open
quantum system, where the environment is made indefinite either through the use
of an interferometer or the introduction of auxiliary qubits. The environments
are modeled by fully connected qubit baths with exact analytical dynamics. As
the system passes through the interferometer or is controlled by auxiliary
qubits, it is propagated along different paths or their superpositions, leading
to distinct interactions with the environment in each path. This results in the
superposition of the environments, which can be detected through specific
measurements that retain certain coherent information about the paths. Our
results demonstrate that the indefiniteness of the environment can
significantly enhance the quantum correlations. However, only the statistical
mixture of the influences from the environments preserves provided that the
path coherence is destructed. We also examine the serviceability of the
indefiniteness as a resource for teleportation and quantum parameter
estimation. Additionally, we discuss how to quantify the indefiniteness and the
ways in which it affects the system's dynamics from the perspective of
wave-particle-entanglement-ignorance complementarity. Overall, our study
highlights the potential benefits of an indefinite environment in quantum
information processing and sheds light on the fundamental principles underlying
its effects.
