scholarly journals Convexity of momentum map, Morse index, and quantum entanglement

2014 ◽  
Vol 26 (03) ◽  
pp. 1450004 ◽  
Author(s):  
Adam Sawicki ◽  
Michał Oszmaniec ◽  
Marek Kuś
Keyword(s):  
Morse Index ◽  
Hausdorff Space ◽  
Quantum Systems ◽  
Total Variance ◽  
Geometric Invariant ◽  
Classical Communication ◽  
Critical Set ◽  
Momentum Map ◽  
Critical Sets

We analyze from the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a non-Hausdorff space. Using geometric invariant theory (GIT) and momentum map geometry, we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many "asymptotically" equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well-defined measure of entanglement) of the state Var [v] attains maximum. We provide an algorithm for finding critical sets of Var [v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using some developments in the momentum map geometry known as the Kirwan–Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of non-SLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCC-invariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning.

A resource-based view of quantum information

2004 ◽  
Vol 4 (6&7) ◽  
pp. 460-466
Author(s):  
C.H. Bennett
Keyword(s):  
Quantum Information ◽  
Source Coding ◽  
Adequate Description ◽  
Tripartite Entanglement ◽  
Classical Communication ◽  
Resource Based View ◽  
Open Questions ◽  
Classical Information Theory

We survey progress in understanding quantum information in terms of equivalences, reducibilities, and asymptotically achievable rates for transformations among nonlocal resources such as classical communication, entanglement, and particular quantum states or channels. In some areas, eg source coding, there are straightforward parallels to classical information theory; in others eg entanglement-assisted communication, new effects and tradeoffs appear that are beginning to be fairly well understood, or the remaining uncertainty has become focussed on a few simple open questions, such as conjectured additivity of the Holevo capacity. In still other areas, e.g. the role of the back communication and the classification of tripartite entanglement, much remains unknown, and it appears unlikely that an adequate description exists in terms of a finite number of resources.

Decoherence, Classical Properties and Entanglement of Quantum Systems

2001 ◽  
Vol 56 (1-2) ◽  
pp. 124-127
Author(s):  
Ph. Blanchard ◽  
R. Olkiewicz
Keyword(s):  
Environmental Monitoring ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Open Quantum ◽  
Pure States

AbstractWe discuss the properties of decoherence and its role in the appearance of classical properties in open quantum systems. In particular, it is used for classification of pure states with respect to their ability to persist despite the environmental monitoring.

Two remarks about Sudoku squares

2006 ◽  
Vol 90 (519) ◽  
pp. 425-430 ◽  
Author(s):  
A. D. Keedwell
Keyword(s):  
Current Problem ◽  
Latin Square ◽  
Latin Squares ◽  
Critical Set ◽  
Critical Sets ◽  
The Given ◽  
A Current ◽  
Defining Sets

Smallest defining setsA standard Sudoku square is a 9 × 9 latin square in which each of the nine 3 × 3 subsquares into which it can be separated contains each of the integers 1 to 9 exactly once.A current problem is to complete such a square when only some of the cells have been filled. These cells are often called ‘givens’. (Such problems are currently (2005) published daily in British newspapers.) In more mathematical terms, the given filled cells constitute a defining set or uniquely completable set for the square if they lead to a unique completion of the square. If, after deletion of any one of these givens, the square can no longer be completed uniquely, the givens form a critical set. The investigation of critical sets for ‘ordinary’ latin squares is a topic of current mathematical interest. (See [1] for more details.)

scholarly journals The tropical momentum map: a classification of toric log symplectic manifolds

2016 ◽  
Vol 367 (3-4) ◽  
pp. 1217-1258 ◽  
Author(s):  
Marco Gualtieri ◽  
Songhao Li ◽  
Álvaro Pelayo ◽  
Tudor S. Ratiu
Keyword(s):  
Symplectic Manifolds ◽  
Momentum Map

scholarly journals Classification of the attacks on quantum systems for the transfer of confidential data

2018 ◽  
Vol 23 (3) ◽  
Author(s):  
Ігор Валерійович Лімарь ◽  
Євген Вікторович Васіліу ◽  
Олександр Миколайович Рябуха
Keyword(s):  
Quantum Systems ◽  
Confidential Data

scholarly journals Interplay of symmetry and topology in science

2014 ◽  
Vol 70 (a1) ◽  
pp. C1419-C1419
Author(s):  
Boris Zhilinskii
Keyword(s):  
Classical System ◽  
Quantum Systems ◽  
Qualitative Theory ◽  
Topological Invariant ◽  
Particle Systems ◽  
Qualitative Description ◽  
Energy Bands ◽  
And Topology ◽  
Finite Particle

