Optimized Correlation Measures in Holography
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Introduction to Quantum Information Theory
Introduction to Quantum Information Theory Carlos Palazuelos Instituto de Ciencias Matemáticas (ICMAT) [email protected] Madrid, Spain March 2013 Contents Chapter 1. A comment on these notes 3 Chapter 2. Postulates of quantum mechanics 5 1. Postulate I and Postulate II 5 2. Postulate III 7 3. Postulate IV 9 Chapter 3. Some basic results 13 1. No-cloning theorem 13 2. Quantum teleportation 14 3. Superdense coding 15 Chapter 4. The density operators formalism 17 1. Postulates of quantum mechanics: The density operators formalism 17 2. Partial trace 19 3. The Schmidt decomposition and purifications 20 4. Definition of quantum channel and its classical capacity 21 Chapter 5. Quantum nonlocality 25 1. Bell’s result: Correlations in EPR 25 2. Tsirelson’s theorem and Grothendieck’s theorem 27 Chapter 6. Some notions about classical information theory 33 1. Shannon’s noiseless channel coding theorem 33 2. Conditional entropy and Fano’s inequality 35 3. Shannon’s noisy channel coding theorem: Random coding 38 Chapter 7. Quantum Shannon Theory 45 1. Von Neumann entropy 45 2. Schumacher’s compression theorem 48 3. Accessible Information and Holevo bound 55 4. Classical capacity of a quantum channel 58 5. A final comment about the regularization 61 Bibliography 63 1 CHAPTER 1 A comment on these notes These notes were elaborated during the first semester of 2013, while I was preparing a course on quantum information theory as a subject for the PhD programme: Investigación Matemática at Universidad Complutense de Madrid. The aim of this work is to present an accessible introduction to some topics in the field of quantum information theory for those people who do not have any background on the field. -
Limitations on Protecting Information Against Quantum Adversaries
Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School April 2020 Limitations on Protecting Information Against Quantum Adversaries Eneet Kaur Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Quantum Physics Commons Recommended Citation Kaur, Eneet, "Limitations on Protecting Information Against Quantum Adversaries" (2020). LSU Doctoral Dissertations. 5208. https://digitalcommons.lsu.edu/gradschool_dissertations/5208 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. LIMITATIONS ON PROTECTING INFORMATION AGAINST QUANTUM ADVERSARIES A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Physics and Astronomy by Eneet Kaur B.Sc., University of Delhi, 2012 M.Sc., Indian Institute of Technology, Roorkee, 2014 May 2020 Acknowledgments First and foremost, I would like to thank my advisor, Mark M. Wilde, for offering me the opportunity to work with him. My discussions with him have been instru- mental in shaping my approach towards research and in writing research papers. I am also thankful to Mark for providing me with the much-needed confidence boost, and for being an incredible mentor. I would also like to thank Jonathan P. Dowling for his encouragement, the fun pizza parties, and for all his stories. I would also like to thank Andreas Winter for hosting me in Barcelona and for his collaborations. -
Efficient and Exact Quantum Compression
Efficient and Exact Quantum Compression a; ;1 b;2 John H. Reif ∗ , Sukhendu Chakraborty aDepartment of Computer Science, Duke University, Durham, NC 27708-0129 bDepartment of Computer Science, Duke University, Durham, NC 27708-0129 Abstract We present a divide and conquer based algorithm for optimal quantum compression / de- compression, using O(n(log4 n) log log n) elementary quantum operations . Our result pro- vides the first quasi-linear time algorithm for asymptotically optimal (in size and fidelity) quantum compression and decompression. We also outline the quantum gate array model to bring about this compression in a quantum computer. Our method uses various classical algorithmic tools to significantly improve the bound from the previous best known bound of O(n3)(R. Cleve, D. P. DiVincenzo, Schumacher's quantum data compression as a quantum computation, Phys. Rev. A, 54, 1996, 2636-2650) for this operation. Key words: Quantum computing, quantum compression, algorithm 1 Introduction 1.1 Quantum Computation Quantum Computation (QC) is a computing model that applies quantum me- chanics to do computation. (Computations and methods not making use of quantum mechanics will be termed classical). A single molecule (or collection of .particles and/or atoms) may have n degrees of freedom known as qubits. Associated with each fixed setting X of the n qubits to boolean values is a basis state denoted a . Quantum mechanics allows for a linear superposition of these basis states to j i ∗ Corresponding author. Email addresses: [email protected] ( John H. Reif), [email protected] (Sukhendu Chakraborty). 1 The work is supported by NSF EMT Grants CCF-0523555 and CCF-0432038. -
Linear Growth of the Entanglement Entropy for Quadratic Hamiltonians and Arbitrary Initial States
Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states Giacomo De Palma∗1 and Lucas Hackl†2 1Scuola Normale Superiore, 56126 Pisa, Italy 2School of Mathematics and Statistics & School of Physics, The University of Melbourne, Parkville, VIC 3010, Australia July 26, 2021 Abstract We prove that the entanglement entropy of any pure initial state of a bipartite bosonic quantum system grows linearly in time with respect to the dynamics induced by any unstable quadratic Hamiltonian. The growth rate does not depend on the ini- tial state and is equal to the sum of certain Lyapunov exponents of the corresponding classical dynamics. This paper generalizes the findings of [Bianchi et al., JHEP 2018, 25 (2018)], which proves the same result in the special case of Gaussian initial states. Our proof is based on a recent generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum systems [De Palma et al., arXiv:2105.05627]. This technique allows us to extend our result to generic mixed initial states, with the squashed entanglement providing the right generalization of the entanglement en- tropy. We discuss several applications of our results to physical systems with (weakly) interacting Hamiltonians and periodically driven quantum systems, including certain quantum field theory models. 1 Introduction arXiv:2107.11064v1 [quant-ph] 23 Jul 2021 Entanglement is a cornerstone of quantum theory and its dynamics has been extensively studied in a wide range of different systems [1{5]. It also provides an important link be- tween classical chaos and quantum chaos in the context of Lyapunov instabilities [6]. -
Unitary Decomposition Implemented in the Openql Programming Language for Quantum Computation A.M
Unitary Decomposition Implemented in the OpenQL programming language for quantum computation A.M. Krol September 13th 36 Electrical Engineering, Mathematics and Computer Sciences LB 01.010 Snijderszaal Unitary Decomposition Implemented in the OpenQL programming language for quantum computation by A.M. Krol to obtain the degree of Master of Science at the Delft University of Technology, to be defended publicly on Friday September 13, 2019 at 15:00. Student number: 4292391 Project duration: November 20, 2018 – September 13, 2019 Thesis committee: Prof. dr. ir. K. Bertels, TU Delft, supervisor Dr. I. Ashraf, TU Delft Dr. M. Möller, TU Delft Dr. Z. Al-Ars TU Delft An electronic version of this thesis is available at http://repository.tudelft.nl/. Preface In this thesis, I explain my implementation of Unitary Decomposition in OpenQL. Unitary Decomposition is an algorithm for translating a unitary matrix into many small unitary matrices, which correspond to a circuit that can be executed on a quantum computer. It is implemented in the quantum programming framework of the QCA-group at TU Delft: OpenQL, a library for Python and C++. Unitary Decomposition is a necessary part in Quantum Associative Memory, an algorithm used in Quantum Genome Sequenc- ing. The implementation is faster than other known implementations, and generates 3 ∗ 2 ∗ (2 − 1) rotation gates for an n-qubit input gate. This is not the least-known nor the theoretical minimum amount, and there are some optimizations that can still be done to make it closer to these numbers. I would like to thank my supervisor, Prof. Koen Bertels, for the great supervision and insightful feedback. -
Quantum Information Chapter 10. Quantum Shannon Theory
Quantum Information Chapter 10. Quantum Shannon Theory John Preskill Institute for Quantum Information and Matter California Institute of Technology Updated June 2016 For further updates and additional chapters, see: http://www.theory.caltech.edu/people/preskill/ph219/ Please send corrections to [email protected] Contents 10 Quantum Shannon Theory 1 10.1 Shannon for Dummies 2 10.1.1 Shannon entropy and data compression 2 10.1.2 Joint typicality, conditional entropy, and mutual infor- mation 6 10.1.3 Distributed source coding 8 10.1.4 The noisy channel coding theorem 9 10.2 Von Neumann Entropy 16 10.2.1 Mathematical properties of H(ρ) 18 10.2.2 Mixing, measurement, and entropy 20 10.2.3 Strong subadditivity 21 10.2.4 Monotonicity of mutual information 23 10.2.5 Entropy and thermodynamics 24 10.2.6 Bekenstein’s entropy bound. 26 10.2.7 Entropic uncertainty relations 27 10.3 Quantum Source Coding 30 10.3.1 Quantum compression: an example 31 10.3.2 Schumacher compression in general 34 10.4 Entanglement Concentration and Dilution 38 10.5 Quantifying Mixed-State Entanglement 45 10.5.1 Asymptotic irreversibility under LOCC 45 10.5.2 Squashed entanglement 47 10.5.3 Entanglement monogamy 48 10.6 Accessible Information 50 10.6.1 How much can we learn from a measurement? 50 10.6.2 Holevo bound 51 10.6.3 Monotonicity of Holevo χ 53 10.6.4 Improved distinguishability through coding: an example 54 10.6.5 Classical capacity of a quantum channel 58 ii Contents iii 10.6.6 Entanglement-breaking channels 62 10.7 Quantum Channel Capacities and Decoupling -
Quantum Computing
