scholarly journals DUALITIES AND POSITIVITY IN THE STUDY OF QUANTUM ENTANGLEMENT

2010 ◽  
Vol 08 (05) ◽  
pp. 721-754 ◽  
Author(s):  
ŁUKASZ SKOWRONEK
Keyword(s):  
Related Problem ◽  
Separable State ◽  
Real Numbers ◽  
Entanglement Witness ◽  
Separable States ◽  
Entanglement Witnesses ◽  
Real Polynomials ◽  
Schmidt Rank ◽  
Families Of Operators ◽  
Cone Duality

We present a survey on mathematical topics relating to separable states and entanglement witnesses. The convex cone duality between separable states and entanglement witnesses is discussed and later generalized to other families of operators, leading to their characterization via multiplicative properties. The condition for an operator to be an entanglement witness is rephrased as a problem of positivity of a family of real polynomials. By solving the latter in a specific case of a three-parameter family of operators, we obtain explicit description of entanglement witnesses belonging to that family. A related problem of block positivity over real numbers is discussed. We also consider a broad family of block positivity tests and prove that they can never be sufficient, which should be useful in case of future efforts in that direction. Finally, we introduce the concept of length of a separable state and present new results concerning relationships between the length and Schmidt rank. In particular, we prove that separable states of length lower or equal to 3 have Schmidt ranks equal to their lengths. We also give an example of a state which has length 4 and Schmidt rank 3.

scholarly journals Distance between Bound Entangled States from Unextendible Product Bases and Separable States

2020 ◽  
Vol 2 (1) ◽  
pp. 49-56 ◽  
Author(s):  
Marcin Wieśniak ◽  
Palash Pandya ◽  
Omer Sakarya ◽  
Bianka Woloncewicz
Keyword(s):  
The State ◽  
Reference State ◽  
Entangled States ◽  
Separable State ◽  
Entanglement Witness ◽  
Separable States ◽  
Entanglement Witnesses

We discuss the use of the Gilbert algorithm to tailor entanglement witnesses for unextendible product basis bound entangled states (UPB BE states). The method relies on the fact that an optimal entanglement witness is given by a plane perpendicular to a line between the reference state, entanglement of which is to be witnessed, and its closest separable state (CSS). The Gilbert algorithm finds an approximation of CSS. In this article, we investigate if this approximation can be good enough to yield a valid entanglement witness. We compare witnesses found with Gilbert algorithm and those given by Bandyopadhyay–Ghosh–Roychowdhury (BGR) construction. This comparison allows us to learn about the amount of entanglement and we find a relationship between it and a feature of the construction of UPBBE states, namely the size of their central tile. We show that in most studied cases, witnesses found with the Gilbert algorithm in this work are more optimal than ones obtained by Bandyopadhyay, Ghosh, and Roychowdhury. This result implies the increased tolerance to experimental imperfections in a realization of the state.

scholarly journals Characterization and properties of weakly optimal entanglement witnesses

2015 ◽  
Vol 15 (13&14) ◽  
pp. 1109-1121
Author(s):  
Bang-Hai Wang ◽  
Hai-Ru Xu ◽  
Steve Campbell ◽  
Simone Severini
Keyword(s):  
Hilbert Space ◽  
Identity Matrix ◽  
Negative Number ◽  
Entanglement Witness ◽  
Geometric Properties ◽  
Expectation Value ◽  
Separable States ◽  
Entanglement Witnesses ◽  
Product Vector

We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the form of $W^{wopt}=\sigma-c_{\sigma}^{max} I$, where $c_{\sigma}^{max}$ is a non-negative number and $I$ is the identity matrix. We show the relation between the weakly optimal witness $W^{wopt}$ and the eigenvalues of the separable states $\sigma$. Further we give an application of weakly optimal witnesses for constructing entanglement witnesses in a larger Hilbert space by extending the result of [P. Badzi\c{a}g {\it et al}, Phys. Rev. A {\bf 88}, 010301(R) (2013)], and we examine their geometric properties.

scholarly journals Exposedness of Choi-Type Entanglement Witnesses and Applications to Lengths of Separable States

2013 ◽  
Vol 20 (04) ◽  
pp. 1350012 ◽  
Author(s):  
Kil-Chan Ha ◽  
Seung-Hyeok Kye
Keyword(s):  
Large Class ◽  
Separable State ◽  
Matrix Algebras ◽  
Separable States ◽  
Linear Maps ◽  
Entanglement Witnesses ◽  
Positive Linear Maps

We present a large class of indecomposable exposed positive linear maps between 3 × 3 matrix algebras. We also construct two-qutrit separable states with lengths ten in the interior of their dual faces. With these examples, we show that the length of a separable state may decrease strictly when we mix it with another separable state.

