scholarly journals JENSEN–SHANNON DIVERGENCE AS A MEASURE OF THE DEGREE OF ENTANGLEMENT

2008 ◽  
Vol 06 (supp01) ◽  
pp. 715-720 ◽  
Author(s):  
A. P. MAJTEY ◽  
A. BORRAS ◽  
M. CASAS ◽  
P. W. LAMBERTI ◽  
A. PLASTINO
Keyword(s):  
Information Theory ◽  
Hilbert Space ◽  
Quantum Information ◽  
Quantum Information Theory ◽  
Quantum States ◽  
Jensen Shannon Divergence ◽  
Degree Of Entanglement

The notion of distance in Hilbert space is relevant in many scenarios. In particular, "distances" between quantum states play a central role in quantum information theory. An appropriate measure of distance is the quantum Jensen Shannon divergence (QJSD) between quantum states. Here we study this distance as a geometrical measure of entanglement and apply it to different families of states.

scholarly journals IMPOSSIBILITY OF PARTIAL SWAPPING OF QUANTUM INFORMATION

2007 ◽  
Vol 05 (04) ◽  
pp. 605-609 ◽  
Author(s):  
I. CHAKRABARTY
Keyword(s):  
Information Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Higher Dimensions ◽  
Quantum Information Theory ◽  
Quantum States ◽  
Two Dimensional ◽  
Bloch Sphere ◽  
Two Parameters

It is a well–known fact that a quantum state |ψ(θ, ϕ)〉 is represented by a point on the Bloch sphere, characterized by two parameters θ and ϕ. Here in this work, we find out another impossible operation in quantum information theory. We name this impossibility as 'Impossibility of partial swapping of quantum information'. By this we mean that if two unknown quantum states are given at the input port, there exists no physical process, consistent with the principles of quantum mechanics, by which we can partially swap either of the two parameters θ and ϕ between these two quantum states. In this work, we provided the impossibility proofs for the qubits (i.e. the quantum states taken from two-dimensional Hilbert space) and this impossible operation can be shown to hold in higher dimensions also.

scholarly journals NATURAL METRIC FOR QUANTUM INFORMATION THEORY

2009 ◽  
Vol 07 (05) ◽  
pp. 1009-1019 ◽  
Author(s):  
P. W. LAMBERTI ◽  
M. PORTESI ◽  
J. SPARACINO
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Quantum Information Theory ◽  
Quantum States ◽  
Square Root ◽  
Mixed States ◽  
Basic Properties ◽  
Illustrative Application

We study in detail a very natural metric for quantum states. This new proposal has two basic ingredients: entropy and purification. The metric for two mixed states is defined as the square root of the entropy of the average of representative purifications of those states. Some basic properties are analyzed and its relation to other distances is investigated. As an illustrative application, the proposed metric is evaluated for one-qubit mixed states.

Thresholds for reduction-related entanglement criteria in quantum information theory

2015 ◽  
Vol 15 (13&14) ◽  
pp. 1165-1184
Author(s):  
Maria A. Jivulescu ◽  
Nicolae Lupa ◽  
Ion Nechita
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Quantum State ◽  
Random Matrix ◽  
Matrix Theory ◽  
Quantum Information Theory ◽  
Quantum States ◽  
Absolute Reduction ◽  
Open Questions ◽  
Pure Quantum State

We consider random bipartite quantum states obtained by tracing out one subsystem from a random, uniformly distributed, tripartite pure quantum state. We compute thresholds for the dimension of the system being traced out, so that the resulting bipartite quantum state satisfies the reduction criterion in different asymptotic regimes. We consider as well the basis-independent version of the reduction criterion (the absolute reduction criterion), computing thresholds for the corresponding eigenvalue sets. We do the same for other sets relevant in the study of absolute separability, using techniques from random matrix theory. Finally, we gather and compare the known values for the thresholds corresponding to different entanglement criteria, and conclude with a list of open questions.

scholarly journals Computable constraints on entanglement-sharing of multipartite quantum states

2010 ◽  
Vol 10 (3&4) ◽  
pp. 223-238
Author(s):  
Y.-C. Ou ◽  
M.S. Byrd
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Quantum Information Theory ◽  
Quantum States ◽  
Many Body ◽  
Pure States ◽  
Arbitrary Dimensions ◽  
Many Body Systems ◽  
Bipartite States ◽  
Entanglement Sharing

\Negativity is regarded as an important measure of entanglement in quantum information theory. In contrast to other measures of entanglement, it is easily computable for bipartite states in arbitrary dimensions. In this paper, based on the negativity and realignment, we provide a set of entanglement-sharing constraints for multipartite states, where the entanglement is not necessarily limited to bipartite and pure states, thus aiding in the quantification of constraints for entanglement-sharing. These may find applications in studying many-body systems.

