Heisenberg Picture Approach to the Stability of Quantum Markov Systems

TL;DR

This paper develops a stability theory for quantum Markov systems using the Heisenberg picture, providing conditions for invariant states and extending classical invariance principles to quantum systems, aiding quantum engineering.

Contribution

It introduces Lyapunov-type stability conditions and proves a quantum invariance principle in the Heisenberg picture, advancing quantum control theory.

Findings

Derived sufficient conditions for stability and existence of invariant quantum states.

Proved the quantum invariance principle extending LaSalle's principle.

Formulated algebraic constraints for quantum system engineering.

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.