Tsirelson S Problems And Entanglement Breaking Rank Download Ebook PDF Epub Online

Author : Jitendra Prakash
Publisher :
Release : 2018
Page : 77
Category : Group theory
ISBN 13 :
Description :


There are two main themes of this thesis. In the first part, we study the various sets of correlations arising in the study of non-local games and Tsirelson's problems. In the second part we introduce the notion of entanglement breaking rank and study its connection with the SIC POVM existence problem. Tsirelson's problems have been open for quite some time and continue to be an active area of research. These problems seek to identify relationships between different sets of quantum correlations arising from different quantum models and their relaxations. One of the goals of this thesis is to show that the set of correlations arising from the tensor product model is not topologically closed, and hence is distinct from the set of correlations arising from the commuting tensor model. We begin our investigation in Chapter 3 by working through an example of a non-local game with three inputs and binary outputs. While this toy example does not help in showing the distinction between the aforementioned sets of correlations, it still provides valuable insights and motivation for a more abstract approach which we take up in the next chapter and which turns out to be fruitful. In Chapter 4, we observe that given a graph we can associate functions (of a real variable) corresponding to different correlation sets. We study the properties of these functions. We show that for the complete graph on five vertices, the assumption that the correlation set obtained through tensor model is closed forces its corresponding function to be piecewise linear on an interval, while the function corresponding to the commuting model is not. This shows that the quantum correlation set is not closed hence solving one of Tsirelson's problem and also improves the results of Slofstra by reducing the number of experiments and outcomes to five and two, respectively. Finally, for the second part of the thesis, we introduce the notion of entanglement breaking rank of an entanglement breaking channel. While technically this notion is not new, this point of view helps us in showing that the entanglement breaking rank of a particular channel on $\mathbb{M}_d$ being $d^2$ is equivalent to the existence of a SIC POVM in dimension $d$. This opens up another approach to tackle this existence problem.


Author : Avi Wigderson
Publisher : Princeton University Press
Release : 2019-10-29
Page : 440
Category : Mathematics
ISBN 13 : 0691192545
Description :


An introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography


Author : Vern Paulsen
Publisher : Cambridge University Press
Release : 2002
Page : 300
Category : Mathematics
ISBN 13 : 9780521816694
Description :


Table of contents


Author : Mark Wilde
Publisher : Cambridge University Press
Release : 2013-04-18
Page : 655
Category : Computers
ISBN 13 : 1107034256
Description :


A self-contained, graduate-level textbook that develops from scratch classical results as well as advances of the past decade.


Author : Troy Lee
Adi Shraibman
Publisher : Now Publishers Inc
Release : 2009
Page : 152
Category : Computers
ISBN 13 : 1601982585
Description :


The communication complexity of a function f(x, y) measures the number of bits that two players, one who knows x and the other who knows y, must exchange to determine the value f(x, y). Communication complexity is a fundamental measure of complexity of functions. Lower bounds on this measure lead to lower bounds on many other measures of computational complexity. This monograph surveys lower bounds in the field of communication complexity. Our focus is on lower bounds that work by first representing the communication complexity measure in Euclidean space. That is to say, the first step in these lower bound techniques is to find a geometric complexity measure, such as rank or trace norm, that serves as a lower bound to the underlying communication complexity measure. Lower bounds on this geometric complexity measure are then found using algebraic and geometric tools.


Author : Pedro W. Lamberti
Gustavo M. Bosyk
Publisher : MDPI
Release : 2019-04-02
Page : 188
Category : Science
ISBN 13 : 3038977543
Description :


Since its conception 90 years ago, the quantum uncertainty principle introduced by Werner Heisenberg lies behind most important features of quantum physics, and its implications have an impact that goes far beyond the physics community. This book focuses on the quantum uncertainty principle, providing an up-to-date examination of recent developments of its applications in quantum information theory. The book brings together several renowned experts working in the foundations of quantum mechanics and quantum information theory. The authors provide different approaches to the study of uncertainty relations and other fundamental aspects of the quantum formalism. Topics addressed include entanglement and Bell inequalities, the application of entropic information measures to the study of uncertainty inequalities, the characterization of deep learning networks in the context of adiabatic quantum computation, and the study of general properties of the set of quantum states. The content of this book will surely benefit both experienced and new researchers specializing in quantum information theory and the foundations of quantum mechanics.


