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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. -
Painstakingly-Edited Typeset Lecture Notes
Physics 239: Topology from Physics Winter 2021 Lecturer: McGreevy These lecture notes live here. Please email corrections and questions to mcgreevy at physics dot ucsd dot edu. Last updated: May 26, 2021, 14:36:17 1 Contents 0.1 Introductory remarks............................3 0.2 Conventions.................................8 0.3 Sources....................................9 1 The toric code and homology 10 1.1 Cell complexes and homology....................... 21 1.2 p-form ZN toric code............................ 23 1.3 Some examples............................... 26 1.4 Higgsing, change of coefficients, exact sequences............. 32 1.5 Independence of cellulation......................... 36 1.6 Gapped boundaries and relative homology................ 39 1.7 Duality.................................... 42 2 Supersymmetric quantum mechanics and cohomology, index theory, Morse theory 49 2.1 Supersymmetric quantum mechanics................... 49 2.2 Differential forms consolidation...................... 61 2.3 Supersymmetric QM and Morse theory.................. 66 2.4 Global information from local information................ 74 2.5 Homology and cohomology......................... 77 2.6 Cech cohomology.............................. 80 2.7 Local reconstructability of quantum states................ 86 3 Quantum Double Model and Homotopy 89 3.1 Notions of `same'.............................. 89 3.2 Homotopy equivalence and cohomology.................. 90 3.3 Homotopy equivalence and homology................... 91 3.4 Morse theory and homotopy -
Front Matter of the Book Contains a Detailed Table of Contents, Which We Encourage You to Browse
Cambridge University Press 978-1-107-00217-3 - Quantum Computation and Quantum Information: 10th Anniversary Edition Michael A. Nielsen & Isaac L. Chuang Frontmatter More information Quantum Computation and Quantum Information 10th Anniversary Edition One of the most cited books in physics of all time, Quantum Computation and Quantum Information remains the best textbook in this exciting field of science. This 10th Anniversary Edition includes a new Introduction and Afterword from the authors setting the work in context. This comprehensive textbook describes such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography, and quantum error-correction. Quantum mechanics and computer science are introduced, before moving on to describe what a quantum computer is, how it can be used to solve problems faster than “classical” computers, and its real-world implementation. It concludes with an in-depth treatment of quantum information. Containing a wealth of figures and exercises, this well-known textbook is ideal for courses on the subject, and will interest beginning graduate students and researchers in physics, computer science, mathematics, and electrical engineering. MICHAEL NIELSEN was educated at the University of Queensland, and as a Fulbright Scholar at the University of New Mexico. He worked at Los Alamos National Laboratory, as the Richard Chace Tolman Fellow at Caltech, was Foundation Professor of Quantum Information Science and a Federation Fellow at the University of Queensland, and a Senior Faculty Member at the Perimeter Institute for Theoretical Physics. He left Perimeter Institute to write a book about open science and now lives in Toronto. ISAAC CHUANG is a Professor at the Massachusetts Institute of Technology, jointly appointed in Electrical Engineering & Computer Science, and in Physics. -
Superconducting Quantum Simulator for Topological Order and the Toric Code
Superconducting Quantum Simulator for Topological Order and the Toric Code Mahdi Sameti,1 Anton Potoˇcnik,2 Dan E. Browne,3 Andreas Wallraff,2 and Michael J. Hartmann1 1Institute of Photonics and Quantum Sciences, Heriot-Watt University Edinburgh EH14 4AS, United Kingdom 2Department of Physics, ETH Z¨urich,CH-8093 Z¨urich,Switzerland 3Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom (Dated: March 20, 2017) Topological order is now being established as a central criterion for characterizing and classifying ground states of condensed matter systems and complements categorizations based on symmetries. Fractional quantum Hall systems and quantum spin liquids are receiving substantial interest because of their intriguing quantum correlations, their exotic excitations and prospects for protecting stored quantum information against errors. Here we show that the Hamiltonian of the central model of this class of systems, the Toric Code, can be directly implemented as an