Lie Theoretic Framework Controls -Qubit Open Quantum Systems Via Lindblad-Kossakowski Lie Algebra Parametrisation

Controlling the behaviour of quantum systems, which are inherently susceptible to environmental noise, presents a significant challenge in quantum technology. Corey O’Meara, alongside colleagues, addresses this problem by developing a powerful mathematical framework based on Lie theory to understand and manipulate open quantum systems. This research establishes a new way to describe how these systems evolve, utilising Lie semigroups and Lie wedges to characterise physically achievable dynamics. The team’s work provides a complete solution to a long-standing problem by identifying the precise conditions under which control strategies lead to predictable, stable behaviour, and importantly, reveals a method for designing systems that settle into specific, desired quantum states, paving the way for more robust and reliable quantum devices.

Dr. K. Hüper, Julius-Maximilians-Universität Würzburg. The dissertation was submitted on 26 June 2014 at the Technische Universität München and accepted by the Faculty of Chemistry on 17 July 2014. This thesis focuses on the Lie theoretical foundations of controlled open quantum systems, describing Markovian open quantum system evolutions by Lie semigroups, whose corresponding infinitesimal generators are part of a special type of convex cone, a Lie wedge. The Lie wedge associated to a given control system therefore consists of all generators of the quantum dynamical semigroup which are physically realisable.

Quantum Control and Open System Dynamics

This extensive bibliography compiles research papers and books concerning quantum control, open quantum systems, Lie theory, and related mathematical and physical concepts. The collection covers key themes including techniques for controlling quantum states and implementing quantum gates, understanding systems interacting with their environment, and employing Lie theory as a mathematical framework for designing control strategies. It also addresses the distinction between Markovian and non-Markovian dynamics in open quantum systems, and explores the use of dissipation as a resource for creating and stabilising quantum states.

Lie Semigroups Describe Dissipative Quantum Dynamics

This work establishes a powerful Lie-theoretic framework for understanding controlled open quantum systems, describing their evolution using Lie semigroups and Lie wedges, geometric objects representing physically realisable dynamics. Researchers successfully parameterised the Lindblad-Kossakowski (LK) Lie algebra for n-qubit systems using a Pauli basis, extending this to the LK-wedge, allowing interpretation of geometric time evolutions induced by Lindblad generators. This reveals a crucial separation between unital and non-unital dissipative dynamics. The team demonstrated that non-unital dynamics can be expressed as affine translation operations, effectively shifts within the system’s state space.

Exploiting this translational component, scientists devised a novel method for engineering unique fixed points using purely dissipative noise, leveraging symmetries to achieve either pure or mixed state fixed points. They successfully generated GHZ states, W states, stabilizer states, and Dicke states as unique solutions. Furthermore, the research reveals that, with full Hamiltonian control, any target state with non-degenerate eigenvalues can be reached through multiple pathways, opening avenues for future optimisation strategies. Detailed analysis of concrete examples, including one- and two-qubit systems, characterises their Lie wedges, providing a unique “fingerprint” for each open quantum system. The team showed that the Lie wedge contains more system-specific information than the system Lie algebra alone, and identified conditions under which these wedges simplify to solutions of time-independent Lindblad master equations, remaining closed under Baker-Campbell-Hausdorff multiplication. This delivers a comprehensive understanding of open quantum system dynamics and provides a powerful toolkit for manipulating and controlling these systems.

Lie Semigroups Define Markovian System Dynamics

This work advances understanding of open quantum systems through a Lie-theoretic approach, describing their evolution using Lie semigroups and Lie wedges. Researchers have established a parametrisation for the Lindblad-Kossakowski Lie algebra, enabling explicit construction of system Lie wedges and algebras. This parametrisation clarifies the conditions under which these wedges generate Markovian semigroups, demonstrating that such behaviour arises only when coherent controls do not influence both the system’s inherent drift and its incoherent dynamics. Furthermore, the research reveals an intuitive separation between unital and non-unital dissipative dynamics, with the latter described by affine translation operations.

Exploiting these translation operators, scientists have developed schemes for engineering fixed points, allowing for the creation of either pure or mixed quantum states as the system’s unique stable configuration. The authors acknowledge that determining the existence of invariant subspaces is crucial for achieving unique fixed points, and they highlight propositions that guarantee their non-existence. Future research directions include applying these findings to engineer specific quantum states useful in quantum information processing, as demonstrated through several illustrative examples. This work provides a powerful theoretical framework for controlling and manipulating open quantum systems, with potential applications in diverse areas of quantum technology.