# Chaotic Maps: Applications to Cryptography and Network Generation for the Graph Laplacian Quantum States

23 Dec 2018-Vol. 307, pp 155-164

TL;DR: The value of maximal Lyapunov exponent of the proposed chaotic map goes beyond 1 and shows more chaotic behaviour than existing one-dimensional chaotic maps, showing that proposed chaotic maps are more effective for cryptographic applications.

Abstract: In this article, we proposed a new chaotic map and is compared with existing chaotic maps such as Logistic map and Tent map. The value of maximal Lyapunov exponent of the proposed chaotic map goes beyond 1 and shows more chaotic behaviour than existing one-dimensional chaotic maps. This shows that proposed chaotic maps are more effective for cryptographic applications. Further, we are using one-dimensional chaotic maps to generate random time series data and define a method to create a network. Lyapunov exponent and entropy of the data are considered to measure the randomness or chaotic behaviour of the time series data. We study the relationship between concurrence (for the two-qubit quantum states) and Lyapunov exponent with respect to initial condition and parameter of the logistic map which is showing how chaos can lead to concurrence based on such Lyapunov exponents.

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TL;DR: Practical application of the generalized Bells theorem in the so-called key distribution process in cryptography is reported, based on the Bohms version of the Einstein-Podolsky-Rosen gedanken experiment andBells theorem is used to test for eavesdropping.

Abstract: Practical application of the generalized Bells theorem in the so-called key distribution process in cryptography is reported. The proposed scheme is based on the Bohms version of the Einstein-Podolsky-Rosen gedanken experiment and Bells theorem is used to test for eavesdropping. © 1991 The American Physical Society.

9,259 citations

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TL;DR: In this article, the authors present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series, which provide a qualitative and quantitative characterization of dynamical behavior.

Abstract: We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow.

8,128 citations

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Williams College

^{1}TL;DR: In this article, an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix was conjectured.

Abstract: The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state $\ensuremath{\rho}$ is the minimum average entanglement of an ensemble of pure states that represents \ensuremath{\rho}. An earlier paper conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula for special states. The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.

6,999 citations

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[...]

TL;DR: In this article, the basic aspects of entanglement including its characterization, detection, distillation, and quantification are discussed, and a basic role of entonglement in quantum communication within distant labs paradigm is discussed.

Abstract: All our former experience with application of quantum theory seems to say:
{\it what is predicted by quantum formalism must occur in laboratory} But the
essence of quantum formalism - entanglement, recognized by Einstein, Podolsky,
Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a
new resource as real as energy This holistic property of compound quantum systems, which involves
nonclassical correlations between subsystems, is a potential for many quantum
processes, including ``canonical'' ones: quantum cryptography, quantum
teleportation and dense coding However, it appeared that this new resource is
very complex and difficult to detect Being usually fragile to environment, it
is robust against conceptual and mathematical tools, the task of which is to
decipher its rich structure This article reviews basic aspects of entanglement including its
characterization, detection, distillation and quantifying In particular, the
authors discuss various manifestations of entanglement via Bell inequalities,
entropic inequalities, entanglement witnesses, quantum cryptography and point
out some interrelations They also discuss a basic role of entanglement in
quantum communication within distant labs paradigm and stress some
peculiarities such as irreversibility of entanglement manipulations including
its extremal form - bound entanglement phenomenon A basic role of entanglement
witnesses in detection of entanglement is emphasized

6,980 citations

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01 Jan 1994

TL;DR: The logistic map, a canonical one-dimensional system exhibiting surprisingly complex and aperiodic behavior, is modeled as a function of its chaotic parameter, and the progression through period-doubling bifurcations to the onset of chaos is considered.

Abstract: We explore several basic aspects of chaos, chaotic systems, and non-linear dynamics through three different setups. The logistic map, a canonical one-dimensional system exhibiting surprisingly complex and aperiodic behavior, is modeled as a function of its chaotic parameter. We consider maps of its phase space, and the progression through period-doubling bifurcations to the onset of chaos. The Feigenbaum ratio of successive bifurcation periods is estimated at 4.674, in good agreement with the accepted value. The Liapunov exponent, governing the exponential growth of small perturbations in chaotic systems, is calculated and its fractal structure compared to the corresponding bifurcation diagram for the logistic map. Using a non-linear p-n junction circuit we analyze the return maps and power spectrums of the resulting time series at various types of system behavior. Similarly, an electronic analog to a ball bouncing on a vertically driven table provides insight into real-world applications of chaotic motion. For both systems we calculate the fractal information dimension and compare with theoretical behavior for dissipative versus Hamiltonian systems. Subject headings: non-linear dynamics; non-linear dynamical systems; fractal dimension; chaos; strange attractors; logistic map

5,372 citations