Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Approaches to open quantum systems : decoherence, localisation and all that

Ting Yu

Published Jan 1, 1998• 0 citations
OpenAlex

Abstract

This thesis is mainly concerned with issues in quantum open systems and the foundations of quantum theory.Chapter I introduces the aim, background and main results which take place in the following chapters.Chapters II and III are used to To the memory of my mother, Guofeng Han-Yu and to

Related papers

Introduction in Quantum Theory of Open Systems

Yu. L. Klimontovich

Abstract In this paper we try to define the author's position with respect to some fundamental problems of quantum theory.

Contributions to Plasma Physics•Jan 1, 1997•17 citations

Numerical Treatment of the Liouville-von Neumann Equation for Quantum Spin Dynamics

Giacomo Mazzi

This thesis is concerned with the design of numerical methods for quantum simulation and the development of improved models for quantum relaxation. Analysis is presented for the treatment of quantum systems using the density matrix formalism. This approach has been developed from the early days of quantum mechanics as a tool to describe from a statistical point of view a large number of identical quantum ensembles. Traditional methods are well established and reliable, but they perform poorly for practical simulation as the system size is scaled up. Ad hoc schemes for nuclear spin dynamics appearing in the literature can be shown to fail in certain situations. The challenge is therefore to identify efficient reduction methods for the quantum system which are also based on a rigorous foundation. The method presented in the thesis, for the time–independent Hamiltonian case, combines a quantum density matrix formalism with a procedure based on Chebyshev polynomials; application of the method to Nuclear Magnetic Resonance (NMR) spectroscopy is considered, and it is shown that the new technique outperforms existing alternatives in term of computational costs. The case of a time–dependent Hamiltonian in NMR simulation is studied as well and some splitting methods are presented. To the author’s knowledge this is the first time such methods have been applied within the NMR framework, and the numerical results show a better error–to–cost rate than traditional methods. In a separate strand of research, formulations for open quantum systems are studied and new dynamical systems approaches are considered for this problem. Motivations This thesis work is mainly focused on nuclear spin dynamics. Nuclear spin dynamics constitutes the basis for NMR, which is a very powerful spectroscopy technique that exploits the interaction between nuclear spins and magnetic fields. The same technique is used to reveal the presence of hydrogen atoms in the blood for Magnetic Resonance Imaging (MRI). Within this framework the role of simulations is extremely important, as it provides a benchmark for studies of new materials, and the development of new magnetic fields. The main computational issue is that with current software for NMR simulation it is extremely expensive to deal with systems made of more than few (7–10) spins. There is therefore a strong need to develop new algorithms capable of simulating larger systems. In recent years NMR simulations have been found to be one of the most favorable candidates for quantum computing. There are two reasons for this: nuclear quantum states maintain extremely long coherences, and it is possible to attain a very strong control on the quantum state via the application of sequences of pulses. In order to develop a proper quantum computer it is fundamental to understand how the entangled states lose coherence and relax back to equilibrium by means of external interactions. This process is described as relaxation in an open quantum system. The theory for such systems has been available for 50 years but there are still substantial limitations in the two main approaches. There are also relatively few numerical approaches for the simulation of such systems, for this reason it is important to develop numerical alternatives for the description of open quantum systems. Thesis Outline The thesis is organized as follow: the first two chapters provide background material to familiarize the reader with fundamental concepts of both quantum mechanics and nuclear spin dynamics; in this part of the thesis no new results are presented. The first chapter introduces the concept of quantum systems and the mathematical environment with which we describe those systems. We also present the main equations we need to solve to determine the dynamics of a quantum system in a statistical framework. In the second chapter we introduce the nuclear spin system, that is the physical system that has been the main reference frame in this work, for both tests and practical applications of the new algorithms. We describe how nuclear spin systems are at the basis of very important applications like NMR spectroscopy and MRI. We present in some detail the physical features of the NMR technique and the equations we need to solve to describe the dynamics of a spin system; we also focus on the relevance of numerical simulations for these systems, and consequently which must be the interest in developing new algorithms, and the major obstacles which must be overcome. In the third chapter we investigate the numerical challenges that arise in simulation of quantum systems, we describe some of the methods that have been developed in the literature, focusing on the performances and the computational costs of them, setting the new developments of this thesis in the proper research frame. We discuss one of the major issues: the evaluation of the matrix exponential. We also present the analysis we have done of a recent method called Zero Track Elimination (ZTE) that has been developed specifically for NMR simulations. This analysis shows the limitations of this method but also gives a mathematical explanation of why–and in which cases–it works. In the fourth chapter we present the main result of the thesis, the development of a new method that directly evaluates the expectation values for a quantum simulation via a different application of the well known Chebyshev expansion. We have proved that this new method can provide an excellent boost in terms of performance, with computational costs that can be reduced by a factor ten in common cases. (The results of this chapter and the new method have been presented in international conferences and recently they have been submitted for publication). We also present some attempts we have made in the application of splitting methods for the evolution of the system in a time dependent environment. To our knowledge this is the first time splitting methods have been used for NMR simulations. The results of this approach are as follows: for a particular splitting technique combined with a Lanczos iteration method it is possible to speed up the calculation by a third if compared with a Lanczos type method whilst keeping the error below a critical threshold. This last approach is still a work in progress especially in terms of developing clever ways to split the Hamiltonian. The last chapter of this thesis deals with simulation of quantum systems interacting with an external environment. After presenting the main theoretical approaches for the description of such systems we then survey several the techniques that are currently used for the numerical implementation of such theories. As a work in progress we present a considerably different new approach we have been developing aiming to overcome some of the issues that arise when treating this kind of system within usual frameworks. This is somewhat speculative work that gives rise to some new directions in the development of a numerical description for open quantum systems. We also present some numerical results. (The main core of this chapter has been presented in international conferences).

