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arXiv:quant-ph/0208078

Quantum information science as an approach to complex quantum systems

Michael A. Nielsen

Published Aug 13, 2002• 8 citations
arXivHTMLPDF

Abstract

What makes quantum information science a science? These notes explore the idea that quantum information science may offer a powerful approach to the study of complex quantum systems. We discuss how to quantify complexity in quantum systems, and argue that there are two qualitatively different types of complex quantum system. We also explore ways of understanding complex quantum dynamics by quantifying the strength of a quantum dynamical operation as a physical resource. This is the text for a talk at the ``Sixth International Conference on Quantum Communication, Measurement and Computing'', held at MIT, July 2002. Viewgraphs for the talk may be found at http://www.qinfo.org/talks/.

DOI
10.48550/arxiv.quant-ph/0208078

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What makes quantum information science a science? These notes explore the idea that quantum information science may offer a powerful approach to the study of complex quantum systems. We discuss how to quantify complexity in quantum systems, and argue that there are two qualitatively different types of complex quantum system. We also explore ways of understanding complex quantum dynamics by quantifying the strength of a quantum dynamical operation as a physical resource. This is the text for a talk at the ``Sixth International Conference on Quantum Communication, Measurement and Computing'', held at MIT, July 2002. Viewgraphs for the talk may be found at http://www.qinfo.org/talks/.
How useful is a quantum dynamical operation for quantum information processing? Motivated by this question, we investigate several strength measures quantifying the resources intrinsic to a quantum operation. We develop a general theory of such strength measures, based on axiomatic considerations independent of state-based resources. The power of this theory is demonstrated with applications to quantum communication complexity, quantum computational complexity, and entanglement generation by unitary operations.
This thesis is an attempt to apply the tools and techniques developed by quantum information science to gain Insight into strongly-interacting many-body quantum systems. The principle concept we make use of to do this is quantum entanglement. The thesis is divided into two parts. Part I is about quantum entanglement. The second part applies the techniques developed to understand quantum entanglement to complex quantum systems. In Part I we explore the rules governing the manipulation of quantum information. More specifically, we take as our premise the idea that quantum entanglement is the quantity responsible for the difference between classical and quantum information. We discuss the basic principles governing entanglement manipulation, focusing on the problem of quantifying quantum entanglement for quantum states. We prove that the probability that a certain class of quantum algorithms succeeds is an entanglement monotone for the initial state of the register used by these algorithms. We provide an easily-computable formula for a good mixed-state bipartite entanglement measure. We then use this formula to investigate a possible law of entanglement, namely, entanglement cannot be shared arbitrarily. In Part II we focus on using the concepts and tools of quantum information science to study complex quantum systems. The principal tool we use to do this is quantum entanglement. We begin with a definition of what we mean by a complex quantum system. This is followed with a study of the quantum entanglement present in a variety of complex quantum systems. We argue that quantum entanglement constitutes a signature of complex quantum behaviour. We conclude this thesis with a discussion of a well-known technique to understand complex quantum systems, namely, the density matrix renormalization group. We show that the success of this method is due to the way it treats the quantum entanglement present in a system.
We argue that complex systems science and the rules of quantum physics are intricately related. We discuss a range of quantum phenomena, such as cryptography, computation and quantum phases, and the rules responsible for their complexity. We identify correlations as a central concept connecting quantum information and complex systems science. We present two examples for the power of correlations: using quantum resources to simulate the correlations of a stochastic process and to implement a classically impossible computational task.
Nowadays quantum information science is in a state of theoretic completeness; much is invested to research quality physical solution for quantum computer and applications, so the field is led by physicists and information science and technology professionals. This field proved its potential and many (including us) believe it is in the right direction to go. In this paper we'll try to present and analyze the philosophy and nature of quantum information science and discuss hot topics around them. Although there is plenty of space for discussion around this field, we will concentrate on some important issues, such as quantum communication, algorithms and cryptography. Communication promises to be one of the directions where huge breakthroughs will be observed (entanglement as communication), algorithms have already shown exponential advantage (Shor's factoring) over classical counterparts and different paradigm of information processing. Cryptography seems to utilize general achievements in quantum information science and make encryption protocols very secure. Quantum computation can be understood as a generalization of classical computation: The possible states of a quantum computer are superpositions of bit strings, where the tensor factor is called a quantum bit or qubit. To evaluate the computational complexity of an algorithm, either classical or quantum, it is necessary to specify a set of elementary operations, the number of which used during the computation quantities the complexity. If we allow arbitrary permutations classically, or arbitrary unitary transformations quantum mechanically, any state can be reached from any other state in a single step. The general kind of quantum computer should be programmed to perform distinctively quantum operations such as generating two bits in a superposition state. An appropriate measurement on the final superposed state of the computer produces a probabilistic behavior of the output, and because of this quantum parallelization does not reduce the average time required for parallelizable computation completion. The development of quantum computing science and technology as well as quantum computers as thinking machines is very dependent on different achievements of modern research in fundamental and applied physics as well as novel mathematical methods and tools.
Quantum information science is an area of study based on the idea that information science depends on quantum effects in physics. It includes theoretical issues in computational models as well as more experimental topics in quantum physics including what can and cannot be done with quantum information. Subfields include: Quantum computing, Quantum error correction, Quantum complexity theory, Quantum cryptography and its generalization, quantum communication, Quantum communication complexity, Quantum entanglement, Quantum dense coding, Quantum teleportation. In the present book, fifteen typical literatures about Quantum Communication published on international authoritative journals were selected to introduce the worldwide newest progress, which contains reviews or original researches on Communication, mechanics, optics, ect. We hope this book can demonstrate advances in Quantum Communication as well as give references to the researchers, students and other related people.