Does Symmetry Imply PPT Property?

TL;DR

This paper investigates the relationship between SPC matrices and PPT property, proving that all SPC matrices in 2x2 systems are PPT and exploring conditions where SPC, PPT, and separability are equivalent.

Contribution

It establishes that all SPC matrices in 2x2 systems are PPT and generalizes tensor rank results, also providing examples where SPC and PPT are equivalent.

Findings

All SPC matrices in M_2⊗M_2 are PPT.

Tensor rank ≤ 3 matrices in M_2⊗M_m are separable.

Existence of SPC matrices in M_3⊗M_3 that are not PPT.

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], T\'oth and G\"uhne 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 $M_2\otimes M_2$ (theorem 4.3). This theorem is a consequence of the fact that every density matrix in $M_2\otimes M_m$, 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 $M_k\otimes M_m$. Although, in $M_3\otimes M_3$, 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 $M_k\otimes M_k$, 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).