In a measurement process, a quantum system is subject to an unavoidable back-action due to the quantum fluctuations of the detector. As a consequence, the system evolves along stochastic quantum trajectories. This features can be used as a tool to engineer and control quantum states, e.g. via feedback mechanisms, as well as to access new properties of open quantum systems. This is possible in actual experiments thanks to the degree of control of some quantum systems. In this talk, I present two application of quantum measurement. First, I will discuss the thermodynamics, i.e. the energy and information exchange, of a qubit coupled to a quantum detector . In particular, I will show how information gained by tracking single quantum trajectories of the qubit can be converted into work using quantum coherent feedback. I show that quantum backaction can lead to a loss of information in imperfect measurements in a superconducting circuit. As a second example, I will show that a time-dependent sequence of measurement can induce a geometric phase equivalent to the Berry phase of driven quantum systems . I will discuss the concomitant probability distribution of such geometric phases and show that, when tuning the measurement strength, the mapping between the measurement sequence and the geometric phase undergoes a topological transition. These findings have the potential to impact the study of measurement-induced state distillation and shed new light on the thermodynamics of open systems.
 Naghiloo et al., Phys. Rev. Lett. 121, 030604 (2018); Ibidem, arXiv:1703.05885  Gebhart et al., arXiv:1905.01147