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

New entanglement Witnesses and Bell Operator for n-qubits related to Hilbert-Schmidt decompositions

Y. Ben-Aryeh, A. Mann

Published May 26, 2019• 0 citations
OpenAlex

Abstract

The use of entanglement witness (EW) for non-full separability and the Bell operator for non-local hidden-variables (LHV) model are analyzed by relating them to the Hilbert-Schmidt (HS) decomposition of n-qubits states and these methods are applied explicitly to some 3 and 4 qubits states. EW for non-full separability (fs) is given by fs parameter minus operator G where the choice of G in the HS decomposition leads to the fs parameter and to the condition for non-separability by using criterions which are different from those used for genuine entanglement. We analyze especially entangled states with probability p mixed with white noise with probability 1-p and find the critical value p(crit.) above which the system is not fully separable. As the choice of EW might not be optimal we add to the analysis of EW explicit construction of fully separable density matrix and find the critical value p below which the system is fully separable. If the two values for p coincide we conclude that this parameter gives the optimal result. Such optimal results are obtained in the present work for some 3 and 4 qubits entangled states mixed with white noise. The use of partial-transpose (PT ) (say relative to qubit A) gives also p value above which the system is not fully separable. The use of EW gives better results (or at least equal) than those obtained by PT .

Related papers

Full-separability and bi-separability of qubits using Bell operators, partial transpose, witnesses and explicit (full/bi) separability

Y. Ben-Aryeh, A. Mann

Using the Hilbert–Schmidt (HS) decomposition we suggest new possible choices of Bell operators and entanglement witnesses (EWs) for [Formula: see text] ([Formula: see text]) qubits systems for (full/bi) separability. The latter give upper bounds for (full/bi) separability. Also using the HS decomposition, we find explicitly (full/bi) separable forms for some qubits states which give lower bounds for (full/bi) separability. When the lower bounds and upper bounds coincide it means that the EW is optimal. In the case of full separability, the positive transpose method can sometimes give optimal results. As concrete examples, we give results for the GHZ(3), [Formula: see text] and cluster [Formula: see text] states.

arXiv:1905.10798•International Journal of Quantum Information•Apr 11, 2022•0 citations

Entanglement Witness and Bell Operator by Hilbert-Schmidt Decomposition

Y. Ben-Aryeh, A. Mann

A new method is developed for obtaining entanglement witnesses (EW) and Bell operators for n-qubit state. Observing that both EW and Bell operators must have nonzero overlap with the given state we suggest constructing them using some of the terms that appear in the Hilbert-Schmidt (HS) decomposition of the state (not necessarily with the same coefficients). We illustrate the method for several 3 and 4 qubits states, and show improvement of the entanglement region compared with known results. Since the Bell operator and the EW may be restricted to few HS terms, separability/nonseparability of the quantum state can be checked by measuring only these terms.

•May 26, 2019•0 citations

Separability and entanglement of n-qubits and a qubit with a qudit using Hilbert-Schmidt decompositions

Y. Ben-Aryeh, A. Mann

Hilbert-Schmidt (HS) decompositions are employed for analyzing systems of n-qubits, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary transformations (PTU) for one qubit from the whole system, are used for indicating inseparability. A sufficient criterion for full separability of the n-qubits and qubit-qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and n-qubits systems, with emphasis on maximally disordered subsystems (MDS) (i.e., density matrices rho(MDS) for which tracing over any subsystem gives the unit density matrix). A sufficient condition that rho(MDS) is not separable is that it has an eigenvalue larger than 1/d for a qubit and a qudit, and larger than 1/2^(n-1) for n-qubits system. The PTU transformation does not change the eigenvalues of the n-qubits MDS density matrices for odd n. Thus the Peres-Horodecki criterion does not give any information about entanglement of these density matrices, but this criterion is useful for indicating inseparability for even n. The changes of the entanglement and separability properties of the GHZ state, the Braid entangled state and the W state by mixing them with white noise are analyzed by the use of the present methods. The entanglement and separability properties of the GHZ-diagonal density matrices, composed of mixture of 8 GHZ density matrices with probabilities p(i), is analyzed as function of these probabilities. In some cases we show that the Peres-Horodecki criterion is both sufficient and necessary.