John H. Reif, Quantum Computing. Published in book “Bio-inspired and Nanoscale Integrated Computing”, Chapter 3, pp. 67-110 (edited by Mary Mehrnoosh Eshaghian-Wilner), Publisher: Wiley, Hoboken, NJ, USA, (February 2009). Chapter 3: Quantum Computing John H. Reif Department of Computer Science Duke University ‡ 3.0 Summary Quantum Computation (QC) is a type of computation where unitary and measurement operations are executed on linear superpositions of basis states. This paper provides a brief introduction to QC. We begin with a discussion of basic models for QC such as quantum TMs, quantum gates and circuits and related complexity results. We then discuss a number of topics in quan- tum information theory, including bounds for quantum communication and I/O complexity, methods for quantum data compression. and quantum er- ‡Surface address: Department of Computer Science, Duke University, Durham, NC 27708-0129. E-mail: [email protected]. This work has been supported by grants from NSF CCF-0432038 and CCF-0523555. A postscript preprint of this Chapter is available online at URL: http://www.cs.duke.edu/∼reif/paper/qsurvey/qsurvey.chapter.pdf. ror correction (that is, techniques for decreasing decoherence errors in QC), Furthermore, we enumerate a number of methodologies and technologies for doing QC. Finally, we discuss resource bounds for QC including bonds for processing time, energy and volume, particularly emphasizing challenges in determining volume bounds for observation apperatus. 3.1 Introduction 3.1.1 Reversible Computations Reversible Computations are computations where each state transformation is a reversible function, so that any computation can be reversed without loss of information. -
Supercomputer Simulations of Transmon Quantum Computers Quantum Simulations of Transmon Supercomputer
IAS Series IAS Supercomputer simulations of transmon quantum computers quantum simulations of transmon Supercomputer 45 Supercomputer simulations of transmon quantum computers Dennis Willsch IAS Series IAS Series Band / Volume 45 Band / Volume 45 ISBN 978-3-95806-505-5 ISBN 978-3-95806-505-5 Dennis Willsch Schriften des Forschungszentrums Jülich IAS Series Band / Volume 45 Forschungszentrum Jülich GmbH Institute for Advanced Simulation (IAS) Jülich Supercomputing Centre (JSC) Supercomputer simulations of transmon quantum computers Dennis Willsch Schriften des Forschungszentrums Jülich IAS Series Band / Volume 45 ISSN 1868-8489 ISBN 978-3-95806-505-5 Bibliografsche Information der Deutschen Nationalbibliothek. Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografe; detaillierte Bibliografsche Daten sind im Internet über http://dnb.d-nb.de abrufbar. Herausgeber Forschungszentrum Jülich GmbH und Vertrieb: Zentralbibliothek, Verlag 52425 Jülich Tel.: +49 2461 61-5368 Fax: +49 2461 61-6103 [email protected] www.fz-juelich.de/zb Umschlaggestaltung: Grafsche Medien, Forschungszentrum Jülich GmbH Titelbild: Quantum Flagship/H.Ritsch Druck: Grafsche Medien, Forschungszentrum Jülich GmbH Copyright: Forschungszentrum Jülich 2020 Schriften des Forschungszentrums Jülich IAS Series, Band / Volume 45 D 82 (Diss. RWTH Aachen University, 2020) ISSN 1868-8489 ISBN 978-3-95806-505-5 Vollständig frei verfügbar über das Publikationsportal des Forschungszentrums Jülich (JuSER) unter www.fz-juelich.de/zb/openaccess. This is an Open Access publication distributed under the terms of the Creative Commons Attribution License 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We develop a simulator for quantum computers composed of superconducting transmon qubits. -
Quantum Information Processing with Superconducting Circuits: a Review
Quantum Information Processing with Superconducting Circuits: a Review G. Wendin Department of Microtechnology and Nanoscience - MC2, Chalmers University of Technology, SE-41296 Gothenburg, Sweden Abstract. During the last ten years, superconducting circuits have passed from being interesting physical devices to becoming contenders for near-future useful and scalable quantum information processing (QIP). Advanced quantum simulation experiments have been shown with up to nine qubits, while a demonstration of Quantum Supremacy with fifty qubits is anticipated in just a few years. Quantum Supremacy means that the quantum system can no longer be simulated by the most powerful classical supercomputers. Integrated classical-quantum computing systems are already emerging that can be used for software development and experimentation, even via web interfaces. Therefore, the time is ripe for describing some of the recent development of super- conducting devices, systems and applications. As such, the discussion of superconduct- ing qubits and circuits is limited to devices that are proven useful for current or near future applications. Consequently, the centre of interest is the practical applications of QIP, such as computation and simulation in Physics and Chemistry. Keywords: superconducting circuits, microwave resonators, Josephson junctions, qubits, quantum computing, simulation, quantum control, quantum error correction, superposition, entanglement arXiv:1610.02208v2 [quant-ph] 8 Oct 2017 Contents 1 Introduction 6 2 Easy and hard problems 8 2.1 Computational complexity . .9 2.2 Hard problems . .9 2.3 Quantum speedup . 10 2.4 Quantum Supremacy . 11 3 Superconducting circuits and systems 12 3.1 The DiVincenzo criteria (DV1-DV7) . 12 3.2 Josephson quantum circuits . 12 3.3 Qubits (DV1) . -