•Jan 1, 2010•1 citation

Decoherence, Einselection, and Its Consequences

Wojciech Hubert Zurek

Chapter 5 explores the consequences of decoherence. We live in a Universe that is fundamentally quantum. Yet, our everyday world appears to be resolutely classical. The aim of Chapter 5 is to discuss how preferred classical states, and, more generally, classical physics, arise, as an excellent approximation, on a macroscopic level of a quantum Universe. We show why quantum theory results in the familiar “classical reality” in open quantum systems, that is, systems interacting with their environments. We shall see how and why, and to what extent, quantum theory accounts for our classical perceptions. We shall not complete this task here—a more detailed analysis of how the information is acquired by observers is needed for that, and this task will be taken up in Part III of the book. Moreover, Chapter 5 shows that not just Newtonian physics but also equilibrium thermodynamics follows from the same symmetries of entanglement that led to Born’s rule (in Chapter 3).

Cambridge University Press eBooks•Mar 6, 2025•0 citations

PART I GENERAL THEORY OF OPEN QUANTUM SYSTEMS

WORLD SCIENTIFIC eBooks•Mar 1, 2008•0 citations

Quantum Information Theory

Kazuyuki Fujii

What are the information processing capabilities of physical systems?As recently as the first half of the 20 th century this question did not even have a definite meaning.What is information, and how would one process it?It took the development of theories of computing (in the 1930s) and information (late in the 1940s) for us to formulate mathematically what it means to compute or communicate.Yet these theories were abstract, based on axiomatic mathematics: what did physical systems have to do with these axioms?Rolf Landauer had the essential insight -"Information is physical" -that information is always encoded in the state of a physical system, whose dynamics on a microscopic level are well-described by quantum physics.This means that we cannot discuss information without discussing how it is represented, and how nature dictates it should behave.Wigner considered the situation from another perspective when he wrote about "the unreasonable effectiveness of mathematics in the natural sciences".Why are the computational techniques of mathematics so astonishingly useful in describing the physical world [1]?One might begin to suspect foul play in the universe's operating principles.Interesting insights into the physics of information accumulated through the 1970s and 1980s -most sensationally in the proposal for a "quantum computer".If we were to mark a particular year in which an explosion of interest took place in information physics, that year would have to be 1994, when Shor showed that a problem of practical interest (factorisation of integers) could be solved easily on a quantum computer.But the applications of information in physics -and vice versa -have been far more widespread than this popular discovery.These applications range from improved experimental technology, more sophisticated measurement techniques, methods for characterising the quantum/classical boundary, tools for quantum chaos, and deeper insight into quantum theory and nature.In this thesis I present a short review of ideas in quantum information theory.The first chapter contains introductory material, sketching the central ideas of probability and information theory.Quantum mechanics is presented at the level of advanced undergraduate knowledge, together with some useful tools for quantum mechanics of open systems.In the second chapter I outline how classical information is represented in quantum systems and what this means for agents trying to extract information from these systems.The final chapter presents a new resource: quantum information.This resource has some bewildering applications which have been discovered in the last ten years, and continually presents us with unexpected insights into quantum theory and the universe.ii This thesis is dedicated to my mother.iii This thesis was begun as an intrepid adventure.It caught

