PPT-squared conjecture
Is entanglement breaking for every pair of compatible finite-dimensional complex completely positive, trace-preserving maps whose Choi matrices have positive partial transpose?
Problem statement
Is entanglement breaking for every pair of compatible finite-dimensional complex completely positive, trace-preserving maps whose Choi matrices have positive partial transpose?
Lean 4
No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.
Background
A PPT channel has a Choi matrix positive under partial transpose. An entanglement-breaking channel destroys entanglement with every reference system, equivalently having a separable Choi matrix. The conjecture asks whether two compatible PPT stages always suffice. It connects channel composition with limits on entanglement swapping across PPT-entangled links.
Established special cases include low dimensions and Gaussian channels, as developed by Christandl–Müller-Hermes–Wolf and related work. The general complex higher-dimensional problem remains the target. The straightforward real-Hilbert-space analogue is false, so the scalar field is not cosmetic.
Recent partial results include symplectic-covariant and Cartan-covariant map classes. An–Lee's July 2026 preprint claims a stronger qutrit composition theorem involving one-copy-undistillable Choi matrices and Schmidt number at most two; it does not settle arbitrary higher dimension.
Important scope update: Park's August preprint claims that every PPT channel has a finite entanglement-breaking index, and gives an index-at-most-three bound for a restricted family. Eventual entanglement breaking is therefore not being listed as the open existence question. A finite number of iterations depending on the map does not establish that two always suffice.
A general proof or one complex finite-dimensional PPT-channel pair whose composition is not entanglement breaking resolves the stated conjecture.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
preprint: Every PPT channel is claimed to become entanglement breaking after finitely many iterations; this does not settle the two-step conjecture. Evidence
preprint: Qutrit composition result beyond the PPT setting, not an arbitrary-dimensional resolution. Evidence
- [1]Christandl, Müller-Hermes & Wolf, 'When Do Composed Maps Become Entanglement Breaking?', Annales Henri Poincaré (2019) link
- [2]Prudhoe, 'Entanglement Breaking Structure of Cartan-Covariant Quantum Channels' (2026 revision, v3) link
- [3]Park, 'k-Positivity and high-dimensional bound entanglement under symplectic group symmetry' (2026 preprint) link
- [4]An & Lee, 'Beyond the Positive Partial Transpose Squared Conjecture: The Qutrit Case' (2026 preprint) link
- [5]Park, 'Every PPT channel has finite entanglement breaking index' (2026 preprint, v2) link
- [6]Chiribella, Davidson, Paulsen & Rahaman, 'Positive maps and entanglement in real Hilbert spaces' (real-analogue counterexample) link
Recorded progress
Add yours →Nothing recorded yet. A pointer to a paper the entry is missing counts.
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Discussion
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