The arithmetic of certain semigroups of positive operators

1968 ◽  
Vol 64 (1) ◽  
pp. 161-166 ◽  
Author(s):  
John Lamperti
Keyword(s):  
Finite Number ◽  
Positive Definite ◽  
Positive Operators ◽  
Linear Operators ◽  
Real Numbers ◽  
Ultraspherical Polynomials ◽  
Real Polynomials ◽  
Fixed Parameter ◽  
Summation Sign ◽  
Image Position

Some time ago, S. Bochner gave an interesting analysis of certain positive operators which are associated with the ultraspherical polynomials (1,2). Let {Pn(x)} denote these polynomials, which are orthogonal on [ − 1, 1 ] with respect to the measureand which are normalized by settigng Pn(1) = 1. (The fixed parameter γ will not be explicitly shown.) A sequence t = {tn} of real numbers is said to be ‘positive definite’, which we will indicate by writing , provided thatHere the coefficients an are real, and the prime on the summation sign means that only a finite number of terms are different from 0. This condition can be rephrased by considering the set of linear operators on the space of real polynomials which have diagonal matrices with respect to the basis {Pn(x)}, and noting that

scholarly journals Optimal Polynomial Prediction Measures and Extremal Polynomial Growth

2020 ◽  
Author(s):  
L. Bos ◽  
N. Levenberg ◽  
J. Ortega-Cerdà
Keyword(s):  
Least Squares ◽  
Related Problem ◽  
Polynomial Growth ◽  
Compact Set ◽  
Extremal Polynomial ◽  
Real Polynomials ◽  
Optimal Polynomial

Abstract We show that the problem of finding the measure supported on a compact set $$K\subset \mathbb {C}$$ K ⊂ C such that the variance of the least squares predictor by polynomials of degree at most n at a point $$z_0\in \mathbb {C}^d\backslash K$$ z 0 ∈ C d \ K is a minimum is equivalent to the problem of finding the polynomial of degree at most n,  bounded by 1 on K,  with extremal growth at $$z_0.$$ z 0 . We use this to find the polynomials of extremal growth for $$[-1,1]\subset \mathbb {C}$$ [ - 1 , 1 ] ⊂ C at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by Erdős (Bull Am Math Soc 53:1169–1176, 1947).

scholarly journals Higher Order Oscillation and Uniform Distribution

2019 ◽  
Vol 14 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Shigeki Akiyama ◽  
Yunping Jiang
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Uniform Distribution ◽  
Differentiable Function ◽  
Mild Condition ◽  
Higher Order ◽  
Real Numbers ◽  
Mobius Function ◽  
Real Polynomials ◽  
Almost All

AbstractIt is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (e2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function g with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number α and almost all real numbers β> 1 (alternatively, for a fixed real number β> 1 and almost all real numbers α) and for all real polynomials Q(x), sequences (αβng(β)+ Q(n)) n∈𝕅 are uniformly distributed modulo 1.

scholarly journals Classical extension of quantum-correlated separable states

2015 ◽  
Vol 13 (02) ◽  
pp. 1550015 ◽  
Author(s):  
G. Bellomo ◽  
A. Plastino ◽  
A. R. Plastino
Keyword(s):  
Separable State ◽  
Separable States ◽  
Classical State ◽  
Classical Extension ◽  
Low Dimensional ◽  
Remarkable Relation

Li and Luo [Phys. Rev. A 78 (2008) 024303] discovered a remarkable relation between discord and entanglement. It establishes that all separable states can be obtained via reduction of a classically-correlated state "living" in a space of larger dimension. Starting from this result, we discuss here an optimal classical extension of separable states and explore this notion for low-dimensional systems. We find that the larger the dimension of the classical extension, the larger the discord in the original separable state. Further, we analyze separable states of maximum discord in ℂ2 ⊗ ℂ2 and their associated classical extensions showing that, from the reduction of a classical state in (ℂ2 ⊗ ℂ3) ⊗ ℂ2, one can obtain a separable state of maximum discord in ℂ2 ⊗ ℂ2.

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.

scholarly journals Characterization of separable states and entanglement witnesses

2001 ◽  
Vol 63 (4) ◽  
Author(s):  
M. Lewenstein ◽  
B. Kraus ◽  
P. Horodecki ◽  
J. I. Cirac
Keyword(s):  
Separable States ◽  
Entanglement Witnesses
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