Quantum Markovian systems, modeled as unitary dilations in the quantum stochastic calculus of Hudson and Parthasarathy, have become standard in current quantum technological applications. This paper investigates the stability theory of such systems. Lyapunov-type conditions in the Heisenberg picture are derived in order to stabilize the evolution of system operators as well as the underlying dynamics of the quantum states. In particular, using the quantum Markov semigroup associated with this quantum stochastic differential equation, we derive sufficient conditions for the existence and stability of a unique and faithful invariant quantum state. Furthermore, this paper proves the quantum invariance principle, which extends the LaSalle invariance principle to quantum systems in the Heisenberg picture. These results are formulated in terms of algebraic constraints suitable for engineering quantum systems that are used in coherent feedback networks.

The purpose of this paper is to investigate the extension of the Luenberger observer design approach to linear quantum stochastic systems to obtain coherent quantum observers. We show how a physically realizable quantum observer can be designed, consistent with the laws of quantum mechanics. The quantum observer has the property that the mean values of the observer variables asymptotically track the corresponding mean values of the plant. In addition, we discuss entanglement of the joint plant-observer state, as well as the implications of the no-cloning theorem. Several examples are considered.

The purpose of this paper is to address the problem of physical realizability for $n$-level quantum systems. We provide necessary and sufficient conditions for quantum stochastic differential equations that ensure the existence of physical parameters characterizing the unitary evolution required by the laws of quantum mechanics. Also, these conditions guarantee the preservation of the commutation and anticommutation relations of the underlying Lie algebra $su(n)$.

The primary challenge in advancing practical quantum technology is the presence of noise, which can lead to decoherence and undermine the advantages of quantum systems. However, it is worth noting that noise can also contain information that can be harnessed to improve performance in certain quantum information tasks. This has been explored for specific types of noise, but the full potential of informative noise within general dynamics remains largely unknown. Here we demonstrate that informative noise can be transformed into a valuable resource for enhancing precision limits in quantum metrology. We provide necessary and sufficient conditions under which the precision achieved in the presence of informative noise can attain Heisenberg scaling. Our research offers a fresh perspective on the role of noise in quantum metrology and opens up avenues for achieving higher precision beyond the previously anticipated ultimate limit.

In this note, the distributed leaderless consensus problem of multiple Euler-Lagrange systems under switching directed graphs using only position measurements is investigated. We adopt a model reference adaptive consensus strategy to assign each agent a reference to track, which transforms the consensus problem into a trajectory tracking problem. Due to the absence of velocity information, a velocity filter and a third-order linear reference model are designed for each agent. The proposed consensus algorithm only requires the position information and only the relative position information is interchanged among the neighboring agents. It is shown that the agents can achieve position consensus under a uniformly connected switching directed graph in the absence of global gains, in which sense the proposed algorithm is fully distributed. The numerical simulations are performed to show the effectiveness of the theoretical results.