Negative eigenvalues of partial transposition of arbitrary bipartite states
Swapan Rana · Physical Review A · 2013
The partial transposition of a two-qubit state has at most one negative eigenvalue and all the eigenvalues lie in $[\ensuremath{-}1/2,1]$. In this Brief Report, we extend this result by Sanpera et al. [A. Sanpera, R. Tarrach, and G. Vidal, Phys. Rev. A 58, 826 (1998)] to arbitrary bipartite states. We show that partial transposition of an $m\ensuremath{\bigotimes}n$ state cannot have more than $(m\ensuremath{-}1)(n\ensuremath{-}1)$ number of negative eigenvalues. Low-dimensional states have been studied to show the tightness of this result and explicit examples have been provided for $mn\ensuremath{\le}9$. It is also shown that all the eigenvalues of partial transposition lie within $[\ensuremath{-}1/2,1]$. Some possible applications are also discussed.