Minimum distance of the boundary of the set of PPT states from the maximally mixed state using the geometry of the positive semidefinite cone

  • Shreya BanerjeeEmail author
  • Aryaman A. Patel
  • Prasanta K. Panigrahi


Using a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices (Patel and Panigrahi in Geometric measure of entanglement based on local measurement, 2016. arXiv:1608.06145), we obtain the minimum distance between the set of bipartite n-qudit density matrices with a positive partial transpose and the maximally mixed state. This minimum distance is obtained as \(\frac{1}{\sqrt{d^n(d^n-1)}}\), which is also the minimum distance within which all quantum states are separable. An idea of the interior of the set of all positive semidefinite matrices has also been provided. A particular class of Werner states has been identified for which the PPT criterion is necessary and sufficient for separability in dimensions greater than six.


Entanglement Separability Partial transpose Werner states PPT criterion Positive semidefinite cone 



The authors want to acknowledge valuable inputs from Prof. Somshubhro Bandyopadhyay (Bose Institute, Kolkata).


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Shreya Banerjee
    • 1
    Email author
  • Aryaman A. Patel
    • 2
  • Prasanta K. Panigrahi
    • 1
  1. 1.Indian Institute of Science Education and Research KolkataMohanpurIndia
  2. 2.National Institute of Technology KarnatakaSurathkal, MangaloreIndia

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