The PPT squared conjecture

From papers

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.

Progress summary

Partially solved

The conjecture is proved in small dimensions and several special settings, but no general proof or counterexample has appeared.

The conjecture says that squaring any PPT map destroys entanglement. It was first proposed by M. Christandl in 2012.

Known results

The conjecture holds for maps on $d=2$ matrices and for all maps on $d=3$ matrices.
For unital or trace-preserving PPT maps, iterates approach the entanglement-breaking maps asymptotically; for unital completely copositive channels, some finite power is entanglement breaking.
It holds for Gaussian quantum channels and certain graph-associated families.
It is proved for broad diagonal-unitary-covariant classes, including Choi-type, depolarizing, dephasing, and amplitude-damping maps.

July 2026 qutrit update

A recent study reiterates that the qutrit case is settled while dimensions $d>3$ remain open; it reports neither a general proof nor a counterexample.

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 with no reported counterexample or proof.

Sources

Equivalent formulations 1

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).

Sources & referencesView supporting material

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.

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