The PPT squared conjecture

About 8 years old · traced to 2018

Let $\mathcal{M}_d$ denote the algebra of complex $d\times d$ matrices, and let $T:\mathcal{M}_d\rightarrow\mathcal{M}_d$ be a linear map. A map is completely positive if all its ampliations are positive, and completely copositive if its composition with transposition is completely positive. A map is entanglement breaking if it sends every bipartite positive matrix to a separable matrix. PPT squared conjecture. If $T$ is completely positive and completely copositive, then

T\circ T

is entanglement breaking. This conjecture asks whether applying a PPT map twice always destroys entanglement; it remains open, with partial progress known for asymptotic versions and additional hypotheses.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

PPT squared conjecture
A linear map is PPT when it is completely positive and its Choi matrix is PPT; a map is entanglement breaking when its Choi matrix is separable. Let $\mathcal{L}_1$ and $\mathcal{L}_2$ be completely positive linear maps.

PPT squared conjecture. If the Choi matrices of $\mathcal{L}_1$ and $\mathcal{L}_2$ are PPT, then the Choi matrix of
$\mathcal{L}_1\circ\mathcal{L}_2$
is separable; equivalently, the composition of any two PPT linear maps is entanglement breaking. The conjecture is resolved in dimensions $d=2$ and $d=3$ , while it remains open in general and has been proved in various restricted settings.
source: Sang-Jun Park, “k-Positivity and high-dimensional bound entanglement under symplectic group symmetries”, arXiv:2602.09860 (2026).

References

Primary source

Matthias Christandl, Alexander Müller-Hermes and Michael M. Wolf, “When Do Composed Maps Become Entanglement Breaking?”, arXiv:1807.01266 (2019).

Additional references

2 papers in this index state this conjecture (2018). The statement above is taken from the most recent of them; the others are arXiv:1805.11570.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Open

The conjecture is proved in low dimensions and special families, but the general higher-dimensional question remains open, with no public proof or counterexample.

Matthias Christandl proposed the conjecture in 2012: squaring every completely positive and completely copositive map should produce an entanglement-breaking map.

Known results

The conjecture holds for $d=2$ and $d=3$ , including compositions with intermediate dimension $3$ .
Kennedy-Manor-Paulsen proved asymptotic entanglement breaking for unital or trace-preserving PPT maps.
Rahaman-Jaques-Paulsen proved that every unital PPT channel becomes entanglement breaking after finitely many iterations.
It holds for Choi-type maps, Gaussian channels, and several further structured families.

July 2026 qutrit update

A July 2026 preprint proves a related composition result for a completely positive map with $1$ -undistillable Choi matrix and another with Schmidt number at most $2$ . It strengthens the qutrit picture but does not resolve dimensions $d>3$ ; no general proof or counterexample is reported.

Current status (as of August 2026): The conjecture is settled for $d\leq 3$ and several restricted classes, while the general case for $d>3$ remains open.

Sources

No solutions have been posted yet.