Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0907.2369 · Physical Review A

Constructing optimal entanglement witnesses

Dariusz Chruściński, Justyna Pytel, Gniewomir Sarbicki

Published Dec 7, 2009• 44 citations
arXivHTMLPDF

Abstract

We provide a class of indecomposable entanglement witnesses. In $4\ifmmode\times\else\texttimes\fi{}4$ case, it reproduces the well-known Breuer-Hall witness. We prove that these witnesses are optimal and atomic, i.e., they are able to detect the ``weakest'' quantum entanglement encoded into states with positive partial transposition. Equivalently, we provide a construction of indecomposable atomic maps in the algebra of $2k\ifmmode\times\else\texttimes\fi{}2k$ complex matrices. It is shown that their structural physical approximations give rise to entanglement breaking channels. This result supports recent conjecture by Korbicz et al. [Phys. Rev. A 78, 062105 (2008)].

DOI
10.1103/physreva.80.062314

Related papers

Constructing optimal entanglement witnesses. II. Witnessing entanglement in 4N×4N systems

Dariusz Chruściński, Justyna Pytel

We provide a class of optimal nondecomposable entanglement witnesses for $4N\ifmmode\times\else\texttimes\fi{}4N$ composite quantum systems or, equivalently, another construction of nondecomposable positive maps in the algebra of $4N\ifmmode\times\else\texttimes\fi{}4N$ complex matrices. This construction provides natural generalization of the Robertson map. It is shown that their structural physical approximations give rise to entanglement breaking channels.

arXiv:1006.3059•Physical Review A•Nov 15, 2010•32 citations

Characterization of optimal entanglement witnesses

Xiaofei Qi, Jinchuan Hou

In this paper, we present a characterization of optimal entanglement witnesses in terms of positive maps and then provide a general method of checking optimality of entanglement witnesses. Applying it, we obtain indecomposable optimal witnesses that have no spanning property. These also provide examples that support a recent conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking maps (separable states).

arXiv:1109.2300•Physical Review A•Feb 27, 2012•27 citations

A class of positive atomic maps

Dariusz Chruściński, Andrzej Kossakowski

We construct a new class of positive indecomposable maps in the algebra of d × d complex matrices. These maps are characterized by the ‘weakest’ positivity property and for this reason they are called atomic. This class provides a new rich family of atomic entanglement witnesses which define an important tool for investigating quantum entanglement. It turns out that they are able to detect states with the ‘weakest’ quantum entanglement.

arXiv:0711.4483•Journal of Physics A Mathematical and Theoretical•May 7, 2008•23 citations

How to construct indecomposable entanglement witnesses

Dariusz Chruściński, Andrzej Kossakowski

We present a very simple method for constructing indecomposable entanglement witnesses out of a given pair—an entanglement witness W and the corresponding state detected by W . This method may be used to produce new classes of atomic witnesses which are able to detect the 'weakest' quantum entanglement. Actually, it works perfectly in the multipartite case, too. Moreover, this method provides a powerful tool for constructing new examples of bound entangled states.

arXiv:0712.1114•Journal of Physics A Mathematical and Theoretical•Mar 26, 2008•37 citations

Optimal witnesses detecting positive-partial-transpose entangled states in Cn⊗Cn

Kil-Chan Ha

Recently, Qi and Hou [Phys. Rev. A 85, 022334 (2012)] provided optimal entanglement witnesses without the spanning property. These witnesses are associated with indecomposable positive linear maps, but whether partial transposes of these witnesses are also optimal has not been evaluated. We show that partial transposes of these entanglement witnesses have the spanning property and that these witnesses are indeed optimal witnesses detecting positive-partial- transposed entangled states (nondecomposable optimal entanglement witnesses).

Physical Review A•Jul 23, 2012•5 citations

Indecomposable entanglement witnesses from symmetric measurements

Katarzyna Siudzińska

We propose a family of positive maps constructed from a recently introduced class of symmetric measurements. These maps are used to define entanglement witnesses, which include other popular approaches with mutually unbiased bases and mutually unbiased measurements. A particular interest is given to indecomposable witnesses that can be used to detect entanglement of quantum states with positive partial transposition. We present several examples for different number of measurements.

arXiv:2202.04046•Feb 8, 2022•0 citations
We provide a class of indecomposable entanglement witnesses. In 4\ifmmode×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 case, it reproduces the well-known Breuer-Hall witness. We prove that these witnesses are optimal and atomic, i.e., they are able to detect the ``weakest'' quantum entanglement encoded into states with positive partial transposition. Equivalently, we provide a construction of indecomposable atomic maps in the algebra of 2k\ifmmode×\else\texttimes\fi2k2k\ifmmode\times\else\texttimes\fi{}2k2k\ifmmode×\else\texttimes\fi2k complex matrices. It is shown that their structural physical approximations give rise to entanglement breaking channels. This result supports recent conjecture by Korbicz et al. [Phys. Rev. A 78, 062105 (2008)].
In this paper, we present a characterization of optimal entanglement witnesses in terms of positive maps and then provide a general method of checking optimality of entanglement witnesses. Applying it, we obtain indecomposable optimal witnesses that have no spanning property. These also provide examples that support a recent conjecture saying that the so-called structural physical approximations to optimal positive maps (optimal entanglement witnesses) give entanglement breaking maps (separable states).
Recently, Qi and Hou [Phys. Rev. A 85, 022334 (2012)] provided optimal entanglement witnesses without the spanning property. These witnesses are associated with indecomposable positive linear maps, but whether partial transposes of these witnesses are also optimal has not been evaluated. We show that partial transposes of these entanglement witnesses have the spanning property and that these witnesses are indeed optimal witnesses detecting positive-partial- transposed entangled states (nondecomposable optimal entanglement witnesses).