Measurement in quantum many-body systems: from measurement induced dynamics to quantum critical metrology

Over the past two decades, there has been a continuous and substantial increase in the scientific community's interest in quantum information. This trend is likely driven by exponential advancements in experimental platforms, which have facilitated the development of synthetic quantum matter systems, i.e., artificially engineered complex quantum systems. These advancements enable the exploration of tunable and controllable phases of matter, thereby expanding the scope of potential applications. As a result, research interest has sparked, ranging from practical applications of these new quantum technologies to fundamental aspects of quantum mechanics. In this context, the growing interest in the research topics addressed in this thesis becomes clear. For example, increasing attention has been directed toward quantum metrology, which fundamentally focuses on obtaining more accurate estimates of physical quantities by exploiting the quantum properties of the underlying system. This field has achieved remarkable results, one of the most notable being the use of quantum properties of light to enhance interferometry, enabling, for instance, the direct detection of gravitational waves. Furthermore, recent years have seen growing interest in multi-parameter quantum metrology. This area leverages the quantum properties of systems to develop metrological schemes capable of estimating several physical quantities in a single experiment, aiming to save experimental resources.On the other hand, phase transitions are a fascinating phenomenon that has captivated the physics community for centuries and continue to do so, particularly in quantum systems. It is unsurprising that critical systems have recently garnered significant attention, especially as tools for metrological estimation. When quantum systems are used as probes for estimating channel parameters, the non-classical resource most often employed to enhance estimations is entanglement. Critical systems are known to exhibit interesting entanglement properties, such as the divergent correlation length at criticality, which allows for volume-like entanglement scaling.Lastly, advancements in experimental platforms have sparked the interest of theoretical physicists, who have begun exploring the properties of monitored quantum many-body systems. In this scenario, we focus on the out-of-equilibrium properties of a many-body system whose unitary evolution is interrupted by a series of measurements. Over the last few years, growing interest in these systems has emerged, as they exhibit intriguing phases characterized by distinct entanglement properties separated by a genuine phase transition induced by measurement.The research presented in this thesis contributes to the previously described topics in several ways. This manuscript comprises two introductory chapters, one dedicated to critical systems, the other to quantum metrology, and four chapters containing original research findings.In the third chapter focusing on multi-parameter quantum metrology, we explore how criticality can help overcome the trade-offs associated with the simultaneous estimation of multiple parameters, which arise due to their incompatibility under the Heisenberg uncertainty principle. To address this, we exploit a scalar index called quantumness, which quantifies the distance between the attainable precision and the optimal precision. Our findings show that, in both integrable and non-integrable spin chain, criticality enables achieving the highest possible precision in a multi-parameter estimation scheme.In the fourth chapter, we start to analyze the properties of monitored quantum systems. First, we are interested in studying whether metrological tools are suitable for detecting and studying measurement-induced phase transitions. To this end, we choose to analyze a prototypical Ising chain in two typical settings. In the first setting, we study an interferometric approach that allows for the reconstruction of the phase diagram of the model by witnessing the entanglement properties of the system. In the second setting, we analyze how the critical properties of the measurement-induced phase transition affect the quantum Fisher information.In the fifth chapter, we investigate the effects of imperfect detection on measurement-induced phase transitions. To this end, we introduce a Lindblad framework that allows us to go from the post-selected no-click limit to the full Lindblad evolution. We analyze how the critical properties of a quasi-free fermionic quadratic model are suppressed as the information loss due, to imperfect measurements, is accounted for. Our findings reveal that the introduction of imperfect monitoring results in a finite correlation length, steering the system away from critical behavior.In the sixth chapter, we analyze the statistics of jumps in a system of free fermions where monitoring is limited to a single site. Remarkably, even this minimal setup generates non-trivial entanglement dynamics. Specifically, we demonstrate the emergence of volume-law entanglement, which is linked to a peculiar jump statistics that shifts from a standard Poissonian distribution to a super-Poissonian regime. This regime is characterized by bunching phenomena interspersed with long dark intervals. These intervals significantly enhance the system's entanglement growth, ultimately leading to volume-law scaling.These results, obtained in fields of growing interest, over the past few years, pave the way for potential new applications in the evolving landscape of quantum technologies.