The positive partial transpose conjecture for n=3

TL;DR

This paper proves the PPT square conjecture for three-dimensional quantum states by showing two-qutrit PPT states have Schmidt rank at most two, advancing understanding of entanglement properties in quantum information theory.

Contribution

The paper provides a proof of the PPT square conjecture for n=3, using properties of two-qutrit PPT states, independent of previous proofs.

Findings

Proved the PPT square conjecture for n=3.

Two-qutrit PPT states have Schmidt rank at most two.

The conjecture remains open for n≥4.

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 M\"uller-Hermes. M\"uller-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.