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Product Vectors in the Ranges of Multi-Partite States with Positive Partial Transposes and Permanents of Matrices

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Abstract

In this paper, we consider a system of homogeneous algebraic equations in complex variables and their conjugates, which arise naturally from the range criterion for separability of PPT states. We examine systematically these equations to get sufficient conditions for the existence of nontrivial solutions. This gives us possible upper bounds of ranks of PPT entangled edge states and their partial transposes. We will focus on the multi-partite cases, which are much more delicate than the bi-partite cases. We use the notion of permanents of matrices as well as techniques from algebraic geometry through the discussion.

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Authors and Affiliations

  1. Department of Mathematics and Research Institute of Mathematics, Seoul National University, Seoul, 151-747, Korea

    Young-Hoon Kiem & Seung-Hyeok Kye

  2. School of Computational Sciences, Korea Institute for Advanced Study, Seoul, 130-722, Korea

    Joohan Na

Authors
  1. Young-Hoon Kiem
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  2. Seung-Hyeok Kye
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  3. Joohan Na
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Correspondence to Joohan Na.

Additional information

Communicated by M. M. Wolf

YHK and JN were partially supported by NRF Grant 2011-0027969; SHK was partially supported by NRFK Grant 2013-004942.

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Kiem, YH., Kye, SH. & Na, J. Product Vectors in the Ranges of Multi-Partite States with Positive Partial Transposes and Permanents of Matrices. Commun. Math. Phys. 338, 621–639 (2015). https://doi.org/10.1007/s00220-015-2385-x

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  • DOI: https://doi.org/10.1007/s00220-015-2385-x

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