Quantum Information Chapter 10. Quantum Shannon Theory
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Quantum Information, Entanglement and Entropy
QUANTUM INFORMATION, ENTANGLEMENT AND ENTROPY Siddharth Sharma Abstract In this review I had given an introduction to axiomatic Quantum Mechanics, Quantum Information and its measure entropy. Also, I had given an introduction to Entanglement and its measure as the minimum of relative quantum entropy of a density matrix of a Hilbert space with respect to all separable density matrices of the same Hilbert space. I also discussed geodesic in the space of density matrices, their orthogonality and Pythagorean theorem of density matrix. Postulates of Quantum Mechanics The basic postulate of quantum mechanics is about the Hilbert space formalism: • Postulates 0: To each quantum mechanical system, has a complex Hilbert space ℍ associated to it: Set of all pure state, ℍ⁄∼ ≔ {⟦푓⟧|⟦푓⟧ ≔ {ℎ: ℎ ∼ 푓}} and ℎ ∼ 푓 ⇔ ∃ 푧 ∈ ℂ such that ℎ = 푧 푓 where 푧 is called phase • Postulates 1: The physical states of a quantum mechanical system are described by statistical operators acting on the Hilbert space. A density matrix or statistical operator, is a positive operator of trace 1 on the Hilbert space. Where is called positive if 〈푥, 푥〉 for all 푥 ∈ ℍ and 〈⋇,⋇〉. • Postulates 2: The observables of a quantum mechanical system are described by self- adjoint operators acting on the Hilbert space. A self-adjoint operator A on a Hilbert space ℍ is a linear operator 퐴: ℍ → ℍ which satisfies 〈퐴푥, 푦〉 = 〈푥, 퐴푦〉 for 푥, 푦 ∈ ℍ. Lemma: The density matrices acting on a Hilbert space form a convex set whose extreme points are the pure states. Proof. Denote by Σ the set of density matrices. -
What Is Quantum Information?
What Is Quantum Information? Robert B. Griffiths Carnegie-Mellon University Pittsburgh, Pennsylvania Research supported by the National Science Foundation References (work by R. B. Griffiths) • \Nature and location of quantum information." Ph◦ys. Rev. A 66 (2002) 012311; quant-ph/0203058 \Channel kets, entangled states, and the location of quan◦ tum information," Phys. Rev. A 71 (2005) 042337; quant-ph/0409106 Consistent Quantum Theory (Cambridge 2002) http://quan◦ tum.phys.cmu.edu/ What Is Quantum Information? 01. 100.01.tex Introduction What is quantum information? Precede by: • What is information? ◦ What is classical information theory? ◦ What is information? Example, newspaper • Symbolic representation of some situation ◦ Symbols in newspaper correlated with situation ◦ Information is about that situation ◦ What Is Quantum Information? 04. 100.04.tex Classical Information Theory Shannon: • \Mathematical Theory of Communication" (1948) ◦ One of major scientific developments of 20th century ◦ Proposed quantitative measure of information ◦ Information entropy • H(X) = p log p − X i i i Logarithmic measure of missing information ◦ Probabilistic model: pi are probabilities ◦ Applies to classical (macroscopic)f g signals ◦ Coding theorem: Bound on rate of transmission of information• through noisy channel What Is Quantum Information? 05. 100.05.tex Quantum Information Theory (QIT) Goal of QIT: \Quantize Shannon" • Extend Shannon's ideas to domain where quantum effects◦ are important Find quantum counterpart of H(X) ◦ We live in a quantum world, so • QIT should be the fundamental info theory ◦ Classical theory should emerge from QIT ◦ Analogy: relativity theory Newton for v c ◦ ! What Is Quantum Information? 06. 100.06.tex QIT: Current Status Enormous number of published papers • Does activity = understanding? ◦ Some topics of current interest: • Entanglement ◦ Quantum channels ◦ Error correction ◦ Quantum computation ◦ Decoherence ◦ Unifying principles have yet to emerge • At least, they are not yet widely recognized ◦ What Is Quantum Information? 07. -
Key Concepts for Future QIS Learners Workshop Output Published Online May 13, 2020
