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/0109081 · Physical Review A

Degree of entanglement for two qubits

Ayman F. Abouraddy, Bahaa E. A. Saleh, Alexander V. Sergienko, Malvin Carl Teich

Published Oct 8, 2001• 81 citations
arXivHTMLPDF

Abstract

We demonstrate that any pure bipartite state of two qubits may be decomposed into a superposition of a maximally entangled state and an orthogonal factorizable one. Although there are many such decompositions, the weights of the two superposed states are, remarkably, unique. We propose a measure of entanglement based on this decomposition. We also demonstrate that this measure is connected to three measures of entanglement previously set forth: maximal violation of Bell's inequality, concurrence, and two-particle visibility.

DOI
10.1103/physreva.64.050101

Related papers

Bell’s inequality and entanglement in qubits

Po-Yao Chang, Su-Kuan Chu, Chen‐Te Ma

We propose an alternative evaluation of quantum entanglement by measuring the maximum violation of the Bell’s inequality without information of the reduced density matrix of a system. This proposal is demonstrated by bridging the maximum violation of the Bell’s inequality and a concurrence of a pure state in an n-qubit system, in which one subsystem only contains one qubit and the state is a linear combination of two product states. We apply this relation to the ground states of four qubits in the Wen-Plaquette model and show that they are maximally entangled. A topological entanglement entropy of the Wen-Plaquette model could be obtained by relating the upper bound of the maximum violation of the Bell’s inequality to the generalized concurrence of a pure state with respect to different bipartitions.

arXiv:1705.06444•Journal of High Energy Physics•Sep 1, 2017•10 citations

Concurrence of superpositions of many states

Seyed Javad Akhtarshenas

In this paper, we use the concurrence vector as a measure of entanglement, and investigate lower and upper bounds on the concurrence of a superposition of bipartite states as a function of the concurrence of the superposed states. We show that the amount of entanglement quantified by the concurrence vector is exactly the same as that quantified by $I$ concurrence, so that our results can be compared to those given in Phys. Rev. A 76, 042328 (2007). We obtain a tighter lower bound in the case in which the two superposed states are orthogonal. We also show that when the two superposed states are not necessarily orthogonal, both lower and upper bounds are, in general, tighter than the bounds given in terms of the $I$ concurrence. An extension of the results to the case with more than two states in the superpositions is also given.

arXiv:1010.3413•Physical Review A•Apr 8, 2011•19 citations

Degree of entanglement for two qubits

Jing‐Ling Chen, Libin Fu, Abraham Albert Ungar +1 more

In this paper, we present a measure to quantify the degree of entanglement for two qubits in a pure state.

arXiv:quant-ph/0204162•Physical Review A•Apr 1, 2002•14 citations

Maximal Violation of Bell Inequality for Any Given Two-Qubit Pure State

Yang Xiang

In the case of bipartite two-qubit systems, we derive an analytical expression of bound Bell operator for any given pure state. Our result not only manifests some properties of Bell inequality, for example, which may be violated by any pure entangled state and only be maximally violated for a maximally entangled state, but also gives the explicit values of maximal violation for any pure state. Finally we point out that any mixed states which can produce maximal violation of Bell inequality must have a maximal concurrence value.

arXiv:1006.3602•Chinese Physics Letters•Dec 1, 2010•4 citations

Entanglement and separability of qubits systems related to measurement processes, Hilbert-Schmidt(HS)decompositions and general Bell states

Y. Ben-Aryeh

Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions in which the density matrices are decomposed into various terms representing certain one qubit, two-qubits,and larger qubits measurements. The present method is more general than previous methods for bipartite systems as it can be used for quantification of entanglement for large n-qubits systems.We demonstrate the use of such method by analyzing three qubits GHZ states, and three qubits general Bell states produced by a certain multiplications of Braid operators, operating on the computational basis of states. Quantum correlations are obtained by measuring all qubits of these systems, while a measurement of a part of the system gives only classical correlations. Quantification of entanglement for these systems is given by the use of HS decompositions.

arXiv:1412.0213•Nov 30, 2014•1 citation

Optimal Lewenstein–Sanpera decomposition for some bipartite systems

Seyed Javad Akhtarshenas, M. A. Jafarizadeh

It is shown that for a given bipartite density matrix and by choosing a suitable separable set (instead of a product set) on the separable–entangled boundary, the optimal Lewenstein–Sanpera (LS) decomposition (with respect to an arbitrary separable set) can be obtained via a direct optimization procedure for a generic entangled density matrix. On the basis of this, we obtain the optimal LS decomposition for some bipartite systems such as 2 ⊗ 2 and 2 ⊗ 3 Bell decomposable (BD) states, a generic two qubit state in Wootters basis, iso-concurrence decomposable states, states obtained from BD states via one-parameter and three-parameter local operations and classical communications (LOCC), d ⊗ d Werner and isotropic states and a one-parameter 3 ⊗ 3 state. We also obtain the optimal decomposition for multi-partite isotropic states. It is shown that in all 2 ⊗ 2 systems considered here the average concurrence of the decomposition is equal to the concurrence. We also show that for some 2 ⊗ 3 Bell decomposable states, the average concurrence of the decomposition is equal to the lower bound of the concurrence of the state presented recently in Lozinski et al (2003 Preprint quant-ph/0302144), so an exact expression for concurrence of these states is obtained. It is also shown that for a d ⊗ d isotropic state where decomposition leads to a separable and an entangled pure state, the average I-concurrence of the decomposition is equal to the I-concurrence of the state.

