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/0703106 · Physical Review A

Bell-state diagonal entanglement witnesses for relativistic and nonrelativistic multispinor systems in arbitrary dimensions

M. A. Jafarizadeh, Rahime Sufiani

Published Jan 17, 2008• 18 citations
arXivHTMLPDF

Abstract

Two kinds of Bell-state diagonal (BSD) entanglement witnesses (EWs) are constructed by using the algebra of Dirac $\ensuremath{\gamma}$ matrices in a space-time of arbitrary dimension $d$, where the first kind can detect some BSD relativistic and nonrelativistic $m$-partite multispinor bound entangled states in a Hilbert space of dimension ${2}^{m\ensuremath{\lfloor}d/2\ensuremath{\rfloor}}$, including the bipartite Bell-type and isoconcurrence-type states in four-dimensional space-time $(d=4)$. By using the connection between the Hilbert-Schmidt measure and the optimal EWs associated with the states, it is shown that, as far as the spin quantum correlations are concerned, the amount of entanglement is not a relativistic scalar and has no invariant meaning. The introduced EWs are manipulated via linear programming (LP) and can be solved exactly by using the simplex method. The decomposability or nondecomposability of these EWs is investigated, the region of nondecomposable EWs of the first kind is partially determined and it is shown that, all of the EWs of the second kind are decomposable. These EWs have the property that in bipartite systems they can determine the region of separable states, i.e., bipartite nondetectable density matrices of the same type as the EWs of the first kind are necessarily separable. Also, multispinor EWs with nonpolygon feasible regions are provided, where the problem is solved by approximate LP, and in contrast to the exactly manipulatable EWs, both the first and second kinds of the optimal approximate EWs can detect some bound entangled states.

DOI
10.1103/physreva.77.012105

Related papers

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

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

A mirrored pair of optimal non-decomposable entanglement witnesses for two qudits does exist

Dariusz Chruściński, Anindita Bera, Joonwoo Bae +1 more

Two approaches can be utilized to handle the separability problem, finding out whether a given bipartite qudit state is separable or not: a direct procedure on the state space or the effective tool of entanglement witnesses (EWs). This contribution studies the structure of EWs. Exploiting the very concept of mirrored EWs, increasing the detection power, we show, in contrast to the conjecture in a recent paper (Bera et al. in Sci Rep 13:10733, 2023), there exist pairs of optimal EWs, which are both non-decomposible, i.e. can detect bound/PPT-entangled states in an optimal way. Since we show that the structure also extends to higher dimensions, our results reveal a further structure of entanglement witnesses.

arXiv:2503.04158•Scientific Reports•Aug 2, 2025•4 citations

Characterizing entanglement with geometric entanglement witnesses

Philipp Krammer

We show how to detect entangled, bound entangled and separable bipartite quantum states of arbitrary dimension and mixedness using geometric entanglement witnesses. These witnesses are constructed using properties of the Hilbert–Schmidt geometry and can be shifted along parameterized lines. The involved conditions are simplified using Bloch decompositions of operators and states. As an example we determine the three different types of states for a family of two-qutrit states that is part of the 'magic simplex', i.e. the set of Bell-state mixtures of arbitrary dimension.

arXiv:0807.4830•Journal of Physics A Mathematical and Theoretical•Jan 15, 2009•13 citations

Quantifying entanglement of two relativistic particles via decomposable optimal entanglement witnesses

M. A. Jafarizadeh, M. Mahdian

The study of Entanglement of Formation of a mixed state of a bipartite system in high-dimensional Hilbert space is not easy in general. So, we focus on determining the amount of entanglement for a bipartite mixed state based on the concept of decomposable optimal entanglement witness (DOEW), that can be calculated as a minimum distance of an entangled state from the edge of positive partial transpose (PPT) states which has the most negative (positive) expectation value for non-PPT (bound) entangled states. We have constructed DOEWs based on the convex optimization method, then by using of it we quantify the degree of entanglement for two spin half particles under the Lorentz transformations. For convenience, we restrict ourselves to 2D momentum subspace and under this constraint when the momentum and the Lorentz boost are parallel, we have shown that the entanglement is not relativistic invariant. Keywords : Relativistic entanglement, Measure of entanglement, Optimal entanglement witnesses, Convex optimization PACS numbers: 03.67.Hk, 03.65.Ta

arXiv:0911.2307•Nov 12, 2009•0 citations

Mirrored entanglement witnesses for multipartite and high-dimensional quantum systems

Jiheon Seong, Anindita Bera, Beatrix C. Hiesmayr +2 more

Abstract Entanglement witnesses (EWs) are a versatile tool to detect entangled states and characterize related properties of entanglement in quantum information theory. The verification of entangled states via EWs relies on the fact that separable states form a convex set; this also means that the framework presented by EWs generally applies to other quantum resources where free resources contain the convexity. A witness W corresponds to an observable satisfying Tr [ W σ sep ] ⩾ 0 for all separable states σ sep ; entangled states are detected once the inequality is violated. Recently, mirrored EWs have been introduced by showing that there exist non-trivial upper bounds to EWs, u W ⩾ Tr [ W σ sep ] ⩾ 0. An upper bound to a witness W signifies the existence of the other one M , called a mirrored EW, such that W + M = u W I ⊗ I . The framework of mirrored EWs shows that a single EW can be even more useful, as it can detect a larger set of entangled states by lower and upper bounds. In this work, we develop and investigate mirrored EWs for multipartite qubit states and also for high-dimensional systems, to find the efficiency and effectiveness of mirrored EWs in detecting entangled states. We provide mirrored EWs for n -partite GHZ states, graph states such as two-colorable states and tripartite bound entangled states. We also show that optimal EWs can be reflected with each other. For bipartite systems, we present mirrored EWs for existing optimal EWs and also construct a mirrored pair of optimal EWs in dimension three. Finally, we generalize mirrored EWs such that a pair of EWs can be connected by another EW, i.e. W + M = K is also an EW. Our results enhance the capability of EWs to detect a larger set of entangled states in multipartite and high-dimensional quantum systems.

