PPT-squared conjecture in arbitrary dimensions

Determine whether the composition of any two PPT maps on M_d is entanglement breaking; equivalently, establish whether the square of every PPT map on M_d is entanglement breaking, thereby resolving the PPT-squared conjecture in full generality.

Background

The PPT-squared conjecture asks whether composing two completely positive and completely copositive maps from M_d to M_d always yields an entanglement-breaking map. The equal-factor formulation asks whether every PPT map becomes entanglement breaking after being composed with itself. The paper proves finite entanglement-breaking index for every PPT map, but explicitly notes that this does not resolve the one-step, two-factor conjecture.

References

Therefore, the original PPT-squared conjecture does not follow from Eq. eq:SP2-bridge and still remains open.

— Every PPT channel has finite entanglement-breaking index  (2608.13551 - Park, 13 Aug 2026) in Section 1, Introduction

Although we have shown that every PPT map has finite EB index, it remains unknown whether there exists a channel-independent bound for the EB index. A natural weaker question is the following: For each $d\geq 4$, does there exist finite number $n(d)\geq 2$ such that $\Phi{n(d)}\in EB$ for every PPT map $\Phi:M_d\to M_d$?

— Every PPT channel has finite entanglement-breaking index  (2608.13551 - Park, 13 Aug 2026) in Section 4, 'Every PPT map is eventually entanglement breaking'