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- W2247460108 abstract "Recently, in [1], the author proved that many results that are true for PPT matrices also hold for another class of matrices with a certain symmetry in their Hermitian Schmidt decompositions. These matrices were called SPC in [1] (definition 1.1). Before that, in [9], Toth and Guhne proved that if a state is symmetric then it is PPT if and only if it is SPC. A natural question appeared: What is the connection between SPC matrices and PPT matrices? Is every SPC matrix PPT? Here we show that every SPC matrix is PPT in $M_2otimes M_2$ (theorem 4.3). This theorem is a consequence of the fact that every density matrix in $M_2otimes M_m$, with tensor rank smaller or equal to 3, is separable (theorem 3.2). This theorem is a generalization of the same result found in [1] for tensor rank 2 matrices in $M_kotimes M_m$. Although, in $M_3otimes M_3$, there exists a SPC matrix with tensor rank 3 that is not PPT (proposition 5.2). We shall also provide a non trivial example of a family of matrices in $M_kotimes M_k$, in which both, the SPC and PPT properties, are equivalent (proposition 6.2). Within this family, there exists a non trivial subfamily in which the SPC property is equivalent to separability (proposition 6.4)." @default.
- W2247460108 created "2016-06-24" @default.
- W2247460108 creator A5088153781 @default.
- W2247460108 date "2014-05-14" @default.
- W2247460108 modified "2023-09-27" @default.
- W2247460108 title "Does Symmetry Imply PPT Property" @default.
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