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On the minimal dimension of range-criterion counterexamples to the PPT² conjecture

A mix of human-written and AI-generated textHuman understanding: all partsquant-ph — Quantum Physicsmath.AG — Algebraic Geometrymath.OA — Operator Algebrasmath-ph — Mathematical Physics

Contributed by Tejasvi Singh Tomar ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

The PPT2^2 conjecture asserts that the composition of two completely positive and completely copositive (PPT) maps is entanglement breaking. A recently announced counterexample, with PPT maps M10→M6→M10\mathrm{M}_{10}\to\mathrm{M}_6\to\mathrm{M}_{10}, is certified by the range criterion: the range of the composite Choi matrix contains no nonzero product vector. We ask how small a counterexample certified in this way can be. For PPT maps Φ ⁣:Mm→Mk\Phi\colon\mathrm{M}_m\to\mathrm{M}_k and Ψ ⁣:Mk→Mn\Psi\colon\mathrm{M}_k\to\mathrm{M}_n with Ψ∘Φ≠0\Psi\circ\Phi\neq0 and max⁡(m,k,n)≤6\max(m,k,n)\le6 we prove that the range of the Choi matrix of Ψ∘Φ\Psi\circ\Phi contains a nonzero product vector whenever k≤5k\le5 or min⁡(m,n)≤5\min(m,n)\le5. Hence the range criterion cannot certify a counterexample on M4\mathrm{M}_4 or M5\mathrm{M}_5, and among the types with all three dimensions at most six only (6,6,6)(6,6,6) remains open. Since the range is spanned by products of Kraus operators, the question is when such a span can avoid the matrices of rank one. The computer-assisted proof combines rank bounds for PPT maps with Chern class and excess intersection computations; its computed facts are checked in Lean 4, modulo classical, quoted or hand-proved inputs. The announced counterexample lifts through exterior powers a map on M4\mathrm{M}_4 built from a pencil of real matrices (a map of pencil type). When the map of pencil type loses rank transversally at finitely many points, its lifts through any Schur functor, exterior powers included, put product vectors into the kernel of the composite Choi matrix. When the composite is also nonzero, these product vectors form an unextendible product set only for exterior powers, and then one of the three dimensions is at least ten.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23715651

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Version history

  1. v1 · 38 pagesSubmitted by Tejasvi Singh TomarPublished on Zenodo: https://doi.org/10.5281/zenodo.23715651CurrentNov 12, 2026