Heisenberg Picture Approach to the Stability of Quantum Markov Systems

Abstract

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.

References in corpus (19)

• Quantum control theory and applications: A survey
• Real-time quantum feedback prepares and stabilizes photon number states
• Dissipative production of a maximally entangled steady state
• Feedback control of quantum state reduction
• Coherent quantum LQG control
• Modeling and Control of Quantum Systems: An Introduction
• Stabilizing feedback controls for quantum systems
• Squeezing Components in Linear Quantum Feedback Networks
• Stabilizing Open Quantum Systems by Markovian Reservoir Engineering
• Quantum Feedback Networks: Hamiltonian Formulation
• Analysis of Lyapunov Method for Control of Quantum States
• Analysis of Lyapunov Method for Control of Quantum Systems
• Advantages of Coherent Feedback for Cooling Quantum Oscillators
• Robust Stability of Uncertain Quantum Systems
• Quantum Coherent Nonlinear Feedbacks with Applications to Quantum Optics on Chip
• The structures of state space concerning Quantum Dynamical Semigroups
• Feedback control in quantum optics: an overview of experimental breakthroughs and areas of application
• The Generalized Lyapunov Theorem and its Application to Quantum Channels
• Stabilization of Stochastic Quantum Dynamics via Open and Closed Loop Control

Cited by in corpus (7)

• Rapid Lyapunov control of finite-dimensional quantum systems
• Robust Learning Control Design for Quantum Unitary Transformations
• Exact stabilization of entangled states in finite time by dissipative quantum circuits
• Coherently tracking the covariance matrix of an open quantum system
• Ground-state Stabilization of Open Quantum Systems by Dissipation
• Stabilizing Quantum States and Automatic Error Correction by Dissipation Control
• Model-Free Quantum Stabilization via Finite-Difference Lyapunov Control