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The PPT-squared conjecture: is the composition of two PPT channels always entanglement breaking?

A linear map F:Ma(C)→Mb(C)F:M_a(\mathbb C)\to M_b(\mathbb C) is PPT if both FF and T∘FT\circ F are completely positive, TT the transpose, and entanglement breaking if (id⊗F)(X)(\mathrm{id}\otimes F)(X) is separable for every positive XX, equivalently its Choi matrix is separable. Christandl conjectured that composing two PPT channels always yields an entanglement-breaking channel; the question was recorded as Problem G of the 2012 BIRS workshop 'Operator structures in quantum information theory', and stated for pairs of PPT maps as Conjecture IV.1 of Christandl, Muller-Hermes and Wolf (2019). Is Φ2∘Φ1\Phi_2\circ\Phi_1 entanglement breaking for all PPT channels Φ1,Φ2\Phi_1,\Phi_2, in particular is Φ∘Φ\Phi\circ\Phi entanglement breaking for every PPT channel Φ\Phi?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Quantum channels: PPT maps and entanglement breaking
Posed by
M. Christandl, recorded as Problem G of the BIRS workshop report 12w5084 (2012); stated as Conjecture IV.1 by M. Christandl, A. Muller-Hermes and M. M. Wolf, Ann. Henri Poincare 20 (2019)
Year posed
2012
Years open
14y
Solved
2026-09-27
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
36 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.3: there is an explicit trace-preserving PPT channel Θ\Theta on M21(C)M_{21}(\mathbb C) with Θ∘Θ\Theta\circ\Theta not entanglement breaking, answering the 2012 channel question negatively with the same channel in both slots. Theorem 1.2: explicit PPT maps Φ1,Φ2\Phi_1,\Phi_2 on M10(C)M_{10}(\mathbb C) (not trace preserving) whose composition has nonzero Choi matrix with no product vector in its range, refuting the unrestricted two-map Conjecture IV.1. The same pair yields the zero-key entangled state of the companion entry. Not shown: minimal dimensions, or whether every PPT channel becomes entanglement breaking after three compositions (the PPT-cubed question).

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The release also supplies Lean statements for both counterexamples, produced as part of the same release.

Verification

No independent mathematician has checked this yet. Checked here: Theorems 1.2 and 1.3 of the TeX source against BIRS Problem G and CMHW Conjecture IV.1, and the Lean statements lean/ComparatorChallenges/DimensionTenChannel.lean (OAI.DimensionTen.exists_channel_fin21) and DimensionTenPair.lean (OAI.DimensionTen.main_pair), whose solution modules exist at the pinned commit. Neither challenge is in the formalization catalogue; the statements were read here and were not rebuilt. exists_channel_fin21 states the headline directly: a trace-preserving PPT linear map on 21×2121\times21 complex matrices whose square is not entanglement breaking, with PPT, trace preservation and entanglement breaking defined from scratch. main_pair states the two-map counterexample in dimension ten with explicitly defined maps. No minimal dimension is claimed.

Sources

Changelog1 change

Discussion