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:1503.00528 · Journal of Physics A Mathematical and Theoretical

Using non-positive maps to characterize entanglement witnesses

Marek Mozrzymas, A. Rutkowski, Michał Studziński

Published Sep 11, 2015• 10 citations
arXivHTMLPDF

Abstract

In this paper we present a new method for entanglement witnesses construction. We show that to construct such an object we can deal with maps which are not positive on the whole domain, but only on a certain sub-domain. In our approach a crucial role is played by such maps which are surjective between sets of rank projectors and the set of rank one projectors acting in the d dimensional space. We argue that our method can be used to check whether a given observable is an entanglement witness. In the second part of this paper we show that an inverse reduction map satisfies this requirement and using it we can obtain a bunch of new entanglement witnesses.

DOI
10.1088/1751-8113/48/39/395302

Related papers

Some Remarks on the Role of Minimal Length of Positive Maps in Constructing Entanglement Witnesses

Andrzej Jamiołkowski

The main objective of this paper is to discuss correspondence between the concept of entanglement witnesses (self-adjoint operators on a composite Hilbert space [Formula: see text] that are, in general, not positive, but are positive on separable states) and positive maps [Formula: see text] which are not completely positive. The notion of minimal length of linear positive map is introduced and the role of this quantity in the constructing of entanglement witnesses is explained.

Open Systems & Information Dynamics•Dec 1, 2004•6 citations

Test for entanglement using physically observable witness operators and positive maps

Kai Chen, Ling-An Wu

Motivated by the Peres-Horodecki criterion and the realignment criterion we develop a more powerful method to identify entangled states for any bipartite system through a universal construction of the witness operator. The method also gives a new family of positive but noncompletely positive maps of arbitrary high dimensions, which provide a much better test than the witness operators themselves. Moreover, we find that there are two types of positive maps that can detect $2\ifmmode\times\else\texttimes\fi{}N$ and $4\ifmmode\times\else\texttimes\fi{}N$ bound entangled states. Since entanglement witnesses are physical observables and may be measured locally our construction could be of great significance for future experiments.

arXiv:quant-ph/0306041•Physical Review A•Feb 20, 2004•50 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

Optimization of entanglement witnesses

Maciej Lewenstein, Barbara Kraus, J. I. Cirac +1 more

An entanglement witness (EW) is an operator that allows the detection of entangled states. We give necessary and sufficient conditions for such operators to be optimal, i.e., to detect entangled states in an optimal way. We show how to optimize general EW, and then we particularize our results to the nondecomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES's). We also present a method to systematically construct and optimize this last class of operators based on the existence of ``edge'' PPTES's, i.e., states that violate the range separability criterion [Phys. Lett. A 232, 333 (1997)] in an extreme manner. This method also permits a systematic construction of nondecomposable positive maps (PM's). Our results lead to a sufficient condition for entanglement in terms of nondecomposable EW's and PM's. Finally, we illustrate our results by constructing optimal EW acting on $H={C}^{2}\ensuremath{\bigotimes}{C}^{4}.$ The corresponding PM's constitute examples of PM's with minimal ``qubit'' domains, or---equivalently---minimal Hermitian conjugate codomains.

arXiv:quant-ph/0005014•Physical Review A•Oct 16, 2000•684 citations

Spectral properties of entanglement witnesses

Gniewomir Sarbicki

Entanglement witnesses are observables which when measured, detect entanglement in a measured composed system. It is shown what kind of relations between eigenvectors of an observable should be fulfilled, to allow an observable to be an entanglement witness. Some restrictions on the signature of entaglement witnesses, based on an algebraic-geometrical theorem will be given. The set of entanglement witnesses is linearly isomorphic to the set of maps between matrix algebras which are positive, but not completely positive. A translation of the results to the language of positive maps is also given. The properties of entanglement witnesses and positive maps express as special cases of general theorems for $k$-Schmidt witnesses and $k$-positive maps. The results are therefore presented in a general framework.

arXiv:0711.1936•Journal of Physics A Mathematical and Theoretical•Aug 13, 2008•29 citations

