scholarly journals Some Remarks on the Entanglement Number

Quanta ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 22-36
Author(s):  
George Androulakis ◽  
Ryan McGaha
Keyword(s):  
Point Of View ◽  
Entanglement Measure ◽  
Mixed States ◽  
Interesting Point ◽  
Finite Dimensional ◽  
Entanglement Measures ◽  
Pure States ◽  
Tree Representations ◽  
Natural Way ◽  
Finite Dimensional Hilbert Space

Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement.Quanta 2020; 9: 22–36.

scholarly journals A Gramian approach to entanglement in bipartite finite dimensional systems: the case of pure states

2020 ◽  
Vol 20 (13&14) ◽  
pp. 1081-1108
Author(s):  
Roman Gielerak ◽  
Marek Sawerwain
Keyword(s):  
Quantitative Measure ◽  
Point Of View ◽  
Gram Matrix ◽  
Matrix Structure ◽  
Matrix Approach ◽  
Mixed States ◽  
Density Matrices ◽  
Finite Dimensional ◽  
Pure States ◽  
Reduced Density

It has been observed that the reduced density matrices of bipartite qudit pure states possess a Gram matrix structure. This observation has opened a possibility of analysing the entanglement in such systems from the purely geometrical point of view. In particular, a new quantitative measure of an entanglement of the geometrical nature, has been proposed. Using the invented Gram matrix approach, a version of a non-linear purification of mixed states describing the system analysed has been presented.

scholarly journals Construction of genuinely entangled subspacesand the associated bounds on entanglement measures for mixed states

2021 ◽  
Author(s):  
Konstantin Antipin
Keyword(s):  
Information Processing ◽  
Quantum Information ◽  
Lower Bounds ◽  
Quantum Information Processing ◽  
Multipartite Entanglement ◽  
Quantum Channels ◽  
Valuable Resource ◽  
Mixed States ◽  
Entanglement Measures ◽  
Pure States

Abstract Genuine entanglement is the strongest form of multipartite entanglement. Genuinely entangled pure states contain entanglement in every bipartition and as such can be regarded as a valuable resource in the protocols of quantum information processing. A recent direction of research is the construction of genuinely entangled subspaces — the class of subspaces consisting entirely of genuinely entangled pure states. In this paper we present methods of construction of such subspaces including those of maximal possible dimension. The approach is based on the composition of bipartite entangled subspaces and quantum channels of certain types. The examples include maximal subspaces for systems of three qubits, four qubits, three qutrits. We also provide lower bounds on two entanglement measures for mixed states, the concurrence and the convex-roof extended negativity, which are directly connected with the projection on genuinely entangled subspaces.

Connections between relative entropy of entanglement and geometric measure of entanglement

2004 ◽  
Vol 4 (4) ◽  
pp. 252-272
Author(s):  
T.-C. Wei ◽  
M. Ericsson ◽  
P.M. Goldbart ◽  
W.J. Munro
Keyword(s):  
Lower Bound ◽  
Relative Entropy ◽  
Upper Bounds ◽  
Mixed States ◽  
Geometric Measure ◽  
Entanglement Of Formation ◽  
Entanglement Measures ◽  
Pure States ◽  
Relative Entropy Of Entanglement ◽  
Geometric Measure Of Entanglement

As two of the most important entanglement measures---the entanglement of formation and the entanglement of distillation---have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures---the relative entropy of entanglement and the geometric measure of entanglement---are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.

