scholarly journals Entanglement Monotones and Measures: An Overview

2017 ◽  
Author(s):  
Volkan Erol
Keyword(s):  
Information Theory ◽  
Theoretical Basis ◽  
Search Algorithm ◽  
Short Review ◽  
Quantum Information Theory ◽  
The Other ◽  
Quantum Mechanical ◽  
Quantum Computers ◽  
Factorization Algorithm ◽  
Quantum Mechanical Systems

Quantum information theory and quantum computing are theoritical basis of quantum computers. Thanks to entanglement, quantum mechanical systems are provisioned to realize many information processing problems faster than classical counterparts. For example, Shor’s factorization algorithm, Grover’s search algorithm, quantum Fourrier transformation, etc. Entanglement, is the theoretical basis providing the expected speedups. It can be view in bipartite or multipartite forms. In order to quantify entanglement, some measures are defined. On the other hand, a general and accepted criterion, which can measure the amount of entanglement of multilateral systems, has not yet been found. In this work, we make a short review of recent research on the topic entanglement monotones and measures with an analitical approach.

scholarly journals Entanglement Monotones and Measures: An Overview

2017 ◽  
Author(s):  
Volkan Erol
Keyword(s):  
Information Processing ◽  
Theoretical Basis ◽  
Search Algorithm ◽  
Quantum Information Theory ◽  
The Other ◽  
Quantum Mechanical ◽  
Quantum Computers ◽  
Factorization Algorithm ◽  
Entanglement Measures ◽  
Quantum Mechanical Systems

Quantum information theory and quantum computing are theoritical basis of quantum computers. Thanks to entanglement, quantum mechanical systems are provisioned to realize many information processing problems faster than classical counterparts. For example, Shor’s factorization algorithm, Grover’s search algorithm, quantum Fourrier transformation, etc. [1]. Entanglement, is the theoretical basis providing the expected speedups. It can be view in bipartite or multipartite forms. In order to quantify entanglement, some measures are defined. On the other hand, a general and accepted criterion, which can measure the amount of entanglement of multilateral systems, has not yet been found. However, since it must be used in many information processing tasks, the production and processing of multilateral quantum entangled systems is at the top of the hot topics of recent years [5-11]. Much of the work in the basic quantum technologies, such as quantum cryptography, communications, and computers, requires multi-partite entangled systems such as GHZ, W [22,30]. It can be suggested that the quantum entanglement criteria reflects the different properties of the systems. Many recent research has been done in entanglement and its related disciplines like entanglement measures and majorization, etc. [62-71]. In this work, we make an overview of recent research on the topic entanglement monotones/measures with an analitical approach.

scholarly journals Randomness: Quantum versus classical

2016 ◽  
Vol 14 (04) ◽  
pp. 1640009 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Information Theory ◽  
Related Problem ◽  
Short Review ◽  
Quantum Information Theory ◽  
Quantum Foundations ◽  
Interpretation Of Quantum Mechanics ◽  
New Wave ◽  
Quantum Randomness ◽  
Information Interpretation ◽  
Random Generators

Recent tremendous development of quantum information theory has led to a number of quantum technological projects, e.g. quantum random generators. This development had stimulated a new wave of interest in quantum foundations. One of the most intriguing problems of quantum foundations is the elaboration of a consistent and commonly accepted interpretation of a quantum state. Closely related problem is the clarification of the notion of quantum randomness and its interrelation with classical randomness. In this short review, we shall discuss basics of classical theory of randomness (which by itself is very complex and characterized by diversity of approaches) and compare it with irreducible quantum randomness. We also discuss briefly “digital philosophy”, its role in physics (classical and quantum) and its coupling to the information interpretation of quantum mechanics (QM).

scholarly journals Numerical study of the SWKB condition of novel classes of exactly solvable systems

2020 ◽  
pp. 2150025
Author(s):  
Yuta Nasuda ◽  
Nobuyuki Sawado
Keyword(s):  
Orthogonal Polynomial ◽  
Jacobi Polynomials ◽  
Numerical Study ◽  
Mechanical Systems ◽  
The Other ◽  
Quantum Mechanical ◽  
Shape Invariance ◽  
Exactly Solvable ◽  
Solvable Quantum ◽  
Quantum Mechanical Systems