analog quantum simulator in lattices of superconducting circuits. The four-body interactions, which lie at its heart, are in our concept realized via Superconducting Quantum Interference Devices (SQUIDs) that are driven by a suitably oscillating flux bias. All physical qubits and coupling SQUIDs can be individually controlled with high precision. Topologically ordered states can be prepared via an adiabatic ramp of the stabilizer interactions. Strings of qubit operators, including the stabilizers and correlations along non-contractible loops, can be read out via a capacitive coupling to read-out resonators. Moreover, the available single qubit operations allow to create and propagate elementary excitations of the Toric Code and to verify their fractional statistics. -
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 -
Chapter 6 Quantum Entropy
Chapter 6 Quantum entropy There is a notion of entropy which quantifies the amount of uncertainty contained in an ensemble of Qbits. This is the von Neumann entropy that we introduce in this chapter. In some respects it behaves just like Shannon's entropy but in some others it is very different and strange. As an illustration let us immediately say that as in the classical theory, conditioning reduces entropy; but in sharp contrast with classical theory the entropy of a quantum system can be lower than the entropy of its parts. The von Neumann entropy was first introduced in the realm of quantum statistical mechanics, but we will see in later chapters that it enters naturally in various theorems of quantum information theory. 6.1 Main properties of Shannon entropy Let X be a random variable taking values x in some alphabet with probabil- ities px = Prob(X = x). The Shannon entropy of X is X 1 H(X) = p ln x p x x and quantifies the average uncertainty about X. The joint entropy of two random variables X, Y is similarly defined as X 1 H(X; Y ) = p ln x;y p x;y x;y and the conditional entropy X X 1 H(XjY ) = py pxjy ln p j y x;y x y 1 2 CHAPTER 6. QUANTUM ENTROPY where px;y pxjy = py The conditional entropy is the average uncertainty of X given that we observe Y = y. It is easily seen that H(XjY ) = H(X; Y ) − H(Y ) The formula is consistent with the interpretation of H(XjY ): when we ob- serve Y the uncertainty H(X; Y ) is reduced by the amount H(Y ). -
Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations
Article Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations Jen-Tsung Hsiang 1,*,† and Bei-Lok Hu 2,† ID 1 Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China 2 Maryland Center for Fundamental Physics and Joint Quantum Institute, University of Maryland, College Park, MD 20742-4111, USA; [email protected] * Correspondence: [email protected]; Tel.: +86-21-3124-3754 † These authors contributed equally to this work. Received: 26 April 2018; Accepted: 30 May 2018; Published: 31 May 2018 Abstract: Identifying or constructing a fine-grained microscopic theory that will emerge under specific conditions to a known macroscopic theory is always a formidable challenge. Thermodynamics is perhaps one of the most powerful theories and best understood examples of emergence in physical sciences, which can be used for understanding the characteristics and mechanisms of emergent processes, both in terms of emergent structures and the emergent laws governing the effective or collective variables. Viewing quantum mechanics as an emergent theory requires a better understanding of all this. In this work we aim at a very modest goal, not quantum mechanics as thermodynamics, not yet, but the thermodynamics of quantum systems, or quantum thermodynamics. We will show why even with this minimal demand, there are many new issues which need be addressed and new rules formulated. The thermodynamics of small quantum many-body systems strongly coupled to a heat bath at low temperatures with non-Markovian behavior contains elements, such as quantum coherence, correlations, entanglement and fluctuations, that are not well recognized in traditional thermodynamics, built on large systems vanishingly weakly coupled to a non-dynamical reservoir. -
Some Trace Inequalities for Exponential and Logarithmic Functions