arXiv:1606.04304•Jun 14, 2016•0 citations

Separability and entanglement of n-qubit and a qubit and a qudit using Hilbert–Schmidt decompositions

Y. Ben-Aryeh, A. Mann

Hilbert–Schmidt (HS) decompositions are employed for analyzing systems of [Formula: see text]-qubit, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary (PTU) transformations for one qubit from the whole system, are used for indicating entanglement/separability. A sufficient criterion for full separability of the [Formula: see text]-qubit and qubit–qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and [Formula: see text]-qubit systems, with emphasis on maximally disordered subsystems (MDS) (i.e. density matrices for which tracing over any subsystem gives the unit density matrix). A sufficient condition that [Formula: see text] (MDS) is not separable is that it has an eigenvalue larger than [Formula: see text] for a qubit and a qudit, and larger than [Formula: see text] for [Formula: see text]-qubit system. The PTU transformation does not change the eigenvalues of the [Formula: see text]-qubit MDS density matrices for odd [Formula: see text]. Thus, the Peres–Horodecki (PH) criterion does not give any information about entanglement of these density matrices. The PH criterion may be useful for indicating inseparability for even [Formula: see text]. The changes of the entanglement and separability properties of the GHZ state, the Braid entangled state and the [Formula: see text] state by mixing them with white noise are analyzed by the use of the present methods. The entanglement and separability properties of the GHZ-diagonal density matrices, composed of mixture of 8[Formula: see text]GHZ density matrices with probabilities [Formula: see text], is analyzed as function of these probabilities. In some cases, we show that the PH criterion is both sufficient and necessary.

International Journal of Quantum Information•Aug 1, 2016•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

The full separability of three-qubit superposition state and its decayed states in white noise environment

Weidong Shen, Xiaoyu Chen

Abstract The research on the separability of quantum states is one of the important research directions in quantum information theory. In this paper, we propose a new entanglement witness to study the full separability of the superposition of three-qubit W and <?CDATA $\bar{{\rm{W}}}$?> W ¯ (the qubit inverted W state) states in white noise environment. The witness accounts for most results of separable condition of the state obtained with numerical calculation. The witness also accounts for the decayed state which is the superposition state with and terms decayed by a factor. The separability criterion from our witness is better than that of the further generalized Wootters formula.

