Impact of control representations on efficiency of local nonholonomic motion planning

2011 ◽  
Vol 59 (2) ◽  
pp. 213-218 ◽  
Author(s):  
I. Duleba
Keyword(s):  
Motion Planning ◽  
State Space ◽  
Three Dimensional ◽  
Nonholonomic Systems ◽  
State Spaces ◽  
Controlled Systems ◽  
Higher Dimensional ◽  
Dimensional State Space ◽  
Optimal Motion Planning ◽  
Nonholonomic Motion Planning

Impact of control representations on efficiency of local nonholonomic motion planning In this paper various control representations selected from a family of harmonic controls were examined for the task of locally optimal motion planning of nonholonomic systems. To avoid dependence of results either on a particular system or a current point in a state space, considerations were carried out in a sub-space of a formal Lie algebra associated with a family of controlled systems. Analytical and simulation results are presented for two inputs and three dimensional state space and some hints for higher dimensional state spaces were given. Results of the paper are important for designers of motion planning algorithms not only to preserve controllability of the systems but also to optimize their motion.

Efficient photon sorter in a high-dimensional state space

2011 ◽  
Vol 11 (3&4) ◽  
pp. 313-325
Author(s):  
Warner A. Miller
Keyword(s):  
Quantum Information ◽  
State Space ◽  
Quantum Key Distribution ◽  
Quantum Information Processing ◽  
Three Dimensional ◽  
High Dimensional ◽  
Coupled Mode Theory ◽  
Coupled Mode ◽  
Dimensional State Space ◽  
Significant Obstacle

An increase in the dimension of state space for quantum key distribution (QKD) can decrease its fidelity requirements while also increasing its bandwidth. A significant obstacle for QKD with qu$d$its ($d\geq 3$) has been an efficient and practical quantum state sorter for photons whose complex fields are modulated in both amplitude and phase. We propose such a sorter based on a multiplexed thick hologram, constructed e.g. from photo-thermal refractive (PTR) glass. We validate this approach using coupled-mode theory with parameters consistent with PTR glass to simulate a holographic sorter. The model assumes a three-dimensional state space spanned by three tilted planewaves. The utility of such a sorter for broader quantum information processing applications can be substantial.

On the Nonlinear Observability of Polynomial Dynamical Systems

2021 ◽  
Vol 1 (1) ◽  
pp. 88-94
Author(s):  
Daniel Gerbet ◽  
Klaus Röbenack
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
State Space ◽  
Polynomial Dynamical Systems ◽  
Higher Dimensional ◽  
Dimensional State Space ◽  
System Properties ◽  
Controllability And Observability ◽  
Important System ◽  
Nonlinear Observability

Controllability and observability are important system properties in control theory. These properties cannot be easily checked for general nonlinear systems. This paper addresses the local and global observability as well as the decomposition with respect to observability of polynomial dynamical systems embedded in a higher-dimensional state-space. These criteria are applied on some example system.

scholarly journals The shape of higher-dimensional state space: Bloch-ball analog for a qutrit

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 485
Author(s):  
Christopher Eltschka ◽  
Marcus Huber ◽  
Simon Morelli ◽  
Jens Siewert
Keyword(s):  
State Space ◽  
Quantum System ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The State ◽  
Level System ◽  
Three Dimensional Model ◽  
System A ◽  
Dimensional State Space ◽  
Geometric Intuition

Geometric intuition is a crucial tool to obtain deeper insight into many concepts of physics. A paradigmatic example of its power is the Bloch ball, the geometrical representation for the state space of the simplest possible quantum system, a two-level system (or qubit). However, already for a three-level system (qutrit) the state space has eight dimensions, so that its complexity exceeds the grasp of our three-dimensional space of experience. This is unfortunate, given that the geometric object describing the state space of a qutrit has a much richer structure and is in many ways more representative for a general quantum system than a qubit. In this work we demonstrate that, based on the Bloch representation of quantum states, it is possible to construct a three dimensional model for the qutrit state space that captures most of the essential geometric features of the latter. Besides being of indisputable theoretical value, this opens the door to a new type of representation, thus extending our geometric intuition beyond the simplest quantum systems.

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