Bull. Math. Sci. https://doi.org/10.1007/s13373-018-0123-3 Some trace inequalities for exponential and logarithmic functions Eric A. Carlen1 · Elliott H. Lieb2 Received: 8 October 2017 / Revised: 17 April 2018 / Accepted: 24 April 2018 © The Author(s) 2018 Abstract Consider a function F(X, Y ) of pairs of positive matrices with values in the positive matrices such that whenever X and Y commute F(X, Y ) = X pY q . Our first main result gives conditions on F such that Tr[X log(F(Z, Y ))]≤Tr[X(p log X + q log Y )] for all X, Y, Z such that TrZ = Tr X. (Note that Z is absent from the right side of the inequality.) We give several examples of functions F to which the theorem applies. Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables X, Y, Z instead of just X, Y alone. The investigation of these logarith- mic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy D(X||Y ) = Tr[X(log X − log Y ]), and two others, the Donald relative entropy DD(X||Y ), and the Belavkin–Stasewski relative entropy DBS(X||Y ). They are known to satisfy DD(X||Y ) ≤ D(X||Y ) ≤ DBS(X||Y ). We prove that the Donald relative entropy provides the sharp upper bound, independent of Z on Tr[X log(F(Z, Y ))] in a number of cases in which F(Z, Y ) is homogeneous of degree 1 in Z and −1inY . -
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. -
{Dоwnlоаd/Rеаd PDF Bооk} Entangled Kindle
ENTANGLED PDF, EPUB, EBOOK Cat Clarke | 256 pages | 06 Jan 2011 | Hachette Children's Group | 9781849163941 | English | London, United Kingdom Entangled PDF Book A Trick of the Tail is the seventh studio album by English progressive rock band Genesis. Why the Name Entangled? Shor, John A. Annals of Physics. Bibcode : arXiv The Living Years. Suhail Bennett, David P. Part of a series on. Examples of entanglement in a Sentence his life is greatly complicated by his romantic entanglements. Take the quiz Forms of Government Quiz Name that government! These four pure states are all maximally entangled according to the entropy of entanglement and form an orthonormal basis linear algebra of the Hilbert space of the two qubits. The researchers used a single source of photon pairs that had been entangle d, which means their quantum states are intrinsically linked and any change or measurement of one is mirrored in the other. For example, an interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit which is now a LOCC operation, since both qubits are in B's lab and then teleporting the qubit back to A. Looking for some great streaming picks? In earlier tests, it couldn't be absolutely ruled out that the test result at one point could have been subtly transmitted to the remote point, affecting the outcome at the second location. Thesis University of California at Berkeley, The Hilbert space of the composite system is the tensor product. -
Lecture 18 — October 26, 2015 1 Overview 2 Quantum Entropy
PHYS 7895: Quantum Information Theory Fall 2015 Lecture 18 | October 26, 2015 Prof. Mark M. Wilde Scribe: Mark M. Wilde This document is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 1 Overview In the previous lecture, we discussed classical entropy and entropy inequalities. In this lecture, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumann entropy. It is the quantum generalization of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state. The von Neumann entropy gives meaning to a notion of the information qubit. This notion is different from that of the physical qubit, which is the description of a quantum state of an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is present in a quantum system. The initial definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information. -
Arxiv:1803.00026V4 [Cond-Mat.Str-El] 8 Feb 2019 3 Variational Quantum-Classical Simulation (VQCS) 5
SciPost Physics Submission Efficient variational simulation of non-trivial quantum states Wen Wei Ho1*, Timothy H. Hsieh2,3, 1 Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA 3 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada *[email protected] February 12, 2019 Abstract We provide an efficient and general route for preparing non-trivial quantum states that are not adiabatically connected to unentangled product states. Our approach is a hybrid quantum-classical variational protocol that incorporates a feedback loop between a quantum simulator and a classical computer, and is experimen- tally realizable on near-term quantum devices of synthetic quantum systems. We find explicit protocols which prepare with perfect fidelities (i) the Greenberger- Horne-Zeilinger (GHZ) state, (ii) a quantum critical state, and (iii) a topologically ordered state, with L variational parameters and physical runtimes T that scale linearly with the system size L. We furthermore conjecture and support numeri- cally that our protocol can prepare, with perfect fidelity and similar operational costs, the ground state of every point in the one dimensional transverse field Ising model phase diagram. Besides being practically useful, our results also illustrate the utility of such variational ans¨atzeas good descriptions of non-trivial states of matter. Contents 1 Introduction 2 2 Non-trivial quantum states