arXiv:quant-ph/0304019•Journal of Physics A Mathematical and General•Feb 11, 2004•10 citations
We demonstrate that any pure bipartite state of two qubits may be decomposed into a superposition of a maximally entangled state and an orthogonal factorizable one. Although there are many such decompositions, the weights of the two superposed states are, remarkably, unique. We propose a measure of entanglement based on this decomposition. We also demonstrate that this measure is connected to three measures of entanglement previously set forth: maximal violation of Bell's inequality, concurrence, and two-particle visibility.
We propose an alternative evaluation of quantum entanglement by measuring the maximum violation of the Bell’s inequality without information of the reduced density matrix of a system. This proposal is demonstrated by bridging the maximum violation of the Bell’s inequality and a concurrence of a pure state in an n-qubit system, in which one subsystem only contains one qubit and the state is a linear combination of two product states. We apply this relation to the ground states of four qubits in the Wen-Plaquette model and show that they are maximally entangled. A topological entanglement entropy of the Wen-Plaquette model could be obtained by relating the upper bound of the maximum violation of the Bell’s inequality to the generalized concurrence of a pure state with respect to different bipartitions.
In this paper, we use the concurrence vector as a measure of entanglement, and investigate lower and upper bounds on the concurrence of a superposition of bipartite states as a function of the concurrence of the superposed states. We show that the amount of entanglement quantified by the concurrence vector is exactly the same as that quantified by III concurrence, so that our results can be compared to those given in Phys. Rev. A 76, 042328 (2007). We obtain a tighter lower bound in the case in which the two superposed states are orthogonal. We also show that when the two superposed states are not necessarily orthogonal, both lower and upper bounds are, in general, tighter than the bounds given in terms of the III concurrence. An extension of the results to the case with more than two states in the superpositions is also given.
In the case of bipartite two-qubit systems, we derive an analytical expression of bound Bell operator for any given pure state. Our result not only manifests some properties of Bell inequality, for example, which may be violated by any pure entangled state and only be maximally violated for a maximally entangled state, but also gives the explicit values of maximal violation for any pure state. Finally we point out that any mixed states which can produce maximal violation of Bell inequality must have a maximal concurrence value.
Separability and entanglement for n-qubits systems are quantified by using Hilbert-Schmidt (HS) decompositions in which the density matrices are decomposed into various terms representing certain one qubit, two-qubits,and larger qubits measurements. The present method is more general than previous methods for bipartite systems as it can be used for quantification of entanglement for large n-qubits systems.We demonstrate the use of such method by analyzing three qubits GHZ states, and three qubits general Bell states produced by a certain multiplications of Braid operators, operating on the computational basis of states. Quantum correlations are obtained by measuring all qubits of these systems, while a measurement of a part of the system gives only classical correlations. Quantification of entanglement for these systems is given by the use of HS decompositions.
It is shown that for a given bipartite density matrix and by choosing a suitable separable set (instead of a product set) on the separable–entangled boundary, the optimal Lewenstein–Sanpera (LS) decomposition (with respect to an arbitrary separable set) can be obtained via a direct optimization procedure for a generic entangled density matrix. On the basis of this, we obtain the optimal LS decomposition for some bipartite systems such as 2 ⊗ 2 and 2 ⊗ 3 Bell decomposable (BD) states, a generic two qubit state in Wootters basis, iso-concurrence decomposable states, states obtained from BD states via one-parameter and three-parameter local operations and classical communications (LOCC), d ⊗ d Werner and isotropic states and a one-parameter 3 ⊗ 3 state. We also obtain the optimal decomposition for multi-partite isotropic states. It is shown that in all 2 ⊗ 2 systems considered here the average concurrence of the decomposition is equal to the concurrence. We also show that for some 2 ⊗ 3 Bell decomposable states, the average concurrence of the decomposition is equal to the lower bound of the concurrence of the state presented recently in Lozinski et al (2003 Preprint quant-ph/0302144), so an exact expression for concurrence of these states is obtained. It is also shown that for a d ⊗ d isotropic state where decomposition leads to a separable and an entangled pure state, the average I-concurrence of the decomposition is equal to the I-concurrence of the state.