Key Concepts for Future QIS Learners Workshop output published online May 13, 2020 Background and Overview On behalf of the Interagency Working Group on Workforce, Industry and Infrastructure, under the NSTC Subcommittee on Quantum Information Science (QIS), the National Science Foundation invited 25 researchers and educators to come together to deliberate on defining a core set of key concepts for future QIS learners that could provide a starting point for further curricular and educator development activities. The deliberative group included university and industry researchers, secondary school and college educators, and representatives from educational and professional organizations. The workshop participants focused on identifying concepts that could, with additional supporting resources, help prepare secondary school students to engage with QIS and provide possible pathways for broader public engagement. This workshop report identifies a set of nine Key Concepts. Each Concept is introduced with a concise overall statement, followed by a few important fundamentals. Connections to current and future technologies are included, providing relevance and context. The first Key Concept defines the field as a whole. Concepts 2-6 introduce ideas that are necessary for building an understanding of quantum information science and its applications. Concepts 7-9 provide short explanations of critical areas of study within QIS: quantum computing, quantum communication and quantum sensing. The Key Concepts are not intended to be an introductory guide to quantum information science, but rather provide a framework for future expansion and adaptation for students at different levels in computer science, mathematics, physics, and chemistry courses. As such, it is expected that educators and other community stakeholders may not yet have a working knowledge of content covered in the Key Concepts. -
Lecture 3: Entropy, Relative Entropy, and Mutual Information 1 Notation 2
EE376A/STATS376A Information Theory Lecture 3 - 01/16/2018 Lecture 3: Entropy, Relative Entropy, and Mutual Information Lecturer: Tsachy Weissman Scribe: Yicheng An, Melody Guan, Jacob Rebec, John Sholar In this lecture, we will introduce certain key measures of information, that play crucial roles in theoretical and operational characterizations throughout the course. These include the entropy, the mutual information, and the relative entropy. We will also exhibit some key properties exhibited by these information measures. 1 Notation A quick summary of the notation 1. Discrete Random Variable: U 2. Alphabet: U = fu1; u2; :::; uM g (An alphabet of size M) 3. Specific Value: u; u1; etc. For discrete random variables, we will write (interchangeably) P (U = u), PU (u) or most often just, p(u) Similarly, for a pair of random variables X; Y we write P (X = x j Y = y), PXjY (x j y) or p(x j y) 2 Entropy Definition 1. \Surprise" Function: 1 S(u) log (1) , p(u) A lower probability of u translates to a greater \surprise" that it occurs. Note here that we use log to mean log2 by default, rather than the natural log ln, as is typical in some other contexts. This is true throughout these notes: log is assumed to be log2 unless otherwise indicated. Definition 2. Entropy: Let U a discrete random variable taking values in alphabet U. The entropy of U is given by: 1 X H(U) [S(U)] = log = − log (p(U)) = − p(u) log p(u) (2) , E E p(U) E u Where U represents all u values possible to the variable. -
Quantum Cryptography: from Theory to Practice
Quantum cryptography: from theory to practice by Xiongfeng Ma A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Thesis Graduate Department of Department of Physics University of Toronto Copyright °c 2008 by Xiongfeng Ma Abstract Quantum cryptography: from theory to practice Xiongfeng Ma Doctor of Philosophy Thesis Graduate Department of Department of Physics University of Toronto 2008 Quantum cryptography or quantum key distribution (QKD) applies fundamental laws of quantum physics to guarantee secure communication. The security of quantum cryptog- raphy was proven in the last decade. Many security analyses are based on the assumption that QKD system components are idealized. In practice, inevitable device imperfections may compromise security unless these imperfections are well investigated. A highly attenuated laser pulse which gives a weak coherent state is widely used in QKD experiments. A weak coherent state has multi-photon components, which opens up a security loophole to the sophisticated eavesdropper. With a small adjustment of the hardware, we