Entanglement witnesses of four-qubit tripartite separable quantum states

2021 ◽  
Author(s):  
Miao Xu ◽  
Wei-feng Zhou ◽  
Feng Chen ◽  
Lizhen Jiang ◽  
Xiao-yu Chen
Keyword(s):  
Necessary Conditions ◽  
Quantum Systems ◽  
Ghz State ◽  
W State ◽  
Necessary Condition ◽  
Entangled State ◽  
Separable State ◽  
Entanglement Witnesses ◽  
Qubit System ◽  
Bipartite Quantum Systems

Abstract A quantum entangled state is easily disturbed by noise and degenerates into a separable state. Comparing to the entanglement of bipartite quantum systems, less progresses have been made for the entanglement of multipartite quantum systems. For tripartite separability of a four-qubit system, we propose two entanglement witnesses, each of which corresponds to a necessary condition of tripartite separability. For the four-qubit GHZ state mixed with a W state and white noise, it is proved that the necessary conditions of tripartite separability are also sufficient at W state side.

Entanglement

2017 ◽  
Author(s):  
Richard Healey
Keyword(s):  
Quantum Theory ◽  
Quantum State ◽  
Physical Condition ◽  
Physical System ◽  
Polarization State ◽  
Quantum Systems ◽  
Ghz State ◽  
Mathematical Representation ◽  
Entangled State ◽  
Physical Relation

Often a pair of quantum systems may be represented mathematically (by a vector) in a way each system alone cannot: the mathematical representation of the pair is said to be non-separable: Schrödinger called this feature of quantum theory entanglement. It would reflect a physical relation between a pair of systems only if a system’s mathematical representation were to describe its physical condition. Einstein and colleagues used an entangled state to argue that its quantum state does not completely describe the physical condition of a system to which it is assigned. A single physical system may be assigned a non-separable quantum state, as may a large number of systems, including electrons, photons, and ions. The GHZ state is an example of an entangled polarization state that may be assigned to three photons.

A matrix realignment method for recognizing entanglement

2003 ◽  
Vol 3 (3) ◽  
pp. 193-202
Author(s):  
K. Chen ◽  
L.-A. Wu
Keyword(s):  
Kronecker Product ◽  
Singular Values ◽  
Quantum Systems ◽  
Entangled States ◽  
Approximation Technique ◽  
Separable State ◽  
Simple Method ◽  
Bipartite Quantum Systems ◽  
The Matrix ◽  
Degree Of Entanglement

Motivated by the Kronecker product approximation technique, we have developed a very simple method to assess the inseparability of bipartite quantum systems, which is based on a realigned matrix constructed from the density matrix. For any separable state, the sum of the singular values of the matrix should be less than or equal to $1$. This condition provides a very simple, computable necessary criterion for separability, and shows powerful ability to identify most bound entangled states discussed in the literature. As a byproduct of the criterion, we give an estimate for the degree of entanglement of the quantum state.

Nonlinear and linear entanglement witnesses for bipartite systems via exact convex optimization

2010 ◽  
Vol 10 (7&8) ◽  
pp. 562-579
Author(s):  
M. Jafarizadeh ◽  
A. Heshmati ◽  
K. Aghayar
Keyword(s):  
Convex Optimization ◽  
Single Particle ◽  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary And Sufficient Conditions ◽  
Feasible Region ◽  
Entanglement Witnesses ◽  
Bipartite Quantum Systems ◽  
Linear And Nonlinear ◽  
Necessary And Sufficient

Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5.