Visualizing extremal positive maps in unital and trace-preserving form

Leif Ove Hansen, Jan Myrheim

We define an entanglement witness in a composite quantum system as an observable having a nonnegative expectation value in every separable state. Then a state is entangled if and only if it has a negative expectation value of some entanglement witness. Equivalent representations of entanglement witnesses are nonnegative biquadratic forms or positive linear maps of Hermitian matrices. As reported elsewhere, we have studied extremal entanglement witnesses in dimension $3\ifmmode\times\else\texttimes\fi{}3$ by constructing numerical examples of generic extremal nonnegative forms. These are so complicated that we do not know how to handle them other than by numerical methods. However, the corresponding extremal positive maps can be presented graphically, as we attempt to do in the present paper. We understand that a positive map is extremal when the image of $\mathcal{D}$, the set of density matrices, fills out $\mathcal{D}$ maximally, in a certain sense. For the graphical presentation of a map we transform it to a standard form where it is unital and trace preserving. We present an iterative algorithm for the transformation, which converges rapidly in all our numerical examples and presumably works for any positive map. This standard form of an entanglement witness is unique up to unitary product transformations.

arXiv:1511.02110•Physical Review A•Oct 5, 2015•3 citations
In this paper we present a new method for entanglement witnesses construction. We show that to construct such an object we can deal with maps which are not positive on the whole domain, but only on a certain sub-domain. In our approach a crucial role is played by such maps which are surjective between sets of rank projectors and the set of rank one projectors acting in the d dimensional space. We argue that our method can be used to check whether a given observable is an entanglement witness. In the second part of this paper we show that an inverse reduction map satisfies this requirement and using it we can obtain a bunch of new entanglement witnesses.
Motivated by the Peres-Horodecki criterion and the realignment criterion we develop a more powerful method to identify entangled states for any bipartite system through a universal construction of the witness operator. The method also gives a new family of positive but noncompletely positive maps of arbitrary high dimensions, which provide a much better test than the witness operators themselves. Moreover, we find that there are two types of positive maps that can detect $2\ifmmode\times\else\texttimes\fi{}N$ and $4\ifmmode\times\else\texttimes\fi{}N$ bound entangled states. Since entanglement witnesses are physical observables and may be measured locally our construction could be of great significance for future experiments.
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).
An entanglement witness (EW) is an operator that allows the detection of entangled states. We give necessary and sufficient conditions for such operators to be optimal, i.e., to detect entangled states in an optimal way. We show how to optimize general EW, and then we particularize our results to the nondecomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES's). We also present a method to systematically construct and optimize this last class of operators based on the existence of ``edge'' PPTES's, i.e., states that violate the range separability criterion [Phys. Lett. A 232, 333 (1997)] in an extreme manner. This method also permits a systematic construction of nondecomposable positive maps (PM's). Our results lead to a sufficient condition for entanglement in terms of nondecomposable EW's and PM's. Finally, we illustrate our results by constructing optimal EW acting on $H={C}^{2}\ensuremath{\bigotimes}{C}^{4}.$ The corresponding PM's constitute examples of PM's with minimal ``qubit'' domains, or---equivalently---minimal Hermitian conjugate codomains.
Entanglement witnesses are observables which when measured, detect entanglement in a measured composed system. It is shown what kind of relations between eigenvectors of an observable should be fulfilled, to allow an observable to be an entanglement witness. Some restrictions on the signature of entaglement witnesses, based on an algebraic-geometrical theorem will be given. The set of entanglement witnesses is linearly isomorphic to the set of maps between matrix algebras which are positive, but not completely positive. A translation of the results to the language of positive maps is also given. The properties of entanglement witnesses and positive maps express as special cases of general theorems for kkk-Schmidt witnesses and kkk-positive maps. The results are therefore presented in a general framework.
We define an entanglement witness in a composite quantum system as an observable having a nonnegative expectation value in every separable state. Then a state is entangled if and only if it has a negative expectation value of some entanglement witness. Equivalent representations of entanglement witnesses are nonnegative biquadratic forms or positive linear maps of Hermitian matrices. As reported elsewhere, we have studied extremal entanglement witnesses in dimension $3\ifmmode\times\else\texttimes\fi{}3$ by constructing numerical examples of generic extremal nonnegative forms. These are so complicated that we do not know how to handle them other than by numerical methods. However, the corresponding extremal positive maps can be presented graphically, as we attempt to do in the present paper. We understand that a positive map is extremal when the image of D\mathcal{D}D, the set of density matrices, fills out D\mathcal{D}D maximally, in a certain sense. For the graphical presentation of a map we transform it to a standard form where it is unital and trace preserving. We present an iterative algorithm for the transformation, which converges rapidly in all our numerical examples and presumably works for any positive map. This standard form of an entanglement witness is unique up to unitary product transformations.