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IEEE Conference on Decision and Control (CDC)

Quantum markovian master equation driven by coherent controls: a controllability analysis

Claudio Altafini

Published Jul 8, 2004• 2 citations
Publisher

Abstract

The behavior of a coherently driven finite dimensional open quantum system (in the Markovian approximation) is studied from a control theoretical perspective. The irreversible character of such a system is formalized by showing that it can be accessible but neither small-time controllable nor controllable in finite time. When the generators of quantum dynamical semigroups are unital, then the reachable sets admit easy characterizations as they monotonically grow in time.

DOI
10.1109/cdc.2003.1272597

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The behavior of a coherently driven finite dimensional open quantum system (in the Markovian approximation) is studied from a control theoretical perspective. The irreversible character of such a system is formalized by showing that it can be accessible but neither small-time controllable nor controllable in finite time. When the generators of quantum dynamical semigroups are unital, then the reachable sets admit easy characterizations as they monotonically grow in time.
Various notions from geometric control theory are used to characterize the behavior of the Markovian master equation for N-level quantum mechanical systems driven by unitary control and to describe the structure of the sets of reachable states. It is shown that the system can be accessible but neither small-time controllable nor controllable in finite time. In particular, if the generators of quantum dynamical semigroups are unital, then the reachable sets admit easy characterizations as they monotonically grow in time. The two level case is treated in detail.
We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway to faithfully represent quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model-parameter and initial-state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov's stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular, explicit results for the synthesis of stabilizing semigroups and noiseless subspaces in finite-dimensional Markovian systems are obtained.
We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway to faithfully represent quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model-parameter and initial-state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov's stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular, explicit results for the synthesis of stabilizing semigroups and noiseless subspaces in finite-dimensional Markovian systems are obtained.
We study reachable sets of coherently controllable open quantum systems with switchable coupling of two models: we show that finite-dimensional systems coupled to a temperature-zero bath allow for approximately generating every quantum state from every initial state. Also studying d-majorization yields an upper bound for non-zero temperature. Then we consider infinite-dimensional open systems following a unital evolution. For arbitrary drift, bounded control Hamiltonians, and noise generated by a compact normal operator, one can approximately reach any state majorized by the initial one.