arXiv:2510.06863•Quantum Science and Technology•Dec 23, 2025•2 citations
Two kinds of Bell-state diagonal (BSD) entanglement witnesses (EWs) are constructed by using the algebra of Dirac \ensuremathγ\ensuremath{\gamma}\ensuremathγ matrices in a space-time of arbitrary dimension ddd, where the first kind can detect some BSD relativistic and nonrelativistic mmm-partite multispinor bound entangled states in a Hilbert space of dimension 2m\ensuremath⌊d/2\ensuremath⌋{2}^{m\ensuremath{\lfloor}d/2\ensuremath{\rfloor}}2m\ensuremath⌊d/2\ensuremath⌋, including the bipartite Bell-type and isoconcurrence-type states in four-dimensional space-time (d=4)(d=4)(d=4). By using the connection between the Hilbert-Schmidt measure and the optimal EWs associated with the states, it is shown that, as far as the spin quantum correlations are concerned, the amount of entanglement is not a relativistic scalar and has no invariant meaning. The introduced EWs are manipulated via linear programming (LP) and can be solved exactly by using the simplex method. The decomposability or nondecomposability of these EWs is investigated, the region of nondecomposable EWs of the first kind is partially determined and it is shown that, all of the EWs of the second kind are decomposable. These EWs have the property that in bipartite systems they can determine the region of separable states, i.e., bipartite nondetectable density matrices of the same type as the EWs of the first kind are necessarily separable. Also, multispinor EWs with nonpolygon feasible regions are provided, where the problem is solved by approximate LP, and in contrast to the exactly manipulatable EWs, both the first and second kinds of the optimal approximate EWs can detect some bound entangled states.
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.
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.
Two approaches can be utilized to handle the separability problem, finding out whether a given bipartite qudit state is separable or not: a direct procedure on the state space or the effective tool of entanglement witnesses (EWs). This contribution studies the structure of EWs. Exploiting the very concept of mirrored EWs, increasing the detection power, we show, in contrast to the conjecture in a recent paper (Bera et al. in Sci Rep 13:10733, 2023), there exist pairs of optimal EWs, which are both non-decomposible, i.e. can detect bound/PPT-entangled states in an optimal way. Since we show that the structure also extends to higher dimensions, our results reveal a further structure of entanglement witnesses.
We show how to detect entangled, bound entangled and separable bipartite quantum states of arbitrary dimension and mixedness using geometric entanglement witnesses. These witnesses are constructed using properties of the Hilbert–Schmidt geometry and can be shifted along parameterized lines. The involved conditions are simplified using Bloch decompositions of operators and states. As an example we determine the three different types of states for a family of two-qutrit states that is part of the 'magic simplex', i.e. the set of Bell-state mixtures of arbitrary dimension.
The study of Entanglement of Formation of a mixed state of a bipartite system in high-dimensional Hilbert space is not easy in general. So, we focus on determining the amount of entanglement for a bipartite mixed state based on the concept of decomposable optimal entanglement witness (DOEW), that can be calculated as a minimum distance of an entangled state from the edge of positive partial transpose (PPT) states which has the most negative (positive) expectation value for non-PPT (bound) entangled states. We have constructed DOEWs based on the convex optimization method, then by using of it we quantify the degree of entanglement for two spin half particles under the Lorentz transformations. For convenience, we restrict ourselves to 2D momentum subspace and under this constraint when the momentum and the Lorentz boost are parallel, we have shown that the entanglement is not relativistic invariant. Keywords : Relativistic entanglement, Measure of entanglement, Optimal entanglement witnesses, Convex optimization PACS numbers: 03.67.Hk, 03.65.Ta
Abstract Entanglement witnesses (EWs) are a versatile tool to detect entangled states and characterize related properties of entanglement in quantum information theory. The verification of entangled states via EWs relies on the fact that separable states form a convex set; this also means that the framework presented by EWs generally applies to other quantum resources where free resources contain the convexity. A witness W corresponds to an observable satisfying Tr [ W σ sep ] ⩾ 0 for all separable states σ sep ; entangled states are detected once the inequality is violated. Recently, mirrored EWs have been introduced by showing that there exist non-trivial upper bounds to EWs, u W ⩾ Tr [ W σ sep ] ⩾ 0. An upper bound to a witness W signifies the existence of the other one M , called a mirrored EW, such that W + M = u W I ⊗ I . The framework of mirrored EWs shows that a single EW can be even more useful, as it can detect a larger set of entangled states by lower and upper bounds. In this work, we develop and investigate mirrored EWs for multipartite qubit states and also for high-dimensional systems, to find the efficiency and effectiveness of mirrored EWs in detecting entangled states. We provide mirrored EWs for n -partite GHZ states, graph states such as two-colorable states and tripartite bound entangled states. We also show that optimal EWs can be reflected with each other. For bipartite systems, we present mirrored EWs for existing optimal EWs and also construct a mirrored pair of optimal EWs in dimension three. Finally, we generalize mirrored EWs such that a pair of EWs can be connected by another EW, i.e. W + M = K is also an EW. Our results enhance the capability of EWs to detect a larger set of entangled states in multipartite and high-dimensional quantum systems.