scholarly journals Separability criterion for multipartite quantum states based on the Bloch representation of density matrices

2008 ◽  
Vol 8 (8&9) ◽  
pp. 773-790
Author(s):  
A.S.M. Hassan ◽  
P.S. Joag
Keyword(s):  
Quantum States ◽  
Necessary Condition ◽  
Multilinear Algebra ◽  
Necessary And Sufficient Condition ◽  
Separability Criterion ◽  
Sufficiency Condition ◽  
Ppt States ◽  
Necessary And Sufficient ◽  
Qubit States ◽  
Bloch Representation

We give a new separability criterion, a necessary condition for separability of N-partite quantum states. The criterion is based on the Bloch representation of a N-partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to arbitrary N-partite quantum states in $\mathcal{H}=\mathcal{H}^{d_1}\otimes \mathcal{H}^{d_2} \otimes \cdots \otimes \mathcal{H}^{d_N}.$ The criterion can test whether a N-partite state is entangled and can be applied to different partitions of the $N$-partite system. We provide examples that show the ability of this criterion to detect entanglement. We show that this criterion can detect bound entangled states. We prove a sufficiency condition for separability of a 3-partite state, straightforwardly generalizable to the case N > 3, under certain condition. We also give a necessary and sufficient condition for separability of a class of N-qubit states which includes N-qubit PPT states.

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 Syarat Perlu dan Cukup untuk Keterbatasan Potensial Riesz di Ruang Morrey Klasik

2013 ◽  
Vol 14 (3) ◽  
pp. 227
Author(s):  
Mohammad Imam Utoyo ◽  
Basuki Widodo ◽  
Toto Nusantara ◽  
Suhariningsih Suhariningsih
Keyword(s):  
Characteristic Function ◽  
Morrey Space ◽  
Necessary Conditions ◽  
Riesz Potential ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Research Results ◽  
Necessary And Sufficient

This script was aimed to determine the necessary conditions for boundedness of Riesz potential in the classical Morrey space. If these results are combined with previous research results will be obtained the necessary and sufficient condition for boundedness of Riesz potential. This necessary condition is obtained through the use of characteristic function as one member of the classical Morrey space.

scholarly journals ON THE FINITENESS PROBLEM FOR AUTOMATON (SEMI)GROUPS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250052 ◽  
Author(s):  
ALI AKHAVI ◽  
INES KLIMANN ◽  
SYLVAIN LOMBARDY ◽  
JEAN MAIRESSE ◽  
MATTHIEU PICANTIN
Keyword(s):  
Decision Problem ◽  
Decision Procedure ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Mealy Automata ◽  
Necessary And Sufficient ◽  
New Criteria

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskiĭ, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure.

The necessary and sufficient condition for clustering of nodes based on the signs of connections in generalized signed networks

2019 ◽  
Vol 33 (10) ◽  
pp. 1950086
Author(s):  
Qi Wang ◽  
Yinhe Wang ◽  
Zilin Gao ◽  
Lili Zhang ◽  
Wenli Wang
Keyword(s):  
Clustering Algorithms ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Signed Networks ◽  
Clustering Problem ◽  
Structural Hole ◽  
Signed Network ◽  
Necessary And Sufficient ◽  
Sufficient And Necessary Condition

This paper investigates the clustering problem for the generalized signed networks. By rigorous derivations, a sufficient and necessary condition for clustering of the nodes in generalized signed networks is proposed in this paper. In order to obtain this condition, the concept of friends group is first introduced for the nodes based on their links’ sign. Then, the unprivileged network is also defined in this paper by employing the concepts of structural hole and broker. Compared with the existing clustering algorithms, the outstanding advantage in this paper is that only the positive or negative (especially, or zero) sign of the links is required regardless of their density or sparsity. We have proved mathematically that a generalized signed network is classifiable if and only if it is an unprivileged network. Finally, two examples associated with numerical simulations are proposed to generate the unprivileged networks.

scholarly journals Summability methods based on the Riemann Zeta function

1988 ◽  
Vol 11 (1) ◽  
pp. 27-36
Author(s):  
Larry K. Chu
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Necessary Condition ◽  
Real Sequence ◽  
Necessary And Sufficient Condition ◽  
Summability Methods ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Summability Matrix ◽  
Necessary And Sufficient