Author : Ingemar Bengtsson
Karol Życzkowski
Publisher : Cambridge University Press
Release : 2017-08-18
Page : 632
Category : Science
ISBN 13 : 1107026253
Description :


Quantum information theory is a branch of science at the frontier of physics, mathematics, and information science, and offers a variety of solutions that are impossible using classical theory. This book provides a detailed introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. The second edition contains new sections and entirely new chapters: the hot topic of multipartite entanglement; in-depth discussion of the discrete structures in finite dimensional Hilbert space, including unitary operator bases, mutually unbiased bases, symmetric informationally complete generalized measurements, discrete Wigner function, and unitary designs; the Gleason and Kochen-Specker theorems; the proof of the Lieb conjecture; the measure concentration phenomenon; and the Hastings' non-additivity theorem. This richly-illustrated book will be useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.


Author : P. A. M. Dirac
Publisher : Lulu Press, Inc
Release : 2019-12-01
Page :
Category : Reference
ISBN 13 : 1794767118
Description :


"The standard work in the fundamental principles of quantum mechanics, indispensable both to the advanced student and to the mature research worker, who will always find it a fresh source of knowledge and stimulation." --Nature "This is the classic text on quantum mechanics. No graduate student of quantum theory should leave it unread"--W.C Schieve, University of Texas


Author : Jordi Tura i Brugués
Publisher : Springer
Release : 2016-12-01
Page : 237
Category : Science
ISBN 13 : 3319495712
Description :


This thesis focuses on the study and characterization of entanglement and nonlocal correlations constrained under symmetries. It includes original results as well as detailed methods and explanations for a number of different threads of research: positive partial transpose (PPT) entanglement in the symmetric states; a novel, experimentally friendly method to detect nonlocal correlations in many-body systems; the non-equivalence between entanglement and nonlocality; and elemental monogamies of correlations. Entanglement and nonlocal correlations constitute two fundamental resources for quantum information processing, as they allow novel tasks that are otherwise impossible in a classical scenario. However, their elusive characterization is still a central problem in quantum information theory. The main reason why such a fundamental issue remains a formidable challenge lies in the exponential growth in complexity of the Hilbert space as well as the space of multipartite correlations. Physical systems of interest, on the other hand, display symmetries that can be exploited to reduce this complexity, opening the possibility that some of these questions become tractable for such systems.


Author : Francois David
Publisher : Springer
Release : 2014-11-06
Page : 157
Category : Science
ISBN 13 : 3319105396
Description :


These lecture notes present a concise and introductory, yet as far as possible coherent, view of the main formalizations of quantum mechanics and of quantum field theories, their interrelations and their theoretical foundations. The “standard” formulation of quantum mechanics (involving the Hilbert space of pure states, self-adjoint operators as physical observables, and the probabilistic interpretation given by the Born rule) on one hand, and the path integral and functional integral representations of probabilities amplitudes on the other, are the standard tools used in most applications of quantum theory in physics and chemistry. Yet, other mathematical representations of quantum mechanics sometimes allow better comprehension and justification of quantum theory. This text focuses on two of such representations: the algebraic formulation of quantum mechanics and the “quantum logic” approach. Last but not least, some emphasis will also be put on understanding the relation between quantum physics and special relativity through their common roots - causality, locality and reversibility, as well as on the relation between quantum theory, information theory, correlations and measurements, and quantum gravity. Quantum mechanics is probably the most successful physical theory ever proposed and despite huge experimental and technical progresses in over almost a century, it has never been seriously challenged by experiments. In addition, quantum information science ha s become an important and very active field in recent decades, further enriching the many facets of quantum physics. Yet, there is a strong revival of the discussions about the principles of quantum mechanics and its seemingly paradoxical aspects: sometimes the theory is portrayed as the unchallenged and dominant paradigm of modern physical sciences and technologies while sometimes it is considered a still mysterious and poorly understood theory, waiting for a revolution. This volume, addressing graduate students and seasoned researchers alike, aims to contribute to the reconciliation of these two facets of quantum mechanics.


