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:quant-ph/0607164 · Physical Review A

Generalized qudit Choi maps

M. A. Jafarizadeh, M. Rezaee, Sodeif Ahadpour

Published Oct 27, 2006• 16 citations
arXivHTMLPDF

Abstract

Following the linear programming prescription of Jafarizadeh et al. [Phys. Rev. A 72, 062106 (2005)], the $d\ensuremath{\bigotimes}d$ Bell diagonal entanglement witnesses are provided. By using Jamiolkowski isomorphism, it is shown that the corresponding positive maps are the generalized qudit Choi maps. Also by manipulating particular $d\ensuremath{\bigotimes}d$ Bell diagonal separable states and constructing corresponding bound entangled states, it is shown that the obtained $d\ensuremath{\bigotimes}d$ Bell states diagonal entanglement witnesses (consequently qudit Choi maps) are nondecomposable in a certain range of their parameters.

DOI
10.1103/physreva.74.042335

Related papers

Bell-state diagonal-entanglement witnesses

M. A. Jafarizadeh, M. Rezaee, S. K. A. Seyed Yagoobi

It has been shown that finding generic Bell-state diagonal-entanglement witnesses for ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$ systems reduces to linear programming if the feasible region is a polygon by itself, and it can be solved approximately via linear programming if the feasible region is encircled by a polygon. Since solving linear programming for the generic case is difficult, multiqubit, $2\ensuremath{\bigotimes}N$ and $3\ensuremath{\bigotimes}3$ systems for the special case of generic Bell-state diagonal-entanglement witnesses for some particular choice of parameters have been considered. We obtain the optimal nondecomposable entanglement witness for a $3\ensuremath{\bigotimes}3$ system for some particular choice of parameters. By proving the optimality of the well-known reduction map and combining it with the optimal and nondecomposable $3\ensuremath{\bigotimes}3$ Bell-state diagonal-entanglement witnesses (named critical entanglement witnesses) the family of optimal and nondecomposable $3\ensuremath{\bigotimes}3$ Bell-state diagonal-entanglement witnesses has also been obtained. Using the approximately critical entanglement witnesses, some $3\ensuremath{\bigotimes}3$ bound entangled states are so detected. So the well-known Choi map as a particular case of the positive map in connection with this witness via Jamiolkowski isomorphism has been considered.

arXiv:quant-ph/0507035•Physical Review A•Dec 6, 2005•27 citations

Manipulating multiqudit entanglement witnesses by using linear programming

M. A. Jafarizadeh, G. Najarbashi, Hessam Habibian

A class of entanglement witnesses (EWs) called reduction-type entanglement witnesses is introduced, which can detect some multipartite entangled states including positive partial transpose ones with Hilbert space of dimension ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$. In fact the feasible regions of these EWs turn out to be convex polygons and hence the manipulation of them reduces to linear programming which can be solved exactly by using the simplex method. The decomposability and nondecomposability of these EWs are studied and it is shown that it has a close connection with eigenvalues and optimality of EWs. Also using the Jamio\l{}kowski isomorphism, the corresponding possible positive maps, including the generalized reduction maps of Hall [Phys. Rev. A 72, 022311 (2005)] are obtained.

arXiv:quant-ph/0611256•Physical Review A•May 21, 2007•22 citations

Constructing all entanglement witnesses from density matrices

Bang-Hai Wang, Dongyang Long

We demonstrate a general procedure to construct entanglement witnesses for any entangled state. This procedure is based on the trace inequality and a general form of entanglement witnesses, which is in the form $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$, where $\ensuremath{\rho}$ is a density matrix, ${c}_{\ensuremath{\rho}}$ is a non-negative number related to $\ensuremath{\rho}$, and $I$ is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamio\l{}kowski isomorphism, that can be reinterpreted as that all quantum states can be obtained by a maximally quantum entangled state pass through certain completely positive maps. Furthermore, we provide the necessary and sufficient condition of the entanglement witness $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$ in operation, as well as in theory.

arXiv:1111.5255•Physical Review A•Jul 25, 2011•5 citations

Optimality for indecomposable entanglement witnesses

Kil-Chan Ha, Seung-Hyeok Kye

We examine various notions related to the optimality for entanglement witnesses arising from Choi-type positive linear maps. We found examples of optimal entanglement witnesses which are nondecomposable but are not ``nondecomposable optimal entanglement witnesses'' in the sense of Lewenstein, Kraus, Cirac, and Horodecki [Phys. Rev. A 62, 052310 (2000)]. We suggest using the terms PPTES witness and optimal PPTES witness in place of ``nondecomposable entanglement witness'' and ``nondecomposable optimal entanglement witnesses'' in order to avoid possible confusion. Here, PPTES refers entangled states with positive partial transposes. We also found examples of nonextremal optimal entanglement witnesses which are indecomposable.

arXiv:1204.6596•Physical Review A•Sep 4, 2012•30 citations

Optimality of generalized Choi maps in $M_3$

Giovanni Scala, Anindita Bera, Gniewomir Sarbicki +1 more

A family of linear positive maps in the algebra of $3 \times 3$ complex matrices proposed recently in Bera et al. arXiv:2212.03807 is further analyzed. It provides a generalization of a seminal Choi nondecomposable extremal map in $M_3$. We investigate when generalized Choi maps are optimal, i.e. cannot be represented as a sum of positive and completely positive maps. This property is weaker than extremality, however, it turns out that it plays a key role in detecting quantum entanglement.

