SEPARABILITY: A NEW APPROACH FROM THE CONIC STRUCTURE OF POSITIVE OPERATORS

2011 ◽  
Vol 09 (supp01) ◽  
pp. 415-422
Author(s):  
D. SALGADO ◽  
J. L. SÁNCHEZ-GÓMEZ ◽  
M. FERRERO
Keyword(s):  
Pure State ◽  
Convex Combination ◽  
Quantum States ◽  
Necessary Condition ◽  
New Approach ◽  
Product Matrix ◽  
Separability Problem ◽  
Multipartite System ◽  
Arbitrary Dimensions ◽  
Bipartite Case

We exploit the cone structure of unnormalized quantum states to reformulate the separability problem. Firstly a convex combination of every quantum state ρ in terms of a state Cρ with the same rank and another one Eρ with lower rank is perfomed, with weights 1 − λρ and λρ, respectively. Secondly a scalar [Formula: see text] is computed. Then ρ is separable if, and only if, [Formula: see text]. The computation of [Formula: see text] has been undergone under the simplest choice for Cρ as a product matrix and Eρ being a pure state, valid for any bipartite and multipartite system in arbitrary dimensions. A necessary condition is also formulated when Eρ is not pure in the bipartite case.

Separability criteria based on the Bloch representation of density matrices

2007 ◽  
Vol 7 (7) ◽  
pp. 624-638
Author(s):  
J. de Vicente
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Pure Product ◽  
Separability Problem ◽  
Arbitrary Dimensions ◽  
Necessary And Sufficient ◽  
Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

scholarly journals CONTRADICTION TO THE TABLE OF D. I. MENDELEEV AND THEIR ELIMINATION

2021 ◽  
Vol 2 (446) ◽  
pp. 14-21
Author(s):  
N.К. Akhmetov ◽  
G.U. Ilyasova ◽  
S. K. Kazybekova
Keyword(s):  
Quantum Number ◽  
Periodic Table ◽  
Chemical Elements ◽  
Principal Quantum Number ◽  
Quantum States ◽  
Outer Electron ◽  
New Approach ◽  
Electronic Configurations ◽  
New Formula ◽  
Electron Shells

The article discusses a new approach to the formation of periods of the Periodic Table of Mendeleev. With the help of the new formula and the first proposed quantum states of the outer electron shells of atoms of chemical elements, the periods of the periodic table are reformatted. It is supposed to reduce the number of periods in the table by introducing the corresponding sub-periods. This is confirmed by the material given in the article. The following description of the order of formation of electron layers is proposed: the principal quantum number (n), then the newly proposed quantum states of electrons («first» and «second»), which in turn constitute the electronic configurations of sub-periods in periods, and only then the remaining quantum orbitals (s, p, d and f).

ONE OF THE POSSIBLE APPROACHES FOR DESIGNING A HOB CUTTER FOR MACHINING A SMALL CYCLOIDAL CONE

2021 ◽  
pp. 97-103
Author(s):  
В.В. Куц ◽  
А.А. Панин ◽  
Д.Н. Тютюнов ◽  
К.В. Жилина
Keyword(s):  
Gear Wheel ◽  
Front Surface ◽  
Necessary Condition ◽  
Machining Accuracy ◽  
Small Range ◽  
New Approach ◽  
Main Requirement ◽  
Number Of Factors ◽  
Gear Wheels ◽  
The Cost

Предлагается краткий обзор промышленного производства червячных фрез. Показано, что повышение качества и производительности изготовления зубчатых колес является необходимым условием снижения себестоимости и расширения объемов производства зубчатых колес на отечественных предприятиях. Главным требованием, предъявляемым к зубьям данной фрезы, является то, чтобы в результате заточки по передней поверхности, которая лежит в осевой плоскости фрезы, профиль зубьев сохранялся до почти полного их износа. Поэтому особое внимание уделяется выбору кривой затылования с учётом целого ряда факторов, способствующих совершенствованию процесса обработки. Отмечено, что затылование имеет ряд преимуществ, в сравнении с острой заточкой фрез. Изложен новый подход к проектированию рабочей оснастки для обработки малых колес циклоидной передачи и исследованы теоретически допустимые интервалы изменения задних углов при затыловании. Установлено, что несмотря на преимущества циклоидальной фрезы перед другими типами затылованных фрез в скорости и точности обработки, она имеет один недостаток - довольно малый промежуток применимости на дуге циклоиды. На основе существующих подходов разработан вариант затылования зубьев червячной фрезы по циклоиде The article provides an overview of the industrial production of hob cutters. We show that improving the quality and productivity of gear wheel manufacturing is a necessary condition for reducing the cost and expanding the production of gear wheels at domestic enterprises. The main requirement for the teeth of this cutter is that, as a result of sharpening on the front surface, which lies in the axial plane of the cutter, the profile of the teeth remains sharp until they are almost completely worn out. Therefore, we paid special attention to the choice of the relief curve, taking into account a number of factors that contribute to the improvement of the processing process. We note that relief has a number of advantages in comparison with sharpening of cutters. We give a new approach to the design of working equipment for processing small cycloidal wheels and investigate the theoretically permissible intervals of variation of the rear angles during relief. We established that despite the advantages of a cycloidal cutter over other types of undercut cutters in terms of speed and machining accuracy, it has one drawback - a rather small range of applicability on the cycloid arc. On the basis of existing approaches, we developed a variant of the relief of the teeth of the worm cutter along the cycloid

scholarly journals Pure States of Maximum Uncertainty with Respect to a Given POVM

2020 ◽  
Vol 27 (01) ◽  
pp. 2050002
Author(s):  
Anna Szymusiak
Keyword(s):  
Shannon Entropy ◽  
Pure State ◽  
Quantum States ◽  
Maximal Amount ◽  
Dimension 2 ◽  
Pure States

One of the differences between classical and quantum world is that in the former we can always perform a measurement that gives certain outcomes for all pure states, while such a situation is not possible in the latter one. The degree of randomness of the distribution of the measurement outcomes can be quantified by the Shannon entropy. While it is well known that this entropy, as a function of quantum states, needs to be minimized by some pure states, we would like to address the question how ‘badly’ can we end by choosing initially any pure state, i.e., which pure states produce the maximal amount of uncertainty under given measurement. We find these maximizers for all highly symmetric POVMs in dimension 2, and for all SIC-POVMs in any dimension.