Journal of Physics Conference Series•Jan 1, 2021•0 citations
The use of entanglement witness (EW) for non-full separability and the Bell operator for non-local hidden-variables (LHV) model are analyzed by relating them to the Hilbert-Schmidt (HS) decomposition of n-qubits states and these methods are applied explicitly to some 3 and 4 qubits states. EW for non-full separability (fs) is given by fs parameter minus operator G where the choice of G in the HS decomposition leads to the fs parameter and to the condition for non-separability by using criterions which are different from those used for genuine entanglement. We analyze especially entangled states with probability p mixed with white noise with probability 1-p and find the critical value p(crit.) above which the system is not fully separable. As the choice of EW might not be optimal we add to the analysis of EW explicit construction of fully separable density matrix and find the critical value p below which the system is fully separable. If the two values for p coincide we conclude that this parameter gives the optimal result. Such optimal results are obtained in the present work for some 3 and 4 qubits entangled states mixed with white noise. The use of partial-transpose (PT ) (say relative to qubit A) gives also p value above which the system is not fully separable. The use of EW gives better results (or at least equal) than those obtained by PT .
Using the Hilbert–Schmidt (HS) decomposition we suggest new possible choices of Bell operators and entanglement witnesses (EWs) for [Formula: see text] ([Formula: see text]) qubits systems for (full/bi) separability. The latter give upper bounds for (full/bi) separability. Also using the HS decomposition, we find explicitly (full/bi) separable forms for some qubits states which give lower bounds for (full/bi) separability. When the lower bounds and upper bounds coincide it means that the EW is optimal. In the case of full separability, the positive transpose method can sometimes give optimal results. As concrete examples, we give results for the GHZ(3), [Formula: see text] and cluster [Formula: see text] states.
A new method is developed for obtaining entanglement witnesses (EW) and Bell operators for n-qubit state. Observing that both EW and Bell operators must have nonzero overlap with the given state we suggest constructing them using some of the terms that appear in the Hilbert-Schmidt (HS) decomposition of the state (not necessarily with the same coefficients). We illustrate the method for several 3 and 4 qubits states, and show improvement of the entanglement region compared with known results. Since the Bell operator and the EW may be restricted to few HS terms, separability/nonseparability of the quantum state can be checked by measuring only these terms.
Hilbert-Schmidt (HS) decompositions are employed for analyzing systems of n-qubits, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary transformations (PTU) for one qubit from the whole system, are used for indicating inseparability. A sufficient criterion for full separability of the n-qubits and qubit-qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and n-qubits systems, with emphasis on maximally disordered subsystems (MDS) (i.e., density matrices rho(MDS) for which tracing over any subsystem gives the unit density matrix). A sufficient condition that rho(MDS) is not separable is that it has an eigenvalue larger than 1/d for a qubit and a qudit, and larger than 1/2^(n-1) for n-qubits system. The PTU transformation does not change the eigenvalues of the n-qubits MDS density matrices for odd n. Thus the Peres-Horodecki criterion does not give any information about entanglement of these density matrices, but this criterion is useful for indicating inseparability for even n. The changes of the entanglement and separability properties of the GHZ state, the Braid entangled state and the W state by mixing them with white noise are analyzed by the use of the present methods. The entanglement and separability properties of the GHZ-diagonal density matrices, composed of mixture of 8 GHZ density matrices with probabilities p(i), is analyzed as function of these probabilities. In some cases we show that the Peres-Horodecki criterion is both sufficient and necessary.
Hilbert–Schmidt (HS) decompositions are employed for analyzing systems of [Formula: see text]-qubit, and a qubit with a qudit. Negative eigenvalues, obtained by partial-transpose (PT) plus local unitary (PTU) transformations for one qubit from the whole system, are used for indicating entanglement/separability. A sufficient criterion for full separability of the [Formula: see text]-qubit and qubit–qudit systems is given. We use the singular value decomposition (SVD) for improving the criterion for full separability. General properties of entanglement and separability are analyzed for a system of a qubit and a qudit and [Formula: see text]-qubit systems, with emphasis on maximally disordered subsystems (MDS) (i.e. density matrices for which tracing over any subsystem gives the unit density matrix). A sufficient condition that [Formula: see text] (MDS) is not separable is that it has an eigenvalue larger than [Formula: see text] for a qubit and a qudit, and larger than [Formula: see text] for [Formula: see text]-qubit system. The PTU transformation does not change the eigenvalues of the [Formula: see text]-qubit MDS density matrices for odd [Formula: see text]. Thus, the Peres–Horodecki (PH) criterion does not give any information about entanglement of these density matrices. The PH criterion may be useful for indicating inseparability for even [Formula: see text]. The changes of the entanglement and separability properties of the GHZ state, the Braid entangled state and the [Formula: see text] state by mixing them with white noise are analyzed by the use of the present methods. The entanglement and separability properties of the GHZ-diagonal density matrices, composed of mixture of 8[Formula: see text]GHZ density matrices with probabilities [Formula: see text], is analyzed as function of these probabilities. In some cases, we show that the PH criterion is both sufficient and necessary.
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.
Abstract The research on the separability of quantum states is one of the important research directions in quantum information theory. In this paper, we propose a new entanglement witness to study the full separability of the superposition of three-qubit W and <?CDATA Wˉ\bar{{\rm{W}}}Wˉ?> W ¯ (the qubit inverted W state) states in white noise environment. The witness accounts for most results of separable condition of the state obtained with numerical calculation. The witness also accounts for the decayed state which is the superposition state with and terms decayed by a factor. The separability criterion from our witness is better than that of the further generalized Wootters formula.