# Lie Algebraic Analysis and Control of Quantum Dynamics

@article{DAlessandro2008LieAA, title={Lie Algebraic Analysis and Control of Quantum Dynamics}, author={Domenico D’Alessandro}, journal={arXiv: Quantum Physics}, year={2008} }

In this paper, we show how to use the analysis of the Lie algebra associated with a quantum mechanical system to study its dynamics and facilitate the design of controls. We give algorithms to decompose the dynamics and describe their application to the control of two coupled spin 1/2's.

## 2 Citations

### General methods to control right-invariant systems on compact Lie groups and multilevel quantum systems

- Mathematics
- 2009

For a right-invariant system on a compact Lie group G, I present two methods to design a control to drive the state from the identity to any element of the group. The first method, under appropriate…

### Corrections, Additions and Comments to the Book: ‘Introduction to Quantum Control and Dynamics’

- Physics
- 2008

This document contains several corrections, additions, improvements and comments to my book ‘Introduction to Quantum Control and Dynamics’. It will be periodically updated. Most of the corrections…

## References

SHOWING 1-10 OF 27 REFERENCES

### Time-optimal Control of Spin Systems

- Mathematics
- 2006

The paper discusses various aspects of time-optimal control of quantum spin systems, modelled as right-invariant systems on a compact Lie group G. The main results are the reduction of such a system…

### Notions of controllability for bilinear multilevel quantum systems

- MathematicsIEEE Trans. Autom. Control.
- 2003

In this note, we define four different notions of controllability of physical interest for multilevel quantum mechanical systems. These notions involve the possibility of driving the evolution…

### Introduction to Lie Algebras and Representation Theory

- Mathematics
- 1973

Preface.- Basic Concepts.- Semisimple Lie Algebras.- Root Systems.- Isomorphism and Conjugacy Theorems.- Existence Theorem.- Representation Theory.- Chevalley Algebras and Groups.- References.-…

### Control of the Evolution of Heisenberg Spin Systems

- MathematicsEur. J. Control
- 2004

An algorithm for the control of the unitary evolution operator for the system of two spin 1 2 ’s interacting through Heisenberg interaction is given and it is shown how the result allows to obtain information on the initial state through a series of evolutions and measurements.

### Quantum computation and quantum information

- PhysicsMathematical Structures in Computer Science
- 2007

This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal…

### Optimal control of two-level quantum systems

- MathematicsProceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334)
- 2000

A comprehensive theory of optimal control for two-level quantum systems is developed, in particular, a classification of normal and abnormal extremals and a proof of regularity of the optimal control functions.

### Degrees of controllability for quantum systems and application to atomic systems

- Mathematics
- 2001

Precise definitions for different degrees of controllability for quantum systems are given, and necessary and sufficient conditions for each type of controllability are discussed. The results are…

### Non-stationary quantum walks on the cycle

- Mathematics
- 2007

We consider quantum walks on the cycle in the non-stationary case where the ‘coin’ operation is allowed to change at each time step. We characterize, in algebraic terms, the set of possible state…

### Uncontrollable quantum systems: A classification scheme based on Lie subalgebras

- Mathematics
- 2009

It is well known that a finite level quantum system is controllable if and only if the Lie algebra of its generators has full rank. When the rank of the Lie algebra is not full, there is a rich…