Comments on "Quantum Control by Decompositions of SU(2)"

The purpose of the paper is to point out some typos and to observe that the main result of V. Ramakrishna, K. Flores, H. Rabitz and R. J. Ober, Phys. Rev. A Volume 62, 054309, 2000 remains valid (and this validity can be verified in a constructive fashion) with only the requirement that the su(2) matrices A and B in the paper in the title be linearly independent. An interpretation of this constructive extension, in terms of Givens rotations of real Euclidean space, is given.Comment: 3 pages, points out typos and some improvements on the paper in the Titl

Note on the Controllability of Molecular Systems

The reachable set for a finite dimensional quantum system is shown to be the orbit of the group corresponding to the internal and control Hamiltonians, even if this group is not compact.Comment: 3 pages, contains a nominal improvement over an earlier result of the Author and collaborators, and also explains why the improvement is only nomina

Control of Switched Networks via Quantum Methods

We illustrate a technique for specifying piecewise constant controls for classes of switched electrical networks, typically used in converting power in a dc-dc converter. This procedure makes use of decompositions of SU(2) to obtain controls that are piecewise constant and can be constrained to be bang-bang with values 0 or 1. Complete results are presented for a third order network first. An example, which shows that the basic strategy is viable for fourth order circuits, is also given. The former evolves on SO(3), while the latter evolves on SO(4). Since the former group is intimately related to SU(2) while the latter is related to SU(2)xSU(2), the methodology of this paper uses factorizations of SU(2). The systems in this paper are single input systems with drift. In this paper, no approximations or other artifices are used to remove the drift. Instead, the drift is important in the determination of the controls. Periodicity arguments are rarely used.Comment: 15 pages, 5 figure

Dilation Theoretic Parametrizations of Positive Matrices with Applications to Quantum Information

This paper, dedicated to the memory of late Professor Tiberiu Constantinescu, discusses two parametrizations of positive matrices. The first, called the Schur-Constantinescu parametrization, is used to construct several examples of separable states (e.g., Hankel states). The second, called the Jacobi parametrization, is used to present an alternative to the Bloch sphere representation of qubits.Comment: Submitted to the Tiberiu Constantinescu Memorial Volum

Castelnuovo-Mumford regularity and Gorensteinness of fiber cone

In this article, we study the Castelnuovo-Mumford regularity and Gorenstein properties of the fiber cone. We obtain upper bounds for the Castelnuovo-Mumford regularity of the fiber cone and obtain sufficient conditions for the regularity of the fiber cone to be equal to that of the Rees algebra. We obtain a formula for the canonical module of the fiber cone and use it to study the Gorenstein property of the fiber cone.Comment: 16 pages; Article is to appear in Communications in Algebr

Fast and Accurate Proper Orthogonal Decomposition using Efficient Sampling and Iterative Techniques for Singular Value Decomposition

In this paper, we propose a computationally efficient iterative algorithm for proper orthogonal decomposition (POD) using random sampling based techniques. In this algorithm, additional rows and columns are sampled and a merging technique is used to update the dominant POD modes in each iteration. We derive bounds for the spectral norm of the error introduced by a series of merging operations. We use an existing theorem to get an approximate measure of the quality of subspaces obtained on convergence of the iteration. Results on various datasets indicate that the POD modes and/or the subspaces are approximated with excellent accuracy with a significant runtime improvement over computing the truncated SVD. We also propose a method to compute the POD modes of large matrices that do not fit in the RAM using this iterative sampling and merging algorithms

Parametrizations of Positive Matrices With Applications

This paper reviews some characterizations of positive matrices and discusses which lead to useful parametrizations. It is argued that one of them, which we dub the Schur-Constantinescu parametrization is particularly useful. Two new applications of it are given. One shows all block-Toeplitz states are PPT. The other application is to relaxation rates.Comment: Submitted for publication to the refereed book ``Mathematics of Quantum Computation and Technology", and is dedicated to the memory of Professor Tiberiu Constantinesc

Applicability of Crisp and Fuzzy Logic in Intelligent Response Generation

This paper discusses the merits and demerits of crisp logic and fuzzy logic with respect to their applicability in intelligent response generation by a human being and by a robot. Intelligent systems must have the capability of taking decisions that are wise and handle situations intelligently. A direct relationship exists between the level of perfection in handling a situation and the level of completeness of the available knowledge or information or data required to handle the situation. The paper concludes that the use of crisp logic with complete knowledge leads to perfection in handling situations whereas fuzzy logic can handle situations imperfectly only. However, in the light of availability of incomplete knowledge fuzzy theory is more effective but may be disadvantageous as compared to crisp logic.Comment: 4 pages, 1 tabl

The Future of Neural Networks

The paper describes some recent developments in neural networks and discusses the applicability of neural networks in the development of a machine that mimics the human brain. The paper mentions a new architecture, the pulsed neural network that is being considered as the next generation of neural networks. The paper also explores the use of memristors in the development of a brain-like computer called the MoNETA. A new model, multi/infinite dimensional neural networks, are a recent development in the area of advanced neural networks. The paper concludes that the need of neural networks in the development of human-like technology is essential and may be non-expendable for it.Comment: 6 pages, 2 figure

Parametrization of Quantum States and Channels

In this manuscript, a parametrization of positive matrices together with some of its many applications in quantum information theory is given.Comment: 19 page