The supersymmetric WKB (SWKB) condition is supposed to be exact for all known exactly solvable quantum mechanical systems with the shape invariance. Recently, it was claimed that the SWKB condition was not exact for the extended radial oscillator, whose eigenfunctions consisted of the exceptional orthogonal polynomial, even the system possesses the shape invariance. In this paper, we examine the SWKB condition for the two novel classes of exactly solvable systems: one has the multi-indexed Laguerre and Jacobi polynomials as the main parts of the eigenfunctions, and the other has the Krein–Adler Hermite, Laguerre and Jacobi polynomials. For all of them, one can always remove the [Formula: see text]-dependency from the condition, and it is satisfied with a certain degree of accuracy.

scholarly journals ${\mathcal N}$-FOLD SUPERSYMMETRIC QUANTUM MECHANICS WITH REFLECTIONS

2012 ◽  
Vol 27 (19) ◽  
pp. 1250102 ◽  
Author(s):  
TOSHIAKI TANAKA
Keyword(s):  
Quantum Mechanics ◽  
Hermitian Matrix ◽  
Mechanical Systems ◽  
Quantum Systems ◽  
Supersymmetric Quantum Mechanics ◽  
The Other ◽  
Quantum Mechanical ◽  
Quantum Mechanical Systems ◽  
Ordinary Quantum

We formulate [Formula: see text]-fold supersymmetry in quantum mechanical systems with reflection operators. As in the cases of other systems, they possess the two significant characters of [Formula: see text]-fold supersymmetry, namely, almost isospectrality and weak quasi-solvability. We construct explicitly the most general one- and two-fold supersymmetric quantum mechanical systems with reflections. In the case of [Formula: see text], we find that there are seven inequivalent such systems, three of which are characterized by three arbitrary functions having definite parity while the other four characterized by two arbitrary functions. In addition, four of the seven inequivalent systems do not reduce to ordinary quantum systems without reflections. Furthermore, in certain particular cases, they are essentially equivalent to the most general two-by-two Hermitian matrix two-fold supersymmetric quantum systems obtained previously by us.

scholarly journals Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systems

2020 ◽  
Vol 80 (7) ◽  
Author(s):  
Sergiu I. Vacaru
Keyword(s):  
Information Theory ◽  
Evolution Equations ◽  
Mechanical Systems ◽  
Von Neumann Entropy ◽  
Quantum Mechanical ◽  
Geometric Flows ◽  
Thermodynamic Models ◽  
Geometric Flow ◽  
Spacelike Hypersurfaces ◽  
Quantum Mechanical Systems

Abstract This work consists an introduction to the classical and quantum information theory of geometric flows of (relativistic) Lagrange–Hamilton mechanical systems. Basic geometric and physical properties of the canonical nonholonomic deformations of G. Perelman entropy functionals and geometric flows evolution equations of classical mechanical systems are described. There are studied projections of such F- and W-functionals on Lorentz spacetime manifolds and three-dimensional spacelike hypersurfaces. These functionals are used for elaborating relativistic thermodynamic models for Lagrange–Hamilton geometric evolution and respective generalized Hamilton geometric flow and nonholonomic Ricci flow equations. The concept of nonholonomic W-entropy is developed as a complementary one for the classical Shannon entropy and the quantum von Neumann entropy. There are considered geometric flow generalizations of the approaches based on classical and quantum relative entropy, conditional entropy, mutual information, and related thermodynamic models. Such basic ingredients and topics of quantum geometric flow information theory are elaborated using the formalism of density matrices and measurements with quantum channels for the evolution of quantum mechanical systems.

scholarly journals QUANTUM COMPUTER: AN APPLIANCE FOR PLAYING MARKET GAMES

2004 ◽  
Vol 02 (04) ◽  
pp. 495-509 ◽  
Author(s):  
EDWARD W. PIOTROWSKI ◽  
JAN SŁADKOWSKI
Keyword(s):  
Game Theory ◽  
Information Theory ◽  
Quantum Information ◽  
Financial Market ◽  
Quantum Computer ◽  
Quantum Information Theory ◽  
Quantum Computers ◽  
Quantum Game ◽  
Recent Developments ◽  
Quantum Strategies

