Entanglement-assisted, noise-assisted, and monitoring-enhanced quantum bath tagging

We analyze the capability of discriminating the statistical nature of a thermal bath in the presence of three different types of side resources: prior entanglement between the probing system and an external (dynamically neutral) memory element, the interaction between the probe and an auxiliary bath, and the continuous monitoring of the system mediated by real-time measurements of the auxiliary bath. We discuss in detail how to obtain improved performances in the discrimination by considering different kinds of interactions, i.e., different jump operators, and different monitoring strategies corresponding to continuous homodyne detection and photodetection. We find that the presence of the auxiliary environment can be beneficial, allowing bath discrimination in regimes where in the standard scenario discrimination is not possible. We then show how additionally monitoring this environment, via either continuous homodyne detection or photodetection, is naturally advantageous for quantum bath tagging, in particular in the long-time limit where a large improvement in the discrimination performance is indeed observed. Our approach can in principle be implemented in a circuit QED setup and paves the way to further developments of quantum probing via continuous monitoring.