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Entanglement of four-qubit rank-2 mixed states

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Abstract

It is known that there are three maximally entangled states \(|\varPhi _1 \rangle = (|0000 \rangle + |1111 \rangle ) / \sqrt{2}\), \(|\varPhi _2 \rangle = (\sqrt{2} |1111 \rangle + |1000 \rangle + |0100 \rangle + |0010 \rangle + |0001 \rangle ) / \sqrt{6}\), and \(|\varPhi _3 \rangle = (|1111 \rangle + |1100 \rangle + |0010 \rangle + |0001 \rangle ) / 2\) in four-qubit system. It is also known that there are three independent measures \(\mathcal{F}^{(4)}_j (j=1,2,3)\) for true four-way quantum entanglement in the same system. In this paper, we compute \(\mathcal{F}^{(4)}_j\) and their corresponding linear monotones \(\mathcal{G}^{(4)}_j\) for three rank-two mixed states \(\rho _j = p |\varPhi _j \rangle \langle \varPhi _j | + (1 - p) |\text{ W }_4 \rangle \langle \text{ W }_4 |\), where \(|\text{ W }_4 \rangle = (|0111 \rangle + |1011 \rangle + |1101 \rangle + |1110 \rangle ) / 2\). We discuss the possible applications of our results briefly.

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Notes

  1. The current status of quantum computer technology was reviewed in Ref. [8].

  2. For complete proof on the connection between SLOCC and local operations, see Appendix A of Ref. [27].

  3. In this paper, we will call \(\tau _3\) three-tangle and \(\tau _3^2\) residual entanglement.

  4. The parameter \(p_1\) is obtained by an equation \(6 p_1 (4 p_1 - 3)^2 = (1 - p_1) (1 + 2 p_1)^2\).

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Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2011-0011971).

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Authors and Affiliations

  1. Department of Electronic Engineering, Kyungnam University, Changwon, 631-701, Korea

    Eylee Jung & DaeKil Park

  2. Department of Physics, Kyungnam University, Changwon, 631-701, Korea

    DaeKil Park

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  1. Eylee Jung
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  2. DaeKil Park
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Correspondence to DaeKil Park.

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Jung, E., Park, D. Entanglement of four-qubit rank-2 mixed states. Quantum Inf Process 14, 3317–3333 (2015). https://doi.org/10.1007/s11128-015-1039-4

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