Recent developments in quantum computation and quantum information theory allow one to extend the scope of game theory for the quantum world. The authors have recently proposed a quantum description of financial market in terms of quantum game theory. This paper contains an analysis of such markets that shows that there are advantages to be gained from using quantum computers and quantum strategies.

scholarly journals Quantifying Disorder in Networks

2011 ◽  
pp. 66-76 ◽  
Author(s):  
Filippo Passerini ◽  
Simone Severini
Keyword(s):  
Information Theory ◽  
Spectral Properties ◽  
Quantum Information Theory ◽  
Von Neumann Entropy ◽  
Connected Components ◽  
Quantum Mechanical ◽  
Von Neumann ◽  
Regularity Properties ◽  
Wide Interface ◽  
Edge Addition

The authors introduce a novel entropic notion with the purpose of quantifying disorder/uncertainty in networks. This is based on the Laplacian and it is exactly the von Neumann entropy of certain quantum mechanical states. It is remarkable that the von Neumann entropy depends on spectral properties and it can be computed efficiently. The analytical results described here and the numerical computations lead us to conclude that the von Neumann entropy increases under edge addition, increases with the regularity properties of the network and with the number of its connected components. The notion opens the perspective of a wide interface between quantum information theory and the study of complex networks at the statistical level.

scholarly journals On the Mathematical Boundaries of Communication with Zero-Capacity Quantum Channels

10.29007/7h1q ◽  
2018 ◽  
Author(s):  
Laszlo Gyongyosi ◽  
Sandor Imre
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
21St Century ◽  
Quantum Information Theory ◽  
Quantum Mechanical ◽  
Quantum Channels ◽  
Classical Systems ◽  
Quantum Relative Entropy ◽  
Mathematical Properties ◽  
Zero Capacity

In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered. These phenomena are purely quantum mechanical and completely unimaginable in classical systems. Recently, the most important discovery in Quantum Information Theory was the possibility of transmitting quantum information over zero-capacity quantum channels. In this work we prove that the possibility of superactivation of quantum channel capacities is determined by the mathematical properties of the quantum relative entropy function.

scholarly journals Recoverability in quantum information theory

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150338 ◽  
Author(s):  
Mark M. Wilde
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Relative Entropy ◽  
Entropy Inequality ◽  
Quantum Information Theory ◽  
The Other ◽  
Complex Interpolation ◽  
The Core ◽  
Quantum Relative Entropy ◽  
Entropy Inequalities

The fact that the quantum relative entropy is non-increasing with respect to quantum physical evolutions lies at the core of many optimality theorems in quantum information theory and has applications in other areas of physics. In this work, we establish improvements of this entropy inequality in the form of physically meaningful remainder terms. One of the main results can be summarized informally as follows: if the decrease in quantum relative entropy between two quantum states after a quantum physical evolution is relatively small, then it is possible to perform a recovery operation, such that one can perfectly recover one state while approximately recovering the other. This can be interpreted as quantifying how well one can reverse a quantum physical evolution. Our proof method is elementary, relying on the method of complex interpolation, basic linear algebra and the recently introduced Rényi generalization of a relative entropy difference. The theorem has a number of applications in quantum information theory, which have to do with providing physically meaningful improvements to many known entropy inequalities.

Quantum information in communication and imaging

2014 ◽  
Vol 12 (04) ◽  
pp. 1430004 ◽  
Author(s):  
David S. Simon ◽  
Gregg Jaeger ◽  
Alexander V. Sergienko
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Quantum Optics ◽  
Experimental Evaluation ◽  
Theoretical Basis ◽  
Quantum Information Theory ◽  
Entanglement Measures ◽  
The Subject

A brief introduction to quantum information theory in the context of quantum optics is presented. After presenting the fundamental theoretical basis of the subject, experimental evaluation of entanglement measures are discussed, followed by applications to communication and imaging.

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