Letters in Mathematical Physics

, Volume 92, Issue 1, pp 67–79 | Cite as

On the Geometric Distance Between Quantum States with Positive Partial Transposition and Private States

  • Jeong San Kim
  • Barry C. Sanders


We prove an analytic positive lower bound for the geometric distance between entangled positive partial transpose (PPT) states of a broad class and any private state that delivers one secure key bit. Our proof holds for any Hilbert space of finite dimension. Although our result is proven for a specific class of PPT states, we show that our bound nonetheless holds for all known entangled PPT states with non-zero distillable key rates, irrespective of whether they are in our special class or not. Thus, our result naturally leads to the conjecture of impossibility of using PPT-bound entangled state in physical implementation of quantum key distribution.

Mathematics Subject Classification (2000)

81P68 46B20 47N50 


bound entangled state private state trace-norm distance positive partial transposition (PPT) 


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Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Institute for Quantum Information ScienceUniversity of CalgaryAlbertaCanada

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