The PPT-squared conjecture

Problem

Must the composition of any two compatible PPT completely positive maps be entanglement breaking? Let \(\Phi:M_{d_1}(\mathbb C)\to M_{d_2}(\mathbb C)\) and \(\Psi:M_{d_2}(\mathbb C)\to M_{d_3}(\mathbb C)\) be completely positive. The Choi operator of \(\Phi\) is

\begin{equation} J(\Phi) :=\sum_{i,j=1}^{d_1}\lvert i\rangle\!\langle j\rvert \otimes\Phi(\lvert i\rangle\!\langle j\rvert), \tag{1} \end{equation}

and \(J(\Psi)\) is defined analogously. In the convention of Eq. (1), a map \(\Phi\) is PPT when

\begin{equation} (T\otimes\operatorname{id})\bigl(J(\Phi)\bigr)\succeq0, \tag{2} \end{equation}

and it is entanglement breaking when \(J(\Phi)\) is separable; the same definitions apply to \(\Psi\). Under condition Eq. (2) for both maps, is \(J(\Psi\circ\Phi)\) necessarily separable? Equivalently, does postselection on any joint measurement outcome on the middle systems of two PPT bipartite states always leave a separable state on the two outer systems?

Source

Christandl, Müller–Hermes, and Wolf formulate the PPT-squared statement as an explicit conjecture about compositions of PPT maps [CMW19].

Progress

The conjecture was recorded in the Banff workshop report on operator structures in quantum information [BIRS12].
Christandl, Müller–Hermes, and Wolf formulated the two-map statement above, proved its equivalence to the self-composition version, and established it in equal dimension \(d=2\), for Gaussian channels, and for several further special classes. Their eventual entanglement-breaking results for repeated composition do not imply the two-composition claim [CMW19].
The equal-dimension conjecture holds for \(d=3\) [CYT19]. Consequently, equal dimension \(d\geq4\) is the first unresolved regime.
The conjecture holds for diagonal-unitary-covariant maps, a class containing Choi-type, depolarizing, dephasing, and amplitude-damping examples [SN22]. This covariance restriction does not cover arbitrary PPT maps.
Other repeated-composition theorems prove eventual entanglement breaking for broad PPT families, but not after exactly two maps [KMP18].
A 2026 qutrit theorem proves a stronger statement for cones larger than the PPT cone: the composition of any completely positive map whose Choi matrix is \(1\)-undistillable with any map whose Choi matrix has Schmidt number at most two is entanglement breaking in either order, and the \(1\)-undistillable cone is exactly the largest qutrit cone with this property. The result also extends to arbitrary selective measurements on the intermediate systems, but its dimension-specific hypothesis leaves higher dimensions open [AL26].
Nechita and Park introduced random covariant channels, including random diagonal-orthogonal-covariant channels, and proved that the composition of two random diagonal-orthogonal-covariant channels is generically entanglement breaking, giving higher-dimensional generic evidence without resolving the unrestricted conjecture [NP26].

Comment

The two-composition claim is settled in equal dimensions two and three, for several structured families, and generically for random covariant channels (Progress above). No proof or counterexample is known for arbitrary compatible finite-dimensional PPT maps, with equal dimension four being the first open square case.

References

[BIRS12]: M. B. Ruskai, M. Junge, D. Kribs, P. Hayden, and A. Winter (organizers), Operator Structures in Quantum Information Theory, Banff International Research Station Workshop Report 12w5084 (2012). workshop report.link
[CMW19]: M. Christandl, A. Müller–Hermes, and M. M. Wolf, “When Do Composed Maps Become Entanglement Breaking?” Annales Henri Poincaré 20, 2295–2322 (2019).DOI arXiv
[CYT19]: L. Chen, Y. Yang, and W.-S. Tang, “Positive-Partial-Transpose Square Conjecture for \(n=3\),” Physical Review A 99, 012337 (2019).DOI arXiv
[SN22]: S. Singh and I. Nechita, “The \(\mathrm{PPT}^2\) Conjecture Holds for All Choi-Type Maps,” Annales Henri Poincaré 23, 3311–3329 (2022).DOI arXiv
[KMP18]: M. Kennedy, N. A. Manor, and V. I. Paulsen, “Composition of PPT Maps,” Quantum Information and Computation 18, 472–480 (2018).arXiv
[AL26]: J. An and S. Lee, “Beyond the Positive Partial Transpose Squared Conjecture: The Qutrit Case,” arXiv:2607.15947 (2026).DOI arXiv
[NP26]: I. Nechita and S.-J. Park, “Random Covariant Quantum Channels,” Annales Henri Poincaré 27, 847–907 (2026).DOI arXiv

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