Institute of Electrical and Electronics Engineers eBooks•Jan 1, 2009•533 citations

Concepts and Methods in the Theory of Open Quantum Systems

Heinz‐Peter Breuer, Francesco Petruccione

arXiv:quant-ph/0302047•Lecture notes in physics•Jan 1, 2003•745 citations
This thesis is concerned with the design of numerical methods for quantum simulation and the development of improved models for quantum relaxation. Analysis is presented for the treatment of quantum systems using the density matrix formalism. This approach has been developed from the early days of quantum mechanics as a tool to describe from a statistical point of view a large number of identical quantum ensembles. Traditional methods are well established and reliable, but they perform poorly for practical simulation as the system size is scaled up. Ad hoc schemes for nuclear spin dynamics appearing in the literature can be shown to fail in certain situations. The challenge is therefore to identify efficient reduction methods for the quantum system which are also based on a rigorous foundation. The method presented in the thesis, for the time–independent Hamiltonian case, combines a quantum density matrix formalism with a procedure based on Chebyshev polynomials; application of the method to Nuclear Magnetic Resonance (NMR) spectroscopy is considered, and it is shown that the new technique outperforms existing alternatives in term of computational costs. The case of a time–dependent Hamiltonian in NMR simulation is studied as well and some splitting methods are presented. To the author’s knowledge this is the first time such methods have been applied within the NMR framework, and the numerical results show a better error–to–cost rate than traditional methods. In a separate strand of research, formulations for open quantum systems are studied and new dynamical systems approaches are considered for this problem. Motivations This thesis work is mainly focused on nuclear spin dynamics. Nuclear spin dynamics constitutes the basis for NMR, which is a very powerful spectroscopy technique that exploits the interaction between nuclear spins and magnetic fields. The same technique is used to reveal the presence of hydrogen atoms in the blood for Magnetic Resonance Imaging (MRI). Within this framework the role of simulations is extremely important, as it provides a benchmark for studies of new materials, and the development of new magnetic fields. The main computational issue is that with current software for NMR simulation it is extremely expensive to deal with systems made of more than few (7–10) spins. There is therefore a strong need to develop new algorithms capable of simulating larger systems. In recent years NMR simulations have been found to be one of the most favorable candidates for quantum computing. There are two reasons for this: nuclear quantum states maintain extremely long coherences, and it is possible to attain a very strong control on the quantum state via the application of sequences of pulses. In order to develop a proper quantum computer it is fundamental to understand how the entangled states lose coherence and relax back to equilibrium by means of external interactions. This process is described as relaxation in an open quantum system. The theory for such systems has been available for 50 years but there are still substantial limitations in the two main approaches. There are also relatively few numerical approaches for the simulation of such systems, for this reason it is important to develop numerical alternatives for the description of open quantum systems. Thesis Outline The thesis is organized as follow: the first two chapters provide background material to familiarize the reader with fundamental concepts of both quantum mechanics and nuclear spin dynamics; in this part of the thesis no new results are presented. The first chapter introduces the concept of quantum systems and the mathematical environment with which we describe those systems. We also present the main equations we need to solve to determine the dynamics of a quantum system in a statistical framework. In the second chapter we introduce the nuclear spin system, that is the physical system that has been the main reference frame in this work, for both tests and practical applications of the new algorithms. We describe how nuclear spin systems are at the basis of very important applications like NMR spectroscopy and MRI. We present in some detail the physical features of the NMR technique and the equations we need to solve to describe the dynamics of a spin system; we also focus on the relevance of numerical simulations for these systems, and consequently which must be the interest in developing new algorithms, and the major obstacles which must be overcome. In the third chapter we investigate the numerical challenges that arise in simulation of quantum systems, we describe some of the methods that have been developed in the literature, focusing on the performances and the computational costs of them, setting the new developments of this thesis in the proper research frame. We discuss one of the major issues: the evaluation of the matrix exponential. We also present the analysis we have done of a recent method called Zero Track Elimination (ZTE) that has been developed specifically for NMR simulations. This analysis shows the limitations of this method but also gives a mathematical explanation of why–and in which cases–it works. In the fourth chapter we present the main result of the thesis, the development of a new method that directly evaluates the expectation values for a quantum simulation