The Squashed Entanglement of a Quantum Channel
The squashed entanglement of a quantum channel Masahiro Takeoka∗y Saikat Guhay Mark M. Wildez January 22, 2014 Abstract This paper defines the squashed entanglement of a quantum channel as the maximum squashed entanglement that can be registered by a sender and receiver at the input and output of a quantum channel, respectively. A new subadditivity inequality for the original squashed entanglement measure of Christandl and Winter leads to the conclusion that the squashed en- tanglement of a quantum channel is an additive function of a tensor product of any two quantum channels. More importantly, this new subadditivity inequality, along with prior results of Chri- standl, Winter, et al., establishes the squashed entanglement of a quantum channel as an upper bound on the quantum communication capacity of any channel assisted by unlimited forward and backward classical communication. A similar proof establishes this quantity as an upper bound on the private capacity of a quantum channel assisted by unlimited forward and backward public classical communication. This latter result is relevant as a limitation on rates achievable in quantum key distribution. As an important application, we determine that these capacities can never exceed log((1 + η)=(1 − η)) for a pure-loss bosonic channel for which a fraction η of the input photons make it to the output on average. The best known lower bound on these capacities is equal to log(1=(1 − η)). Thus, in the high-loss regime for which η 1, this new upper bound demonstrates that the protocols corresponding to the above lower bound are nearly optimal. -
Chapter 5 Quantum Information Theory
Chapter 5 Quantum Information Theory Quantum information theory is a rich subject that could easily have occupied us all term. But because we are short of time (I’m anxious to move on to quantum computation), I won’t be able to cover this subject in as much depth as I would have liked. We will settle for a brisk introduction to some of the main ideas and results. The lectures will perhaps be sketchier than in the first term, with more hand waving and more details to be filled in through homework exercises. Perhaps this chapter should have been called “quantum information theory for the impatient.” Quantum information theory deals with four main topics: (1) Transmission of classical information over quantum channels (which we will discuss). (2) The tradeoff between acquisition of information about a quantum state and disturbance of the state (briefly discussed in Chapter 4 in connec- tion with quantum cryptography, but given short shrift here). (3) Quantifying quantum entanglement (which we will touch on briefly). (4) Transmission of quantum information over quantum channels. (We will discuss the case of a noiseless channel, but we will postpone discus- sion of the noisy channel until later, when we come to quantum error- correcting codes.) These topics are united by a common recurring theme: the interpretation and applications of the Von Neumann entropy. 1 2 CHAPTER 5. QUANTUM INFORMATION THEORY 5.1 Shannon for Dummies Before we can understand Von Neumann entropy and its relevance to quan- tum information, we must discuss Shannon entropy and its relevance to clas- sical information. -
Quantum Information Theory
Quantum Information Theory Jun-Ting Hsieh Bingbin Liu Stanford University Stanford University [email protected] [email protected] 1 Introduction In our current digital world, everything, from computing devices to communication channels, is represented as 0s and 1s. The “information” that these bits can present has been well studied in the field of classical information theory. For example, we know the capacity and limits of data compression and the rate of reliable communication over noisy channels. Recently, perhaps due to the excitement of quantum computation, these ideas in information theory have been extended to the quantum world, leading to the field of quantum information theory. In this paper, we would like to introduce the quantum counterparts of some of the key ideas we studied in the course, including the quantum entropy measure, quantum source encoding, and touch briefly on quantum channels. 1.1 How is quantum different than classical? Compared to the classical world, quantum mechanics has several “weird” properties that make things much more convoluted. Thus, we first introduce some major differences between quantum and classical information, which are the reasons that quantum information is so interesting yet harder to analyze. Refer to Section 2 for a more detailed introduction of quantum mechanics. Qubit. The quantum version of the classical bit is the qubit. A classical bit is either 0 or 1, while a qubit can be superposition of 0 and 1. This means that it is both 0 and 1, and if we perform a measurement we will get 0 and 1 with certain probabilities. Imagine that every bit in our computer is both 0 and 1 at the same time, and if we even try to read from it, we not only get a probabilistic answer but also collapse that qubit state! Entanglement.