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IEEE International Workshop on Workload Characterization (WWC)

Controllability of open quantum systems: The two level case

Claudio Altafini

Published Apr 23, 2004• 8 citations
Publisher

Abstract

We study the quantum Markovian master equation for a two level system in presence of coherent control. We show that its behavior is essentially determined by the complete positivity assumption of the quantum dynamical semigroup and that unitary control cannot modify its irreversibility character. Control theory provides several controllability-type condition's able to capture the peculiarity of such driven system.

DOI
10.1109/phycon.2003.1236992

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We study the quantum Markovian master equation for a two level system in presence of coherent control. We show that its behavior is essentially determined by the complete positivity assumption of the quantum dynamical semigroup and that unitary control cannot modify its irreversibility character. Control theory provides several controllability-type condition's able to capture the peculiarity of such driven system.
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We study the concepts of complete positivity and positivity in a two-level open quantum system whose dynamics are governed by a time-local quantum master equation. We establish necessary and sufficient conditions on the time-dependent relaxation rates to ensure complete positivity and positivity of the dynamical map. We discuss their relations with the non-Markovian behavior of the open system. We also analyze the long-time asymptotic behavior of the dynamics as a function of the rates. We show under which conditions on the rates the system tends to an equilibrium state. Different examples illustrate this general study.
Quantum control of an open system is demonstrated employing a thermodynamically consistent master equation. In this framework, the open system dynamics depend on the control protocol due to the dressing of the system by the drive. This interrelation serves as the key element for control. The influence of the external drive is incorporated within the dynamical equation, enabling an indirect control of the dissipation. The control paradigm is displayed by analyzing entropy changing state to state transformations, heating and cooling N-level systems. Following, we study the generation of quantum non-unitary maps via coherent control. These include both reset maps with complete memory loss and single-qubit unitary maps under dissipative conditions.
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.