A COMPARATIVE STUDY OF NEGATIVITY AND CONCURRENCE BASED ON SPIN COHERENT STATES

2010 ◽  
Vol 21 (03) ◽  
pp. 291-305 ◽  
Author(s):  
K. BERRADA ◽  
M. El BAZ ◽  
H. ELEUCH ◽  
Y. HASSOUNI
Keyword(s):  
Comparative Study ◽  
Coherent States ◽  
Mixed States ◽  
Qubit System ◽  
Entanglement Measures ◽  
Pure States ◽  
Spin Coherent States

In this paper, we investigate two different entanglement measures, the negativity and concurrence, in the case of pure and mixed states of two-qubit system basing on the spin coherent states. For two-qubit pure states, the negativity is the same as the concurrence. For mixed states, using a simplified expression of concurrence in Wootters' measure of entanglement, we write the bounds of Verstraete et al.1as a function of some new parameters and we compare the both measures for a class of mixed states.

scholarly journals Wave–Particle–Entanglement–Ignorance Complementarity for General Bipartite Systems

Entropy ◽  
2020 ◽  
Vol 22 (8) ◽  
pp. 813
Author(s):  
Wei Wu ◽  
Jin Wang
Keyword(s):  
Experimental Investigations ◽  
Entanglement Measure ◽  
Mixed States ◽  
Practical Applications ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Bipartite System ◽  
Wave Particle Duality ◽  
Wide Range ◽  
The Difference

Wave–particle duality as the defining characteristic of quantum objects is a typical example of the principle of complementarity. The wave–particle–entanglement (WPE) complementarity, initially developed for two-qubit systems, is an extended form of complementarity that combines wave–particle duality with a previously missing ingredient, quantum entanglement. For two-qubit systems in mixed states, the WPE complementarity was further completed by adding yet another piece that characterizes ignorance, forming the wave–particle–entanglement–ignorance (WPEI) complementarity. A general formulation of the WPEI complementarity can not only shed new light on fundamental problems in quantum mechanics, but can also have a wide range of experimental and practical applications in quantum-mechanical settings. The purpose of this study is to establish the WPEI complementarity for general multi-dimensional bipartite systems in pure or mixed states, and extend its range of applications to incorporate hierarchical and infinite-dimensional bipartite systems. The general formulation is facilitated by well-motivated generalizations of the relevant quantities. When faced with different directions of extensions to take, our guiding principle is that the formulated complementarity should be as simple and powerful as possible. We find that the generalized form of the WPEI complementarity contains unequal-weight averages reflecting the difference in the subsystem dimensions, and that the tangle, instead of the squared concurrence, serves as a more suitable entanglement measure in the general scenario. Two examples, a finite-dimensional bipartite system in mixed states and an infinite-dimensional bipartite system in pure states, are studied in detail to illustrate the general formalism. We also discuss our results in connection with some previous work. The WPEI complementarity for general finite-dimensional bipartite systems may be tested in multi-beam interference experiments, while the second example we studied may facilitate future experimental investigations on complementarity in infinite-dimensional bipartite systems.

Superselection Sectors, Pure States, Bose and Fermi Distributions by Completely Positive Quantum-Motions

2005 ◽  
Vol 12 (01) ◽  
pp. 23-35 ◽  
Author(s):  
Klaus Dietz
Keyword(s):  
Hilbert Spaces ◽  
Mixed States ◽  
Completely Positive ◽  
Absolute Value ◽  
Zero Modes ◽  
Finite Dimensional ◽  
Constants Of Motion ◽  
Pure States ◽  
Damped Oscillator

The connection of the operators V, building up the Kossakowski-Lindblad generator, with the asymptotic states of the corresponding completely positive quantum-maps is discussed. Maps leading to decoherence are constructed, the importance of zero-modes in the absolute value [Formula: see text] of V for the generation of pure states from arbitrary mixed states is illustrated. The universal rôle of equipartite states appears when unitary V are chosen. The 'damped oscillator model' is generalized to yield Bose and Fermi distributions as asymptotic states for systems described by a Hamiltonian and other constants of motion. Calculations are performed in finite dimensional Hilbert spaces.

scholarly journals MIXING AND DECOHERENCE TO NEAREST SEPARABLE STATES

2009 ◽  
Vol 07 (04) ◽  
pp. 829-846
Author(s):  
AVIJIT LAHIRI ◽  
GAUTAM GHOSH ◽  
SANKHASUBHRA NAG
Keyword(s):  
Quantum Correlation ◽  
Entangled States ◽  
Entanglement Measure ◽  
Measuring Apparatus ◽  
Separable State ◽  
Mixed States ◽  
Classical Correlation ◽  
Separable States ◽  
System A ◽  
Pure States