scholarly journals SINGULAR VALUE DECOMPOSITION AND MATRIX REORDERINGS IN QUANTUM INFORMATION THEORY

2011 ◽  
Vol 22 (09) ◽  
pp. 897-918 ◽  
Author(s):  
JAROSŁAW ADAM MISZCZAK
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Singular Value Decomposition ◽  
Computing System ◽  
Quantum Information Theory ◽  
Singular Value ◽  
Quantum States ◽  
Quantum Channels ◽  
Basic Functions ◽  
Value Decomposition

We review Schmidt and Kraus decompositions in the form of singular value decomposition using operations of reshaping, vectorization and reshuffling. We use the introduced notation to analyze the correspondence between quantum states and operations with the help of Jamiołkowski isomorphism. The presented matrix reorderings allow us to obtain simple formulae for the composition of quantum channels and partial operations used in quantum information theory. To provide examples of the discussed operations, we utilize a package for the Mathematica computing system implementing basic functions used in the calculations related to quantum information theory.

scholarly journals A general formulation based on algebraic spinors for the quantum computation

2020 ◽  
Vol 17 (14) ◽  
pp. 2050206
Author(s):  
Marco A. S. Trindade ◽  
Sergio Floquet ◽  
J. David M. Vianna
Keyword(s):  
Information Theory ◽  
Hilbert Space ◽  
Quantum Information ◽  
Quantum Computation ◽  
Quantum Teleportation ◽  
Clifford Algebras ◽  
Quantum Information Theory ◽  
Quantum Gates ◽  
Entangled States ◽  
Space Formulation

In this work, we explore the structure of Clifford algebras and the representations of the algebraic spinors in quantum information theory. Initially, we present a general formulation through elements of minimal left ideals in tensor products of Clifford algebras. Posteriorly, we perform some applications in quantum computation: qubits, entangled states, quantum gates, representations of the braid group, quantum teleportation, Majorana operators and supersymmetry. Finally, we discuss advantages compared to standard Hilbert space formulation.

scholarly journals Some applications of the Hermit-Hadamard inequality for log-convex functions in quantum divergence

2021 ◽  
Author(s):  
Fatemeh Hassanzad ◽  
Hossien Mehri-Dehnavi ◽  
Hamzeh Agahi
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Relative Entropy ◽  
Convex Functions ◽  
Quantum Information Theory ◽  
Density Matrices ◽  
Hadamard Inequality ◽  
Quantum Relative Entropy ◽  
Nonlinear Anal ◽  
Jensen Shannon Divergence

One of the beautiful and very simple inequalities for a convex function is the Hermit-Hadamard inequality [S. Mehmood, et. al. Math. Methods Appl. Sci., 44 (2021) 3746], [S. Dragomir, et. al., Math. Methods Appl. Sci., in press]. The concept of log-convexity is a stronger property of convexity. Recently, the refined Hermit-Hadamard’s inequalities for log-convex functions were introduced by researchers [C. P. Niculescu, Nonlinear Anal. Theor., 75 (2012) 662]. In this paper, by the Hermit-Hadamard inequality, we introduce two parametric Tsallis quantum relative entropy, two parametric Tsallis-Lin quantum relative entropy and two parametric quantum Jensen-Shannon divergence in quantum information theory. Then some properties of quantum Tsallis-Jensen-Shannon divergence for two density matrices are investigated by this inequality. \newline \textbf{Keywords:} \textit{ Hermit-Hadamard’s inequality; log-convexity; Density matrices; Quantum relative entropy; Tsallis quantum relative entropy; quantum Jensen-Shannon divergence divergence.

scholarly journals A Knot Theoretic Extension of the Bloch Sphere Representation for Qubits in Hilbert Space and Its Application to Contextuality and Many-Worlds Theories

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1135
Author(s):  
Stefan Heusler ◽  
Paul Schlummer ◽  
Malte S. Ubben
Keyword(s):  
Information Theory ◽  
Hilbert Space ◽  
Quantum Information ◽  
Knot Theory ◽  
Quantum Physics ◽  
Quantum Information Theory ◽  
Entangled States ◽  
Theoretic Model ◽  
Bloch Sphere ◽  
Many Worlds

We argue that the usual Bloch sphere is insufficient in various aspects for the representation of qubits in quantum information theory. For example, spin flip operations with the quaternions I J K = e 2 π i 2 = − 1 and J I K = + 1 cannot be distinguished on the Bloch sphere. We show that a simple knot theoretic extension of the Bloch sphere representation is sufficient to track all unitary operations for single qubits. Next, we extend the Bloch sphere representation to entangled states using knot theory. As applications, we first discuss contextuality in quantum physics—in particular the Kochen-Specker theorem. Finally, we discuss some arguments against many-worlds theories within our knot theoretic model of entanglement. The key ingredients of our approach are symmetries and geometric properties of the unitary group.

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