will prove that the decoy state method can close this loophole and substantially improve the QKD performance. We also propose a few practical decoy state protocols, study statistical fluctuations and perform experimental demonstrations. Moreover, we will apply the methods from entanglement distillation protocols based on two-way classical communication to improve the decoy state QKD performance. Fur- thermore, we study the decoy state methods for other single photon sources, such as triggering parametric down-conversion (PDC) source. Note that our work, decoy state protocol, has attracted a lot of scienti¯c and media interest. The decoy state QKD becomes a standard technique for prepare-and-measure QKD schemes. -
Pilot Quantum Error Correction for Global
Pilot Quantum Error Correction for Global- Scale Quantum Communications Laszlo Gyongyosi*1,2, Member, IEEE, Sandor Imre1, Member, IEEE 1Quantum Technologies Laboratory, Department of Telecommunications Budapest University of Technology and Economics 2 Magyar tudosok krt, H-1111, Budapest, Hungary 2Information Systems Research Group, Mathematics and Natural Sciences Hungarian Academy of Sciences H-1518, Budapest, Hungary *[email protected] Real global-scale quantum communications and quantum key distribution systems cannot be implemented by the current fiber and free-space links. These links have high attenuation, low polarization-preserving capability or extreme sensitivity to the environment. A potential solution to the problem is the space-earth quantum channels. These channels have no absorption since the signal states are propagated in empty space, however a small fraction of these channels is in the atmosphere, which causes slight depolarizing effect. Furthermore, the relative motion of the ground station and the satellite causes a rotation in the polarization of the quantum states. In the current approaches to compensate for these types of polarization errors, high computational costs and extra physical apparatuses are required. Here we introduce a novel approach which breaks with the traditional views of currently developed quantum-error correction schemes. The proposed solution can be applied to fix the polarization errors which are critical in space-earth quantum communication systems. The channel coding scheme provides capacity-achieving communication over slightly depolarizing space-earth channels. I. Introduction Quantum error-correction schemes use different techniques to correct the various possible errors which occur in a quantum channel. In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered [12-16], [19-22] however the efficient error- correction in quantum systems is still a challenge. -
Package 'Infotheo'
Package ‘infotheo’ February 20, 2015 Title Information-Theoretic Measures Version 1.2.0 Date 2014-07 Publication 2009-08-14 Author Patrick E. Meyer Description This package implements various measures of information theory based on several en- tropy estimators. Maintainer Patrick E. Meyer <[email protected]> License GPL (>= 3) URL http://homepage.meyerp.com/software Repository CRAN NeedsCompilation yes Date/Publication 2014-07-26 08:08:09 R topics documented: condentropy . .2 condinformation . .3 discretize . .4 entropy . .5 infotheo . .6 interinformation . .7 multiinformation . .8 mutinformation . .9 natstobits . 10 Index 12 1 2 condentropy condentropy conditional entropy computation Description condentropy takes two random vectors, X and Y, as input and returns the conditional entropy, H(X|Y), in nats (base e), according to the entropy estimator method. If Y is not supplied the function returns the entropy of X - see entropy. Usage condentropy(X, Y=NULL, method="emp") Arguments X data.frame denoting a random variable or random vector where columns contain variables/features and rows contain outcomes/samples. Y data.frame denoting a conditioning random variable or random vector where columns contain variables/features and rows contain outcomes/samples. method The name of the entropy estimator. The package implements four estimators : "emp", "mm", "shrink", "sg" (default:"emp") - see details. These estimators require discrete data values - see discretize. Details • "emp" : This estimator computes the entropy of the empirical probability distribution. • "mm" : This is the Miller-Madow asymptotic bias corrected empirical estimator. • "shrink" : This is a shrinkage estimate of the entropy of a Dirichlet probability distribution. • "sg" : This is the Schurmann-Grassberger estimate of the entropy of a Dirichlet probability distribution. -
Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Caltech Theses and Dissertations Analysis of quantum error-correcting codes: symplectic lattice codes and toric codes Thesis by James William Harrington Advisor John Preskill In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2004 (Defended May 17, 2004) ii c 2004 James William Harrington All rights Reserved iii Acknowledgements I can do all things through Christ, who strengthens me. Phillipians 4:13 (NKJV) I wish to acknowledge first of all my parents, brothers, and grandmother for all of their love, prayers, and support. Thanks to my advisor, John Preskill, for his generous support of my graduate studies, for introducing me to the studies of quantum error correction, and for encouraging me to pursue challenging questions in this fascinating field. Over the years I have benefited greatly from stimulating discussions on the subject of quantum information with Anura Abeyesinge, Charlene Ahn, Dave Ba- con, Dave Beckman, Charlie Bennett, Sergey Bravyi, Carl Caves, Isaac Chenchiah, Keng-Hwee Chiam, Richard Cleve, John Cortese, Sumit Daftuar, Ivan Deutsch, Andrew Doherty, Jon Dowling, Bryan Eastin, Steven van Enk, Chris Fuchs, Sho- hini Ghose, Daniel Gottesman, Ted Harder, Patrick Hayden, Richard Hughes, Deborah Jackson, Alexei Kitaev, Greg Kuperberg, Andrew Landahl, Chris Lee, Debbie Leung, Carlos Mochon, Michael Nielsen, Smith Nielsen, Harold Ollivier, Tobias Osborne, Michael Postol, Philippe Pouliot, Marco Pravia, John Preskill, Eric Rains, Robert Raussendorf, Joe Renes, Deborah Santamore, Yaoyun Shi, Pe- ter Shor, Marcus Silva, Graeme Smith, Jennifer Sokol, Federico Spedalieri, Rene Stock, Francis Su, Jacob Taylor, Ben Toner, Guifre Vidal, and Mas Yamada. -
A Mini-Introduction to Information Theory
A Mini-Introduction To Information Theory Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract This article consists of a very short introduction to classical and quantum information theory. Basic properties of the classical Shannon entropy and the quantum von Neumann entropy are described, along with related concepts such as classical and quantum relative entropy, conditional entropy, and mutual information. A few more detailed topics are considered in the quantum case. arXiv:1805.11965v5 [hep-th] 4 Oct 2019 Contents 1 Introduction 2 2 Classical Information Theory 2 2.1 ShannonEntropy ................................... .... 2 2.2 ConditionalEntropy ................................. .... 4 2.3 RelativeEntropy .................................... ... 6 2.4 Monotonicity of Relative Entropy . ...... 7 3 Quantum Information Theory: Basic Ingredients 10 3.1 DensityMatrices .................................... ... 10 3.2 QuantumEntropy................................... .... 14 3.3 Concavity ......................................... .. 16 3.4 Conditional and Relative Quantum Entropy . ....... 17 3.5 Monotonicity of Relative Entropy . ...... 20 3.6 GeneralizedMeasurements . ...... 22 3.7 QuantumChannels ................................... ... 24 3.8 Thermodynamics And Quantum Channels . ...... 26 4 More On Quantum Information Theory 27 4.1 Quantum Teleportation and Conditional Entropy . ......... 28 4.2 Quantum Relative Entropy And Hypothesis Testing . ......... 32 4.3 Encoding -
Models of Quantum Complexity Growth