Author : Vern I. Paulsen
Mrinal Raghupathi
Publisher : Cambridge University Press
Release : 2016-04-11
Page : 182
Category : Mathematics
ISBN 13 : 1107104092
Description :


A unique introduction to reproducing kernel Hilbert spaces, covering the fundamental underlying theory as well as a range of applications.


Author : Ved Prakash Gupta
Prabha Mandayam
Publisher : Springer
Release : 2015-05-28
Page : 139
Category : Science
ISBN 13 : 3319167189
Description :


This book provides readers with a concise introduction to current studies on operator-algebras and their generalizations, operator spaces and operator systems, with a special focus on their application in quantum information science. This basic framework for the mathematical formulation of quantum information can be traced back to the mathematical work of John von Neumann, one of the pioneers of operator algebras, which forms the underpinning of most current mathematical treatments of the quantum theory, besides being one of the most dynamic areas of twentieth century functional analysis. Today, von Neumann’s foresight finds expression in the rapidly growing field of quantum information theory. These notes gather the content of lectures given by a very distinguished group of mathematicians and quantum information theorists, held at the IMSc in Chennai some years ago, and great care has been taken to present the material as a primer on the subject matter. Starting from the basic definitions of operator spaces and operator systems, this text proceeds to discuss several important theorems including Stinespring’s dilation theorem for completely positive maps and Kirchberg’s theorem on tensor products of C*-algebras. It also takes a closer look at the abstract characterization of operator systems and, motivated by the requirements of different tensor products in quantum information theory, the theory of tensor products in operator systems is discussed in detail. On the quantum information side, the book offers a rigorous treatment of quantifying entanglement in bipartite quantum systems, and moves on to review four different areas in which ideas from the theory of operator systems and operator algebras play a natural role: the issue of zero-error communication over quantum channels, the strong subadditivity property of quantum entropy, the different norms on quantum states and the corresponding induced norms on quantum channels, and, lastly, the applications of matrix-valued random variables in the quantum information setting.


Author : Benjamin Schumacher
Michael Westmoreland
Publisher : Cambridge University Press
Release : 2010-03-25
Page :
Category : Science
ISBN 13 : 1139487205
Description :


A new and exciting approach to the basics of quantum theory, this undergraduate textbook contains extensive discussions of conceptual puzzles and over 800 exercises and problems. Beginning with three elementary 'qubit' systems, the book develops the formalism of quantum theory, addresses questions of measurement and distinguishability, and explores the dynamics of quantum systems. In addition to the standard topics covered in other textbooks, it also covers communication and measurement, quantum entanglement, entropy and thermodynamics, and quantum information processing. This textbook gives a broad view of quantum theory by emphasizing dynamical evolution, and exploring conceptual and foundational issues. It focuses on contemporary topics, including measurement, time evolution, open systems, quantum entanglement, and the role of information.


Author : Remi Chauvin
Christine Lepetit
Publisher : Springer
Release : 2016-04-19
Page : 586
Category : Science
ISBN 13 : 3319290223
Description :


This is the first edited volume that features two important frameworks, Hückel and quantum chemical topological analyses. The contributors, which include an array of academics of international distinction, describe recent applications of such topological methods to various fields and topics that provide the reader with the current state-of-the-art and give a flavour of the wide range of their potentialities.


Author : Harald Atmanspacher
Ulrich Müller-Herold
Publisher : Springer
Release : 2016-09-20
Page : 154
Category : Science
ISBN 13 : 3319435736
Description :


This book reflects on the significant and highly original scientific contributions of Hans Primas. A professor of chemistry at ETH Zurich from 1962 to 1995, Primas continued his research activities until his death in 2014. Over these 50 years and more, he worked on the foundations of nuclear magnetic resonance spectroscopy, contributed to a number of significant issues in theoretical chemistry, helped to clarify central topics in quantum theory and the philosophy of physics, suggested innovative ways of addressing interlevel relations in the philosophy of science, and introduced cutting-edge approaches in the flourishing young field of scientific studies of consciousness. His work in these areas of research and its continuing impact is described by noted experts, colleagues, and collaborators of Primas. All authors contextualize their contributions to facilitate the mutual dialog between these fields.


