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.

scholarly journals Golod–Shafarevich-Type Theorems and Potential Algebras

2018 ◽  
Vol 2019 (15) ◽  
pp. 4822-4844 ◽  
Author(s):  
Natalia Iyudu ◽  
Agata Smoktunowicz
Keyword(s):  
Linear Growth ◽  
Complete Classification ◽  
Koszul Complex ◽  
Model Program ◽  
Minimal Model Program ◽  
Infinite Dimensional ◽  
Homogeneous Potential ◽  
Finite Dimensional ◽  
The Difference

Abstract Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree $n\geqslant 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class $\mathcal {P}_{n}$ of potential algebras with homogeneous potential of degree $n+1\geqslant 4$, the minimal Hilbert series is $H_{n}=\frac {1}{1-2t+2t^{n}-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.

scholarly journals COMPLEMENTARITY IN GENERIC OPEN QUANTUM SYSTEMS

2010 ◽  
Vol 24 (24) ◽  
pp. 2485-2509 ◽  
Author(s):  
SUBHASHISH BANERJEE ◽  
R. SRIKANTH
Keyword(s):  
Lower Bound ◽  
Uniform Distribution ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Atomic Systems ◽  
Information Theoretic ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The Difference ◽  
Positive Operator Valued Measure

We develop a unified, information theoretic interpretation of the number-phase complementarity that is applicable both to finite-dimensional (atomic) and infinite-dimensional (oscillator) systems, with number treated as a discrete Hermitian observable and phase as a continuous positive operator valued measure (POVM). The relevant uncertainty principle is obtained as a lower bound on entropy excess, X, the difference between the entropy of one variable, typically the number, and the knowledge of its complementary variable, typically the phase, where knowledge of a variable is defined as its relative entropy with respect to the uniform distribution. In the case of finite-dimensional systems, a weighting of phase knowledge by a factor μ (> 1) is necessary in order to make the bound tight, essentially on account of the POVM nature of phase as defined here. Numerical and analytical evidence suggests that μ tends to 1 as the system dimension becomes infinite. We study the effect of non-dissipative and dissipative noise on these complementary variables for an oscillator as well as atomic systems.

scholarly journals Flattening, squeezing and the existence of random attractors

2006 ◽  
Vol 463 (2077) ◽  
pp. 163-181 ◽  
Author(s):  
Peter E Kloeden ◽  
José A Langa
Keyword(s):  
Dynamical Systems ◽  
Modern Theory ◽  
Qualitative Properties ◽  
Infinite Dimensional ◽  
Squeezing Property ◽  
Finite Dimensional ◽  
Bounded Absorbing Set ◽  
Skew Product Flows ◽  
The Difference ◽  
Autonomous Dynamical Systems

The study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems. In the deterministic case, the properties of flattening and squeezing in infinite-dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor. The flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories. It is shown here that the flattening property is implied by the squeezing property and is in fact weaker, since the attractor in a system with the flattening property can be infinite-dimensional, whereas it is always finite-dimensional in a system with the squeezing property. The flattening property is then generalized to random dynamical systems, for which it is called the pullback flattening property. It is shown to be weaker than the random squeezing property, but equivalent to pullback asymptotic compactness and pullback limit-set compactness, and thus implies the existence of a random attractor. The results are also valid for deterministic non-autonomous dynamical systems formulated as skew-product flows.

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 Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class

1999 ◽  
Vol 67 (2) ◽  
pp. 157-184 ◽  
Author(s):  
A. Caranti ◽  
S. Mattarei
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Loop Algebra ◽  
Maximal Class ◽  
Graded Lie Algebras ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Wide Range ◽  
Graded Lie Algebra

AbstractWe investigate a class of infinite-dimensional, modular, graded Lie algebra in which the homogeneous components have dimension at most two. A subclass of these algebras can be obtained via a twisted loop algebra construction from certain finite-dimensional, simple Lie algebras of Albert-Frank type.Another subclass of these algebras is strictly related to certain graded Lie algebras of maximal class, and exhibits a wide range of behaviours.

