PPT-squared conjecture: is composing two PPT channels entanglement breaking
In plain words
Some noisy channels are so weak at carrying entanglement that their partial transpose stays positive. The conjecture says that applying two such channels in a row always destroys all entanglement.
Precise statement
For all quantum channels $\Phi_1, \Phi_2$ on d x d matrices whose Choi matrices have positive partial transpose, is $\Phi_2$ composed with $\Phi_1$ entanglement breaking? Proved for $d \le 3$ (2018-2019) and in an asymptotic form under repeated composition; open for $d \ge 4$.
What would settle it
A proof for all d, or an explicit pair of PPT channels whose composition is not entanglement breaking.
Status in the literature
Unverified note
A 2026 preprint proves that every PPT channel has finite entanglement-breaking index (arXiv:2608.13551); the two-fold statement remains open for $d \ge 4$.