Characterization of Extreme Points of Multi-Stochastic Tensors

2016 ◽  
Vol 16 (3) ◽  
pp. 459-474 ◽  
Author(s):  
Rihuan Ke ◽  
Wen Li ◽  
Mingqing Xiao
Keyword(s):  
Operation Research ◽  
Convex Combination ◽  
Sufficient Conditions ◽  
Extreme Points ◽  
Stochastic Matrices ◽  
Third Order ◽  
New Approach ◽  
3 Dimensional ◽  
Necessary And Sufficient

AbstractStochastic matrices play an important role in the study of probability theory and statistics, and are often used in a variety of modeling problems in economics, biology and operation research. Recently, the study of tensors and their applications became a hot topic in numerical analysis and optimization. In this paper, we focus on studying stochastic tensors and, in particular, we study the extreme points of a set of multi-stochastic tensors. Two necessary and sufficient conditions for a multi-stochastic tensor to be an extreme point are established. These conditions characterize the “generators” of multi-stochastic tensors. An algorithm to search the convex combination of extreme points for an arbitrary given multi-stochastic tensor is developed. Based on our obtained results, some expression properties for third-order and n-dimensional multi-stochastic tensors (${n=3}$ and 4) are derived, and all extreme points of 3-dimensional and 4-dimensional triply-stochastic tensors can be produced in a simple way. As an application, a new approach for the partially filled square problem under the framework of multi-stochastic tensors is given.

scholarly journals THE CHSH-TYPE INEQUALITIES FOR INFINITE-DIMENSIONAL QUANTUM SYSTEMS

2013 ◽  
Vol 27 (21) ◽  
pp. 1350151
Author(s):  
YU GUO
Keyword(s):  
Pure State ◽  
Bell Inequalities ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Pure States ◽  
Sufficient And Necessary Condition

By establishing CHSH operators and CHSH-type inequalities, we show that any entangled pure state in infinite-dimensional systems is entangled in a 2⊗2 subspace. We find that, for infinite-dimensional systems, the corresponding properties are similar to that of the two-qubit case: (i) The CHSH-type inequalities provide a sufficient and necessary condition for separability of pure states; (ii) The CHSH operators satisfy the Cirel'son inequalities; (iii) Any state which violates one of these Bell inequalities is distillable.

scholarly journals Concurrence of Mixed Bipartite Quantum States in Arbitrary Dimensions

2004 ◽  
Vol 92 (16) ◽  
Author(s):  
Florian Mintert ◽  
Marek Kuś ◽  
Andreas Buchleitner
Keyword(s):  
Quantum States ◽  
Arbitrary Dimensions

scholarly journals Experimental study of quantum coherence decomposition and trade-off relations in a tripartite system

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Zhe Ding ◽  
Ran Liu ◽  
Chandrashekar Radhakrishnan ◽  
Wenchao Ma ◽  
Xinhua Peng ◽  
...  
Keyword(s):  
Experimental Study ◽  
Nuclear Spin ◽  
Quantum Coherence ◽  
Quantum Systems ◽  
Spin Systems ◽  
Quantum States ◽  
Product State ◽  
Trade Off ◽  
Global Coherence ◽  
Multipartite System

AbstractQuantum coherence is the most fundamental of all quantum quantifiers, underlying other well-known quantities such as entanglement. It can be distributed in a multipartite system in various ways—for example, in a bipartite system it can exist within subsystems (local coherence) or collectively between the subsystems (global coherence), and exhibits a trade-off relation. In this paper, we experimentally verify these coherence trade-off relations in adiabatically evolved nuclear spin systems using an NMR spectrometer. We study the full set of coherence trade-off relations between the original state, the bipartite product state, the tripartite product state, and the decohered product state. We also experimentally verify the monogamy inequality and show that both the quantum systems are polygamous during the evolution. We find that the properties of the state in terms of coherence and monogamy are equivalent. This illustrates the utility of using coherence as a characterization tool for quantum states.

A new geometrical method for portfolio optimization

2021 ◽  
Vol 8 (3) ◽  
pp. 400-409
Author(s):  
F. Butin ◽  
Keyword(s):  
Portfolio Optimization ◽  
Euclidean Distance ◽  
Convex Combination ◽  
Short Selling ◽  
Minimal Risk ◽  
Geometrical Method ◽  
New Approach ◽  
Optimization Framework ◽  
The One ◽  
Better Than

Risk aversion plays a significant and central role in investors’ decisions in the process of developing a portfolio. In this portfolio optimization framework, we determine the portfolio that possesses the minimal risk by using a new geometrical method. For this purpose, we elaborate an algorithm that enables us to compute any Euclidean distance to a standard simplex. With this new approach, we can treat the case of portfolio optimization without short-selling in its entirety, and we also recover in geometrical terms the well-known results on portfolio optimization with allowed short-selling. Then, we apply our results to determine which convex combination of the CAC 40 stocks possesses the lowest risk. Thus, we not only obtain a very low risk compared to the index, but we also get a rate of return that is almost three times better than the one of the index.

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