Separability and entanglement of n-qubit and a qubit and a qudit using Hilbert–Schmidt decompositions

2016 ◽  
Vol 14 (05) ◽  
pp. 1650030 ◽  
Author(s):  
Y. Ben-Aryeh ◽  
A. Mann
Keyword(s):  
Singular Value ◽  
Ghz State ◽  
Entangled State ◽  
Sufficient Criterion ◽  
Density Matrices ◽  
Negative Eigenvalues ◽  
Qubit System ◽  
Partial Transpose ◽  
Unit Density ◽  
Value Decomposition

Hilbert–Schmidt (HS) decompositions are employed for analyzing systems of [Formula: see text]-qubit, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary (PTU) transformations for one qubit from the whole system, are used for indicating entanglement/separability. A sufficient criterion for full separability of the [Formula: see text]-qubit and qubit–qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and [Formula: see text]-qubit systems, with emphasis on maximally disordered subsystems (MDS) (i.e. density matrices for which tracing over any subsystem gives the unit density matrix). A sufficient condition that [Formula: see text] (MDS) is not separable is that it has an eigenvalue larger than [Formula: see text] for a qubit and a qudit, and larger than [Formula: see text] for [Formula: see text]-qubit system. The PTU transformation does not change the eigenvalues of the [Formula: see text]-qubit MDS density matrices for odd [Formula: see text]. Thus, the Peres–Horodecki (PH) criterion does not give any information about entanglement of these density matrices. The PH criterion may be useful for indicating inseparability for even [Formula: see text]. The changes of the entanglement and separability properties of the GHZ state, the Braid entangled state and the [Formula: see text] state by mixing them with white noise are analyzed by the use of the present methods. The entanglement and separability properties of the GHZ-diagonal density matrices, composed of mixture of 8[Formula: see text]GHZ density matrices with probabilities [Formula: see text], is analyzed as function of these probabilities. In some cases, we show that the PH criterion is both sufficient and necessary.

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.

Certain Fourier Transforms of Distributions: II

1954 ◽  
Vol 6 ◽  
pp. 186-189 ◽  
Author(s):  
Eugene Lukacs ◽  
Otto Szász
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Necessary Conditions ◽  
Laplace Transforms ◽  
Characteristic Functions ◽  
Necessary Condition ◽  
Multiple Roots ◽  
Not Given ◽  
Class Of Functions

In an earlier paper (1), published in this journal, a necessary condition was given which the reciprocal of a polynomial without multiple roots must satisfy in order to be a characteristic function. This condition is, however, valid for a wider class of functions since it can be shown (2, theorem 2 and corollary to theorem 3) that it holds for all analytic characteristic functions. The proof given in (1) is elementary and has some methodological interest since it avoids the use of theorems on singularities of Laplace transforms. Moreover the method used in (1) yields some additional necessary conditions which were not given in (1) and which do not seem to follow easily from the properties of analytic characteristic functions.

Eccentricity in irregular multistory buildings

1986 ◽  
Vol 13 (1) ◽  
pp. 46-52 ◽  
Author(s):  
V. W.-T. Cheung ◽  
W. K. Tso
Keyword(s):  
Dynamic Analysis ◽  
Necessary Conditions ◽  
Vertical Axis ◽  
Building Code ◽  
Necessary Condition ◽  
Practical Procedure ◽  
Multistory Buildings ◽  
Plane Frame ◽  
Code Procedure ◽  
Code Provisions

To evaluate the seismic torsional effect on multistory buildings, the concept of eccentricity is extended from single-story buildings to multistory buildings by defining the locations of the centers of rigidity at each floor. A practical procedure to locate the centers of rigidity and hence floor eccentricity is introduced. This procedure depends on the use of plane frame computer programs only and is suitable for use in design offices. The seismic torsional provisions in the National Building Code of Canada 1985 (NBCC 1985) explicitly emphasize that the code provisions apply to buildings where the centres of rigidity lie on a vertical axis only. By means of examples, it verifies the claim of NBCC 1985. Also, it shows that, for buildings with centers of rigidity scattered from a vertical axis, the code procedure may or may not apply. Therefore, one should interpret the condition of centers of rigidity located along a vertical axis to be a sufficient, but not a necessary, condition for the NBCC 85 code provisions to be applicable. Until the necessary conditions are known, dynamic analysis remains the most reliable method to assign the torsional effects to various portions of the building. Key words: building code, center of rigidity, dynamic analysis, eccentricity, irregular, multistory, seismic, torsion.

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.

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