via a different application of the well known Chebyshev expansion. We have proved that this new method can provide an excellent boost in terms of performance, with computational costs that can be reduced by a factor ten in common cases. (The results of this chapter and the new method have been presented in international conferences and recently they have been submitted for publication). We also present some attempts we have made in the application of splitting methods for the evolution of the system in a time dependent environment. To our knowledge this is the first time splitting methods have been used for NMR simulations. The results of this approach are as follows: for a particular splitting technique combined with a Lanczos iteration method it is possible to speed up the calculation by a third if compared with a Lanczos type method whilst keeping the error below a critical threshold. This last approach is still a work in progress especially in terms of developing clever ways to split the Hamiltonian. The last chapter of this thesis deals with simulation of quantum systems interacting with an external environment. After presenting the main theoretical approaches for the description of such systems we then survey several the techniques that are currently used for the numerical implementation of such theories. As a work in progress we present a considerably different new approach we have been developing aiming to overcome some of the issues that arise when treating this kind of system within usual frameworks. This is somewhat speculative work that gives rise to some new directions in the development of a numerical description for open quantum systems. We also present some numerical results. (The main core of this chapter has been presented in international conferences).
Chapter 5 explores the consequences of decoherence. We live in a Universe that is fundamentally quantum. Yet, our everyday world appears to be resolutely classical. The aim of Chapter 5 is to discuss how preferred classical states, and, more generally, classical physics, arise, as an excellent approximation, on a macroscopic level of a quantum Universe. We show why quantum theory results in the familiar “classical reality” in open quantum systems, that is, systems interacting with their environments. We shall see how and why, and to what extent, quantum theory accounts for our classical perceptions. We shall not complete this task here—a more detailed analysis of how the information is acquired by observers is needed for that, and this task will be taken up in Part III of the book. Moreover, Chapter 5 shows that not just Newtonian physics but also equilibrium thermodynamics follows from the same symmetries of entanglement that led to Born’s rule (in Chapter 3).
What are the information processing capabilities of physical systems?As recently as the first half of the 20 th century this question did not even have a definite meaning.What is information, and how would one process it?It took the development of theories of computing (in the 1930s) and information (late in the 1940s) for us to formulate mathematically what it means to compute or communicate.Yet these theories were abstract, based on axiomatic mathematics: what did physical systems have to do with these axioms?Rolf Landauer had the essential insight -"Information is physical" -that information is always encoded in the state of a physical system, whose dynamics on a microscopic level are well-described by quantum physics.This means that we cannot discuss information without discussing how it is represented, and how nature dictates it should behave.Wigner considered the situation from another perspective when he wrote about "the unreasonable effectiveness of mathematics in the natural sciences".Why are the computational techniques of mathematics so astonishingly useful in describing the physical world [1]?One might begin to suspect foul play in the universe's operating principles.Interesting insights into the physics of information accumulated through the 1970s and 1980s -most sensationally in the proposal for a "quantum computer".If we were to mark a particular year in which an explosion of interest took place in information physics, that year would have to be 1994, when Shor showed that a problem of practical interest (factorisation of integers) could be solved easily on a quantum computer.But the applications of information in physics -and vice versa -have been far more widespread than this popular discovery.These applications range from improved experimental technology, more sophisticated measurement techniques, methods for characterising the quantum/classical boundary, tools for quantum chaos, and deeper insight into quantum theory and nature.In this thesis I present a short review of ideas in quantum information theory.The first chapter contains introductory material, sketching the central ideas of probability and information theory.Quantum mechanics is presented at the level of advanced undergraduate knowledge, together with some useful tools for quantum mechanics of open systems.In the second chapter I outline how classical information is represented in quantum systems and what this means for agents trying to extract information from these systems.The final chapter presents a new resource: quantum information.This resource has some bewildering applications which have been discovered in the last ten years, and continually presents us with unexpected insights into quantum theory and the universe.ii This thesis is dedicated to my mother.iii This thesis was begun as an intrepid adventure.It caught