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The PPT² conjecture holds for spin-factor maps

A mix of human-written and AI-generated textHuman understanding: all partsquant-ph — Quantum Physicsmath.OA — Operator Algebrasmath.FA — Functional Analysis

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 maps is entanglement breaking. A counterexample in dimension ten has recently been announced, and for maps on M4\mathrm{M}_4 the conjecture is open. Let S⊂M4\mathcal{S}\subset\mathrm{M}_4 be the real span of the identity γ0\gamma_0 and five pairwise anticommuting Hermitian unitaries γ1,…,γ5\gamma_1,\dots,\gamma_5. It is a spin factor whose positive cone is a Lorentz cone. We prove the conjecture for all compositions Ψ∘Φ\Psi\circ\Phi of completely positive maps Φ ⁣:M4→Mk\Phi\colon\mathrm{M}_4\to\mathrm{M}_k and Ψ ⁣:Mk→M4\Psi\colon\mathrm{M}_k\to\mathrm{M}_4, with kk arbitrary, such that the ranges of Ψ\Psi and of the adjoint Φ∗\Phi^{*} lie in invertible congruences of the complex span of S\mathcal{S}. Such maps are automatically completely copositive, and the class contains maps whose Choi matrices are entangled. A theorem of Hildebrand on separability for pairs of Lorentz cones reduces the claim to one trace inequality between the coefficient matrices of the two Choi matrices. We derive this inequality from the operator inequality ∑j=15FjF0−1Fj⪯F0\sum_{j=1}^{5}F_jF_0^{-1}F_j\preceq F_0, valid whenever ∑μ=05Fμ⊗γμ⪰0\sum_{\mu=0}^{5}F_\mu\otimes\gamma_\mu\succeq 0 and F0≻0F_0\succ 0, which in turn follows from the Kadison–Schwarz inequality. As a corollary, Ψ∘Φ\Psi\circ\Phi is entanglement breaking whenever Ψ\Psi is such a map and Φ\Phi is a completely positive map with Kraus operators of rank at most two.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23483252

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Version history

  1. v1 · 10 pagesSubmitted by Tejasvi Singh TomarPublished on Zenodo: https://doi.org/10.5281/zenodo.23483252CurrentNov 12, 2026