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 Review Note on the Applications of Linear Operators in Hilbert Space

2021 ◽  
Author(s):  
Karthic Mohan ◽  
Jananeeswari Narayanamoorthy
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Linear Transformations ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Wide Range ◽  
The Subject ◽  
Adjoint Operators ◽  
User Friendly ◽  
Adjoint Operation

Hilbert Spaces are the closest generalization to infinite dimensional spaces of the Euclidean Spaces. We Consider Linear transformations defined in a normed space and we see that all of them are Continuous if the Space is finite Dimensional Hilbert Space Provide a user-friendly framework for the study of a wide range of subjects from Fourier Analysis to Quantum Mechanics. The adjoint of an Operator is defined and the basic properties of the adjoint operation are established. This allows the introduction of self Adjoint Operators are the subject of the section.

A New Detector System For The HB5 Stem Instrument

1985 ◽  
Vol 43 ◽  
pp. 134-135
Author(s):  
J.M. Cowley
Keyword(s):  
Dark Field ◽  
Detector System ◽  
Electron Holography ◽  
Small Specimen ◽  
Bright Field ◽  
Multiple Images ◽  
Practical Applications ◽  
Wide Range ◽  
Diffraction Patterns ◽  
Operational Modes

The HB5 STEM instrument at ASU has been modified previously to include an efficient two-dimensional detector incorporating an optical analyser device and also a digital system for the recording of multiple images. The detector system was built to explore a wide range of possibilities including in-line electron holography, the observation and recording of diffraction patterns from very small specimen regions (having diameters as small as 3Å) and the formation of both bright field and dark field images by detection of various portions of the diffraction pattern. Experience in the use of this system has shown that sane of its capabilities are unique and valuable. For other purposes it appears that, while the principles of the operational modes may be verified, the practical applications are limited by the details of the initial design.

Comparative Effects of High-Tech Visual Scene Displays and Low-Tech Isolated Picture Symbols on Engagement From Students With Multiple Disabilities

2019 ◽  
Vol 50 (4) ◽  
pp. 693-702 ◽  
Author(s):  
Christine Holyfield ◽  
Sydney Brooks ◽  
Allison Schluterman
Keyword(s):  
Language Learning ◽  
Augmentative And Alternative Communication ◽  
Visual Analysis ◽  
Multiple Disabilities ◽  
Visual Scene ◽  
Future Research ◽  
High Tech ◽  
Single Subject ◽  
Wide Range ◽  
The Difference

Purpose Augmentative and alternative communication (AAC) is an intervention approach that can promote communication and language in children with multiple disabilities who are beginning communicators. While a wide range of AAC technologies are available, little is known about the comparative effects of specific technology options. Given that engagement can be low for beginning communicators with multiple disabilities, the current study provides initial information about the comparative effects of 2 AAC technology options—high-tech visual scene displays (VSDs) and low-tech isolated picture symbols—on engagement. Method Three elementary-age beginning communicators with multiple disabilities participated. The study used a single-subject, alternating treatment design with each technology serving as a condition. Participants interacted with their school speech-language pathologists using each of the 2 technologies across 5 sessions in a block randomized order. Results According to visual analysis and nonoverlap of all pairs calculations, all 3 participants demonstrated more engagement with the high-tech VSDs than the low-tech isolated picture symbols as measured by their seconds of gaze toward each technology option. Despite the difference in engagement observed, there was no clear difference across the 2 conditions in engagement toward the communication partner or use of the AAC. Conclusions Clinicians can consider measuring engagement when evaluating AAC technology options for children with multiple disabilities and should consider evaluating high-tech VSDs as 1 technology option for them. Future research must explore the extent to which differences in engagement to particular AAC technologies result in differences in communication and language learning over time as might be expected.

Sign in / Sign up

Export Citation Format

Share Document