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Exact quadbit–quadbit APPT maximum purity

Abstract

The literal 4-by-4 APPT purity maximum is 2/27 and is attained by the explicit diagonal spectrum (1/6,1/18,…,1/18).

Theorem 1.1 (Unit dot product gives real squared norm one).

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/AbsolutePPT/QuadbitMaximumPurity.norm_of_dot_eq_one (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Jennifer Ahiable; Naga Bhavya Teja Kothakonda; Andreas Winter (2026). The geometry of absolute separability and other convex matrix properties from spectrum. DOI: 10.48550/arXiv.2608.03390. URL: https://arxiv.org/abs/2608.03390v2.

Acknowledgement. Anh T. Tran (2026). Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay. DOI: 10.48550/arXiv.2609.18568. URL: https://arxiv.org/abs/2609.18568v1.

Commentary.

For every finite index type I and complex vector v, star v dot v = 1 implies that the sum of squared coordinate norms is the real number one. This converts unitary preservation of the complex dot product to the normalization used by the separable rank-one ray.

Theorem 1.2 (Separable matrices have positive partial transpose).

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/AbsolutePPT/QuadbitMaximumPurity.separableCone_partialTranspose_posSemidef (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Jennifer Ahiable; Naga Bhavya Teja Kothakonda; Andreas Winter (2026). The geometry of absolute separability and other convex matrix properties from spectrum. DOI: 10.48550/arXiv.2608.03390. URL: https://arxiv.org/abs/2608.03390v2.

Acknowledgement. Anh T. Tran (2026). Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay. DOI: 10.48550/arXiv.2609.18568. URL: https://arxiv.org/abs/2609.18568v1.

Commentary.

For all natural local dimensions, second-factor partial transpose preserves positivity of a matrix in the separable cone. Each product of two positive matrices becomes the Kronecker product of the first factor and the transpose of the second, both positive; finite sums preserve positivity.

Definition 1.3 (Upper bound, supremum and explicit attaining state).

Formalization. D5/S3/Quantum/Entanglement/AbsolutePPT/QuadbitMaximumPurity.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. Jennifer Ahiable; Naga Bhavya Teja Kothakonda; Andreas Winter (2026). The geometry of absolute separability and other convex matrix properties from spectrum. DOI: 10.48550/arXiv.2608.03390. URL: https://arxiv.org/abs/2608.03390v2.

Acknowledgement. Anh T. Tran (2026). Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay. DOI: 10.48550/arXiv.2609.18568. URL: https://arxiv.org/abs/2609.18568v1.

Commentary.

Ahiable–Kothakonda–Winter, arXiv:2608.03390v2, Conjecture 6.7, page 29, conjectures that the inscribed absolute-PPT polytope has the same maximum purity as the full APPT set, attained at the spectra in Eq. (44). This claim is precisely its m=n=4 sector: rho is a complex matrix on Fin 4 × Fin 4, positive semidefinite with trace one, and APPT means the second-factor partial transpose of U rho Uᴴ is positive semidefinite for every global unitary U. Purity is the real part of trace(rho*rho). It asserts the universal upper bound, equality of the real supremum with 2/27, and a density matrix equal to the prescribed diagonal state, with one complex diagonal entry 1/6 and fifteen entries 1/18. It does not classify all maximizers, address other local dimensions, or equate APPT with absolute separability.

Theorem 1.4 (Exact maximum in the 4-by-4 sector).

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/AbsolutePPT/QuadbitMaximumPurity.result (✓ std3). ∎

Resolves. Problems/ahiable-kothakonda-winter-2026-appt-quadbit-maximum-purity (proved) by D5/S3/Quantum/Entanglement/AbsolutePPT/QuadbitMaximumPurity.result.

Source. Repository-derived.

Acknowledgement. Jennifer Ahiable; Naga Bhavya Teja Kothakonda; Andreas Winter (2026). The geometry of absolute separability and other convex matrix properties from spectrum. DOI: 10.48550/arXiv.2608.03390. URL: https://arxiv.org/abs/2608.03390v2.

Acknowledgement. Anh T. Tran (2026). Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay. DOI: 10.48550/arXiv.2609.18568. URL: https://arxiv.org/abs/2609.18568v1.

Commentary.

The proof derives necessary rank-three Bell-corner matrices from the literal APPT predicate. With t the tenth decreasing eigenvalue, positivity gives t≤1/13. Inserting two copies of t for t≤1/18, or deleting t for t≥1/18, preserves both boundary matrices after positive normalization. The original dimension-18 and dimension-15 sharp spectral certificates then give the upper bound 2/27. These are reused certificate theorems, with no copied certificates and no assumption of Hildebrand sufficiency in local dimension four. The known low-rank separable ray proves APPT for the explicit attaining state. This resolves only the stated 4-by-4 sector; worldwide priority remains outside the formal theorem.

References