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Inseparability criteria based on bipartitions of \(N\)-qubit systems

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Abstract

We present a new set of inequalities to detect inseparability in \(N\)-qubit states. These are based on negative partial transposition and involve collective qubit–qubit correlations of any two partitions of the entire system. They reveal the rich texture of partial separability for different partitions and can discriminate between GHZ-type and W-type entanglement, as well. We introduce a new concept of relative entanglement of two different systems and two different partitions in a qubit-ensemble. The criteria in its generic form are also applicable to non-symmetric states and states with odd or even \(N\).

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Notes

  1. It must be borne in mind that our criteria are based on Peres proposition [2]. Hence, they cannot detect entanglement in those states which exhibit positive partial transposition in all bipartitions

  2. A complete optimal set of such criteria can be found in Tóth et al. [11, 12]

  3. It is known that a state which exhibits negative partial transpose in all bipartitions is not necessarily genuine multipartite entangled, see Acin et al. [21]. However, we do not consider the most general form of GHZ and W states and it still remains open to find criteria for these states in such forms

  4. Note that the inequality (15) looks similar to the one as proposed for two bosonic modes in Eq. (8) in Hillery and Zubairy [22]

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Acknowledgments

The author thanks Dr. Shubhrangshu Dasgupta and the Referees for their insightful comments on this work.

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Authors and Affiliations

  1. Department of Physics, Indian Institute of Technology Ropar, Rupnagar, 140001, Punjab, India

    Asoka Biswas

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  1. Asoka Biswas
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Correspondence to Asoka Biswas.

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Sponsored by the Department of Science and Technology, Govt. of India, under Fast Track Scheme (Project No. SR/FTP/PS-71/2009).

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Biswas, A. Inseparability criteria based on bipartitions of \(N\)-qubit systems. Quantum Inf Process 14, 979–988 (2015). https://doi.org/10.1007/s11128-015-0920-5

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