arXiv:2312.02814•Dec 5, 2023•0 citations

Class of bound entangled states of N+N qubits revealed by nondecomposable maps

Marco Piani

We introduce a family of linear maps which are positive but not completely positive. We exhibit examples of nondecomposable maps and ${2}^{N}\ensuremath{\bigotimes}{2}^{N}$, $N\ensuremath{\geqslant}2$, bound entangled states---e.g., nondistillable bipartite states of $N+N$ qubits. Such states, as the standard Bell diagonal states, are diagonal in a maximally entangled basis and, apart remaining positive under partial transposition, are not detected by the realignment criterion.

arXiv:quant-ph/0411098•Physical Review A•Jan 31, 2006•32 citations
Following the linear programming prescription of Jafarizadeh et al. [Phys. Rev. A 72, 062106 (2005)], the d\ensuremath⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂d Bell diagonal entanglement witnesses are provided. By using Jamiolkowski isomorphism, it is shown that the corresponding positive maps are the generalized qudit Choi maps. Also by manipulating particular d\ensuremath⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂d Bell diagonal separable states and constructing corresponding bound entangled states, it is shown that the obtained d\ensuremath⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂d Bell states diagonal entanglement witnesses (consequently qudit Choi maps) are nondecomposable in a certain range of their parameters.
It has been shown that finding generic Bell-state diagonal-entanglement witnesses for ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$ systems reduces to linear programming if the feasible region is a polygon by itself, and it can be solved approximately via linear programming if the feasible region is encircled by a polygon. Since solving linear programming for the generic case is difficult, multiqubit, $2\ensuremath{\bigotimes}N$ and $3\ensuremath{\bigotimes}3$ systems for the special case of generic Bell-state diagonal-entanglement witnesses for some particular choice of parameters have been considered. We obtain the optimal nondecomposable entanglement witness for a $3\ensuremath{\bigotimes}3$ system for some particular choice of parameters. By proving the optimality of the well-known reduction map and combining it with the optimal and nondecomposable $3\ensuremath{\bigotimes}3$ Bell-state diagonal-entanglement witnesses (named critical entanglement witnesses) the family of optimal and nondecomposable $3\ensuremath{\bigotimes}3$ Bell-state diagonal-entanglement witnesses has also been obtained. Using the approximately critical entanglement witnesses, some $3\ensuremath{\bigotimes}3$ bound entangled states are so detected. So the well-known Choi map as a particular case of the positive map in connection with this witness via Jamiolkowski isomorphism has been considered.
A class of entanglement witnesses (EWs) called reduction-type entanglement witnesses is introduced, which can detect some multipartite entangled states including positive partial transpose ones with Hilbert space of dimension ${d}_{1}\ensuremath{\bigotimes}{d}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{d}_{n}$. In fact the feasible regions of these EWs turn out to be convex polygons and hence the manipulation of them reduces to linear programming which can be solved exactly by using the simplex method. The decomposability and nondecomposability of these EWs are studied and it is shown that it has a close connection with eigenvalues and optimality of EWs. Also using the Jamio\l{}kowski isomorphism, the corresponding possible positive maps, including the generalized reduction maps of Hall [Phys. Rev. A 72, 022311 (2005)] are obtained.
We demonstrate a general procedure to construct entanglement witnesses for any entangled state. This procedure is based on the trace inequality and a general form of entanglement witnesses, which is in the form $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$, where $\ensuremath{\rho}$ is a density matrix, ${c}_{\ensuremath{\rho}}$ is a non-negative number related to $\ensuremath{\rho}$, and III is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamio\l{}kowski isomorphism, that can be reinterpreted as that all quantum states can be obtained by a maximally quantum entangled state pass through certain completely positive maps. Furthermore, we provide the necessary and sufficient condition of the entanglement witness $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$ in operation, as well as in theory.
We examine various notions related to the optimality for entanglement witnesses arising from Choi-type positive linear maps. We found examples of optimal entanglement witnesses which are nondecomposable but are not ``nondecomposable optimal entanglement witnesses'' in the sense of Lewenstein, Kraus, Cirac, and Horodecki [Phys. Rev. A 62, 052310 (2000)]. We suggest using the terms PPTES witness and optimal PPTES witness in place of ``nondecomposable entanglement witness'' and ``nondecomposable optimal entanglement witnesses'' in order to avoid possible confusion. Here, PPTES refers entangled states with positive partial transposes. We also found examples of nonextremal optimal entanglement witnesses which are indecomposable.
Optimality of generalized Choi maps in M3M_3M3​
A family of linear positive maps in the algebra of 3×33 \times 33×3 complex matrices proposed recently in Bera et al. arXiv:2212.03807 is further analyzed. It provides a generalization of a seminal Choi nondecomposable extremal map in M3M_3M3​. We investigate when generalized Choi maps are optimal, i.e. cannot be represented as a sum of positive and completely positive maps. This property is weaker than extremality, however, it turns out that it plays a key role in detecting quantum entanglement.
We introduce a family of linear maps which are positive but not completely positive. We exhibit examples of nondecomposable maps and ${2}^{N}\ensuremath{\bigotimes}{2}^{N}$, $N\ensuremath{\geqslant}2$, bound entangled states---e.g., nondistillable bipartite states of N+NN+NN+N qubits. Such states, as the standard Bell diagonal states, are diagonal in a maximally entangled basis and, apart remaining positive under partial transposition, are not detected by the realignment criterion.