arXiv:quant-ph/0406006 · Physical Review A

Characterization of entanglement of more than two qubits with Bell inequalities and global entanglement

Jochen Endrejat, H. Büttner

Published Jan 4, 2005• 21 citations

Abstract

It is shown that the entanglement-structure of 3- and $4\text{\ensuremath{-}}\text{qubit}$ states can be characterized by optimized operators of the Mermin-Klyshko type. It is possible to discriminate between pure $2\text{\ensuremath{-}}\text{qubit}$ entanglements and higher entanglements. A comparison with a global entanglement measure is made.

DOI: 10.1103/physreva.71.012305

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arXiv:quant-ph/0406006 · Physical Review A

Characterization of entanglement of more than two qubits with Bell inequalities and global entanglement

Jochen Endrejat, H. Büttner

Published Jan 4, 2005• 21 citations

arXiv HTML PDF

Abstract

It is shown that the entanglement-structure of 3- and

4\text{\ensuremath{-}}\text{qubit}

states can be characterized by optimized operators of the Mermin-Klyshko type. It is possible to discriminate between pure

2\text{\ensuremath{-}}\text{qubit}

entanglements and higher entanglements. A comparison with a global entanglement measure is made.

DOI: 10.1103/physreva.71.012305

Related papers

Characterization of multiqubit pure-state entanglement

Zeqian Chen, Quanhua Xu

arXiv:quant-ph/0505166Physical Review AMar 16, 20062 citations

Detecting entanglement of n qubits by observing fluctuations of Bell operators

Zeqian Chen, Quanhua Xu

It is presented a criterion of efficiently detecting all pure entangled states of n qubits by observing fluctuation of Bell's operators in place of expectation in Bell's inequalities. The entanglement witnesses are some of MK Bell operators corresponding to Mermin-Klyshko inequalities, which are easily constructed by resolving a maximum problem and using a computable method, respectively. We illustrate generalized GHZ states, generalized W and pure three-qubit states, respectively, for seeing how good our criterion is to detect the entanglement of $n$ qubits. Therefore, we eventually resolve the problem of efficiently detecting pure entangled states of n qubits.It is presented a criterion of efficiently detecting all pure entangled states of n qubits by observing fluctuation of Bell's operators in place of expectation in Bell's inequalities. The entanglement witnesses are some of MK Bell operators corresponding to Mermin-Klyshko inequalities, which are easily constructed by resolving a maximum problem and using a computable method, respectively. We illustrate generalized GHZ states, generalized W and pure three-qubit states, respectively, for seeing how good our criterion is to detect the entanglement of $n$ qubits. Therefore, we eventually resolve the problem of efficiently detecting pure entangled states of n qubits.

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We consider 3‐qubit entangled pure states of the form α|000〉+β|111〉 and explore nonlocal correlations between the qubits by analyzing a Bell inequality belonging to the Mermin‐Klyshko (MK) set of N‐qubit inequalities. We find an expression for the maximum expectation value of the MK Bell operator and show that it does not increase smoothly with the amount of entanglement between the qubits. A subset of these tripartite entangled states does not violate the bound of 2 imposed by local hidden variable theories.We consider 3‐qubit entangled pure states of the form α|000〉+β|111〉 and explore nonlocal correlations between the qubits by analyzing a Bell inequality belonging to the Mermin‐Klyshko (MK) set of N‐qubit inequalities. We find an expression for the maximum expectation value of the MK Bell operator and show that it does not increase smoothly with the amount of entanglement between the qubits. A subset of these tripartite entangled states does not violate the bound of 2 imposed by local hidden variable theories.

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It is shown that the entanglement-structure of 3- and

4\text{\ensuremath{-}}\text{qubit}

states can be characterized by optimized operators of the Mermin-Klyshko type. It is possible to discriminate between pure

2\text{\ensuremath{-}}\text{qubit}

entanglements and higher entanglements. A comparison with a global entanglement measure is made.

n

qubits. Therefore, we eventually resolve the problem of efficiently detecting pure entangled states of n qubits.

For two qubits in a pure state there exists a one-to-one relation between the entanglement measure (the concurrence

\mathcal{C}

) and the maximal violation

\mathcal{M}

of a Bell inequality. No such relation exists for the three-qubit analog of

\mathcal{C}

(the tangle $\ensuremath{\tau}$), but we have found that numerical data is consistent with a simple set of upper and lower bounds for $\ensuremath{\tau}$ given

\mathcal{M}

. The bounds on $\ensuremath{\tau}$ become tighter with increasing

\mathcal{M}

, so they are of practical use. The Svetlichny form of the Bell inequality gives tighter bounds than the Mermin form. We show that the bounds can be tightened further if the tangle is replaced by an entanglement monotone that can identify both the W state and the Greenberger-Horne-Zeilinger state.