The positive partial transpose conjecture for n=3

Abstract

We present the PPT square conjecture introduced by M. Christandl. We prove the conjecture in the case $n=3$ as a consequence of the fact that two-qutrit PPT states have Schmidt at most two. Our result in Lemma 3 is independent from the proof found Müller-Hermes. Müller-Hermes announced that this conjecture is true for the states on $\mathbb{C}_3\otimes\mathbb{C}_3$ \cite{hermes} recently. The PPT square conjecture in the case $n\ge4$ is still open.