Author : Phillip Kaye
Raymond Laflamme
Publisher : Oxford University Press
Release : 2007
Page : 274
Category : Computers
ISBN 13 : 0198570007
Description :


The authors provide an introduction to quantum computing. Aimed at advanced undergraduate and beginning graduate students in these disciplines, this text is illustrated with diagrams and exercises.


Author : Chérif F. Matta
Russell J. Boyd
Publisher : John Wiley & Sons
Release : 2007-06-27
Page : 567
Category : Science
ISBN 13 : 3527610693
Description :


This book distills the knowledge gained from research into atoms in molecules over the last 10 years into a unique, handy reference. Throughout, the authors address a wide audience, such that this volume may equally be used as a textbook without compromising its research-oriented character. Clearly structured, the text begins with advances in theory before moving on to theoretical studies of chemical bonding and reactivity. There follow separate sections on solid state and surfaces as well as experimental electron densities, before finishing with applications in biological sciences and drug-design. The result is a must-have for physicochemists, chemists, physicists, spectroscopists and materials scientists.


Author : Zhengcheng Zhang
Sheng Shui Zhang
Publisher : Springer
Release : 2015-06-24
Page : 712
Category : Technology & Engineering
ISBN 13 : 3319154583
Description :


This book updates the latest advancements in new chemistries, novel materials and system integration of rechargeable batteries, including lithium-ion batteries and batteries beyond lithium-ion and addresses where the research is advancing in the near future in a brief and concise manner. The book is intended for a wide range of readers from undergraduates, postgraduates to senior scientists and engineers. In order to update the latest status of rechargeable batteries and predict near research trend, we plan to invite the world leading researchers who are presently working in the field to write each chapter of the book. The book covers not only lithium-ion batteries but also other batteries beyond lithium-ion, such as lithium-air, lithium-sulfur, sodium-ion, sodium-sulfur, magnesium-ion and liquid flow batteries.


Author : Alexei Yu. Kitaev
Alexander Shen
Publisher : American Mathematical Soc.
Release : 2002
Page : 257
Category : Mathematics
ISBN 13 : 0821832298
Description :


This book presents a concise introduction to an emerging and increasingly important topic, the theory of quantum computing. The development of quantum computing exploded in 1994 with the discovery of its use in factoring large numbers--an extremely difficult and time-consuming problem when using a conventional computer. In less than 300 pages, the authors set forth a solid foundation to the theory, including results that have not appeared elsewhere and improvements on existing works. The book starts with the basics of classical theory of computation, including NP-complete problems and the idea of complexity of an algorithm. Then the authors introduce general principles of quantum computing and pass to the study of main quantum computation algorithms: Grover's algorithm, Shor's factoring algorithm, and the Abelian hidden subgroup problem. In concluding sections, several related topics are discussed (parallel quantum computation, a quantum analog of NP-completeness, and quantum error-correcting codes). This is a suitable textbook for a graduate course in quantum computing. Prerequisites are very modest and include linear algebra, elements of group theory and probability, and the notion of an algorithm (on a formal or an intuitive level). The book is complete with problems, solutions, and an appendix summarizing the necessary results from number theory.


Author : Maria Schuld
Francesco Petruccione
Publisher : Springer
Release : 2018-08-30
Page : 287
Category : Science
ISBN 13 : 3319964240
Description :


Quantum machine learning investigates how quantum computers can be used for data-driven prediction and decision making. The books summarises and conceptualises ideas of this relatively young discipline for an audience of computer scientists and physicists from a graduate level upwards. It aims at providing a starting point for those new to the field, showcasing a toy example of a quantum machine learning algorithm and providing a detailed introduction of the two parent disciplines. For more advanced readers, the book discusses topics such as data encoding into quantum states, quantum algorithms and routines for inference and optimisation, as well as the construction and analysis of genuine ``quantum learning models''. A special focus lies on supervised learning, and applications for near-term quantum devices.