This paper is a study of summability methods that are based on the Riemann Zeta function. A limitation theorem is proved which gives a necessary condition for a sequencexto be zeta summable. A zeta summability matrixZtassociated with a real sequencetis introduced; a necessary and sufficient condition on the sequencetsuch thatZtmapsl1tol1is established. Results comparing the strength of the zeta method to that of well-known summability methods are also investigated.

scholarly journals THE FINITE SUM OF THE PRODUCTS OF TWO TOEPLITZ OPERATORS

2009 ◽  
Vol 86 (1) ◽  
pp. 45-60 ◽  
Author(s):  
XUANHAO DING
Keyword(s):  
Toeplitz Operator ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rank Perturbation ◽  
Sum Of Products ◽  
Finite Sum ◽  
Necessary And Sufficient

AbstractWe consider in this paper the question of when the finite sum of products of two Toeplitz operators is a finite-rank perturbation of a single Toeplitz operator on the Hardy space over the unit disk. A necessary condition is found. As a consequence we obtain a necessary and sufficient condition for the product of three Toeplitz operators to be a finite-rank perturbation of a single Toeplitz operator.

scholarly journals On Modelling AM/AM and AM/PM Conversions via Volterra Series

2016 ◽  
Vol 62 (3) ◽  
pp. 267-272
Author(s):  
Andrzej Borys ◽  
Wiesław Sieńko
Keyword(s):  
Power Amplifiers ◽  
Transfer Functions ◽  
Volterra Series ◽  
Polynomial Coefficient ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Nonlinear Transfer ◽  
Necessary And Sufficient ◽  
Nonzero Polynomial

Abstract In this paper, we present the expressions, not published up to now, that describe the AM/AM and AM/PM conversions of communication power amplifiers (PAs) via the Volterra series based nonlinear transfer functions. Furthermore, we present a necessary and sufficient condition of occurrence of the nonzero values of AM/PM conversion in PAs. Moreover, it has been shown that Saleh’s approach and related ones, which foresee nonzero level of AM/PM conversion, are not models without memory. It has been also shown that using a polynomial description of a PA does not lead to a nonzero AM/PM conversion. Moreover, a necessary condition of occurrence of an AM/AM conversion in this kind of modelling is existence of at least one nonzero polynomial coefficient associated with its odd terms of degree greater than one.

scholarly journals Some results concerning frames in Banach spaces

2007 ◽  
Vol 38 (3) ◽  
pp. 267-276 ◽  
Author(s):  
S. K. Kaushik
Keyword(s):  
Finite Number ◽  
Banach Spaces ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linearly Independent ◽  
Banach Frame ◽  
Minimal Sequence ◽  
Definition Of ◽  
Necessary And Sufficient

A necessary and sufficient condition for the associated sequence of functionals to a complete minimal sequence to be a Banach frame has been given. We give the definition of a weak-exact Banach frame, and observe that an exact Banach frame is weak-exact. An example of a weak-exact Banach frame which is not exact has been given. A necessary and sufficient condition for a Banach frame to be a weak-exact Banach frame has been obtained. Finally, a necessary condition for the perturbation of a retro Banach frame by a finite number of linearly independent vectors to be a retro Banach frame has been given.

Analysing Willerding’s formula for solving the planar three point resection problem

2019 ◽  
Vol 13 (1) ◽  
pp. 27-31
Author(s):  
Gerhard Heindl
Keyword(s):  
Mathematical Problem ◽  
Autonomous Robot ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Robot Localization ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Abstract The main mathematical problem in the planar three point resection problem of surveying is to find a necessary and sufficient condition that a point on a plane is uniquely determined if the directions from that point to three given points are known only up to an unknown offset. It will be shown in this paper that such a condition is the applicability of Willerding’s formula for computing the point in question, and that this condition is equivalent to the well-known necessary condition that the four considered points are not located on a circle or a line. Because of this result, the easy evaluation and the short derivation of Willerding’s formula, it can be expected, that this formula will be widely used in surveying, and also for autonomous robot localization.

Sign in / Sign up

Export Citation Format

Share Document