Models of quantum complexity growth Fernando G.S.L. Brand~ao,a;b;c;d Wissam Chemissany,b Nicholas Hunter-Jones,* e;b Richard Kueng,* b;c John Preskillb;c;d aAmazon Web Services, AWS Center for Quantum Computing, Pasadena, CA bInstitute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125 cDepartment of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125 dWalter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 ePerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5 *Corresponding authors: [email protected] and [email protected] Abstract: The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high energy physics. The quantum complexity of a unitary transformation or quantum state is defined as the size of the shortest quantum computation that executes the unitary or prepares the state. It is reasonable to expect that the complexity of a quantum state governed by a chaotic many- body Hamiltonian grows linearly with time for a time that is exponential in the system size; however, because it is hard to rule out a short-cut that improves the efficiency of a computation, it is notoriously difficult to derive lower bounds on quantum complexity for particular unitaries or states without making additional assumptions. To go further, one may study more generic models of complexity growth. We provide a rigorous connection between complexity growth and unitary k-designs, ensembles which capture the randomness of the unitary group. This connection allows us to leverage existing results about design growth to draw conclusions about the growth of complexity. -
Entropic Uncertainty Relations and Their Applications
REVIEWS OF MODERN PHYSICS, VOLUME 89, JANUARY–MARCH 2017 Entropic uncertainty relations and their applications Patrick J. Coles* Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, N2L3G1 Waterloo, Ontario, Canada Mario Berta† Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA Marco Tomamichel‡ School of Physics, The University of Sydney, Sydney, NSW 2006, Australia Stephanie Wehner§ QuTech, Delft University of Technology, 2628 CJ Delft, Netherlands (published 6 February 2017) Heisenberg’s uncertainty principle forms a fundamental element of quantum mechanics. Uncertainty relations in terms of entropies were initially proposed to deal with conceptual shortcomings in the original formulation of the uncertainty principle and, hence, play an important role in quantum foundations. More recently, entropic uncertainty relations have emerged as the central ingredient in the security analysis of almost all quantum cryptographic protocols, such as quantum key distribution and two-party quantum cryptography. This review surveys entropic uncertainty relations that capture Heisenberg’s idea that the results of incompatible measurements are impossible to predict, covering both finite- and infinite-dimensional measurements. These ideas are then extended to incorporate quantum correlations between the observed object and its environment, allowing for a variety of recent, more general formulations of the uncertainty principle. Finally, various applications are discussed, ranging from entanglement witnessing to wave-particle duality to quantum cryptography. DOI: 10.1103/RevModPhys.89.015002 CONTENTS 1. Shannon entropy 9 2. Rényi entropies 10 I. Introduction 2 3. Maassen-Uffink proof 10 A. Scope of this review 4 4. Tightness and extensions 10 II. -
Some Open Problems in Quantum Information Theory
Some Open Problems in Quantum Information Theory Mary Beth Ruskai∗ Department of Mathematics, Tufts University, Medford, MA 02155 [email protected] June 14, 2007 Abstract Some open questions in quantum information theory (QIT) are described. Most of them were presented in Banff during the BIRS workshop on Operator Structures in QIT 11-16 February 2007. 1 Extreme points of CPT maps In QIT, a channel is represented by a completely-positive trace-preserving (CPT) map Φ: Md1 7→ Md2 , which is often written in the Choi-Kraus form X † X † Φ(ρ) = AkρAk with AkAk = Id1 . (1) k k The state representative or Choi matrix of Φ is 1 X Φ(|βihβ|) = d |ejihek|Φ(|ejihek|) (2) jk where |βi is a maximally entangled Bell state. Choi [3] showed that the Ak can be obtained from the eigenvectors of Φ(|βihβ|) with non-zero eigenvalues. The operators Ak in (1) are known to be defined only up to a partial isometry and are often called Kraus operators. When a minimal set is obtained from Choi’s prescription using eigenvectors of (2), they are defined up to mixing of those from degenerate eigenvalues ∗Partially supported by the National Science Foundation under Grant DMS-0604900 1 and we will refer to them as Choi-Kraus operators. Choi showed that Φ is an extreme † point of the set of CPT maps Φ : Md1 7→ Md2 if and only if the set {AjAk} is linearly independent in Md1 . This implies that the Choi matrix of an extreme CPT map has rank at most d1.