We consider a class of entangled states of a quantum system (S) and a second system (A) where pure states of the former are correlated with mixed states of the latter, and work out the entanglement measure with reference to the nearest separable state. Such "pure-mixed" entanglement is expected when the system S interacts with a macroscopic measuring apparatus in a quantum measurement, where the quantum correlation is destroyed in the process of environment-induced decoherence whereafter only the classical correlation between S and A remains, the latter being large compared to the former. We present numerical evidence that the entangled S–A state drifts towards the nearest separable state through decoherence, with an additional tendency of equimixing among relevant groups of apparatus states.

scholarly journals Distribution of entanglement and correlations in all finite dimensions

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 64 ◽  
Author(s):  
Christopher Eltschka ◽  
Jens Siewert
Keyword(s):  
Conservation Laws ◽  
Pure State ◽  
Body Problem ◽  
Particle System ◽  
A Priori ◽  
Necessary Condition ◽  
Many Body ◽  
Mixed States ◽  
Finite Dimensional ◽  
Entanglement Measures

The physics of a many-particle system is determined by the correlations in its quantum state. Therefore, analyzing these correlations is the foremost task of many-body physics. Any 'a priori' constraint for the properties of the global vs. the local states-the so-called marginals-would help in order to narrow down the wealth of possible solutions for a given many-body problem, however, little is known about such constraints. We derive an equality for correlation-related quantities of any multipartite quantum system composed of finite-dimensional local parties. This relation defines a necessary condition for the compatibility of the marginal properties with those of the joint state. While the equality holds both for pure and mixed states, the pure-state version containing only entanglement measures represents a fully general monogamy relation for entanglement. These findings have interesting implications in terms of conservation laws for correlations, and also with respect to topology.

ENTANGLEMENT MEASURE OF BIPARTITE SYSTEM STATES

2010 ◽  
Vol 07 (06) ◽  
pp. 1051-1064 ◽  
Author(s):  
K. BERRADA ◽  
Y. HASSOUNI
Keyword(s):  
Coherent States ◽  
Linear Entropy ◽  
Entanglement Measure ◽  
Mixed States ◽  
Maximal Entanglement ◽  
Bipartite System ◽  
Pure States ◽  
System States ◽  
Entangled Coherent States

Linear entropy as a measure of entanglement is applied to explain conditions for minimal and maximal entanglement of bipartite nonorthogonal pure states. We formulate this measure in terms of the amplitudes of coherent states in the case of entangled coherent states and calculate the conditions. We generalize this formalism to the case of bipartite mixed states and show that the entanglement measure is also a function of the probabilities.

scholarly journals Classification of all alternatives to the Born rule in terms of informational properties

Quantum ◽  
2017 ◽  
Vol 1 ◽  
pp. 15 ◽  
Author(s):  
Thomas D. Galley ◽  
Lluis Masanes
Keyword(s):  
Quantum Theory ◽  
Two Dimensional ◽  
Born Rule ◽  
Mixed States ◽  
Structure And Dynamics ◽  
Finite Dimensional ◽  
Information Causality ◽  
Pure States ◽  
Probabilistic Theories

The standard postulates of quantum theory can be divided into two groups: the first one characterizes the structure and dynamics of pure states, while the second one specifies the structure of measurements and the corresponding probabilities. In this work we keep the first group of postulates and characterize all alternatives to the second group that give rise to finite-dimensional sets of mixed states. We prove a correspondence between all these alternatives and a class of representations of the unitary group. Some features of these probabilistic theories are identical to quantum theory, but there are important differences in others. For example, some theories have three perfectly distinguishable states in a two-dimensional Hilbert space. Others have exotic properties such as lack of bit symmetry, the violation of no simultaneous encoding (a property similar to information causality) and the existence of maximal measurements without phase groups. We also analyze which of these properties single out the Born rule.

Sign in / Sign up

Export Citation Format

Share Document