Qualitative methods in natural science are based mainly on simultaneous use of symmetry and topology arguments. The idea of the present talk is to demonstrate how the corresponding mathematical tools (based on symmetry and topology arguments) initially applied to describe classification of different phases of matter and transitions between them are extended to construct qualitative theory of finite particle systems and more general dynamical systems. I start with reminding basic notions and tools associated with application of group action ideas to physics as initiated and developed by Louis Michel (1923-1999) [1,2]. Then geometric combinatorial and topological ideas are used to give qualitative description of singularities of dynamical integrable classical system and their quantum analogs. Quantum monodromy and its various generalizations as well as description of energy bands of isolated finite particle quantum systems in terms of topological invariant, Chern number [3], will be discussed on concrete molecular and atomic examples.

scholarly journals Application of Spectroscopic UV-Vis and FT-IR Screening Techniques Coupled with Multivariate Statistical Analysis for Red Wine Authentication: Varietal and Vintage Year Discrimination

Molecules ◽  
2019 ◽  
Vol 24 (22) ◽  
pp. 4166 ◽  
Author(s):  
Elisabeta-Irina Geană ◽  
Corina Teodora Ciucure ◽  
Constantin Apetrei ◽  
Victoria Artem
Keyword(s):  
Discriminant Analysis ◽  
Principal Component ◽  
Total Variance ◽  
Red Wines ◽  
Linear Discriminant ◽  
Vis Spectroscopy ◽  
Ft Ir ◽  
Wine Authentication ◽  
Analytical Approaches

One of the most important issues in the wine sector and prevention of adulterations of wines are discrimination of grape varieties, geographical origin of wine, and year of vintage. In this experimental research study, UV-Vis and FT-IR spectroscopic screening analytical approaches together with chemometric pattern recognition techniques were applied and compared in addressing two wine authentication problems: discrimination of (i) varietal and (ii) year of vintage of red wines produced in the same oenological region. UV-Vis and FT-IR spectra of red wines were registered for all the samples and the principal features related to chemical composition of the samples were identified. Furthermore, for the discrimination and classification of red wines a multivariate data analysis was developed. Spectral UV-Vis and FT-IR data were reduced to a small number of principal components (PCs) using principal component analysis (PCA) and then partial least squares discriminant analysis (PLS-DA) and linear discriminant analysis (LDA) were performed in order to develop qualitative classification and regression models. The first three PCs used to build the models explained 89% of the total variance in the case of UV-Vis data and 98% of the total variance for FR-IR data. PLS-DA results show that acceptable linear regression fits were observed for the varietal classification of wines based on FT-IR data. According to the obtained LDA classification rates, it can be affirmed that UV-Vis spectroscopy works better than FT-IR spectroscopy for the discrimination of red wines according to the grape variety, while classification of wines according to year of vintage was better for the LDA based FT-IR data model. A clear discrimination of aged wines (over six years) was observed. The proposed methodologies can be used as accessible tools for the wine identity assurance without the need for costly and laborious chemical analysis, which makes them more accessible to many laboratories.

To Classify the Groundwater Hydrographs by Factor Analysis Method

2012 ◽  
Vol 518-523 ◽  
pp. 4097-4103
Author(s):  
Hwa Sheng Gau ◽  
Chung Yi Chung ◽  
Shao Wei Liao ◽  
Wen Liang Lai
Keyword(s):  
Factor Analysis ◽  
Water Table ◽  
Unconfined Aquifer ◽  
Groundwater Table ◽  
Total Variance ◽  
Analysis Method ◽  
Pingtung Plain ◽  
Table Data ◽  
Three Components

This study is using Factor Analysis method to analyze the hydrographs of groundwater table for classification of recharge zone. The water table data are taken from 37 wells which located on unconfined aquifer in Pingtung plain. The result shows that 93% of total variance can be explained by three components. The 1th component is related to rainfall; the 2th and 3th are related to recharge from TungKang Basin and Kaoping basin, respectively.

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