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Exact qutrit–qudit APPT maximum purity

Abstract

For every n at least three, the literal qutrit APPT purity supremum equals the larger candidate purity and is attained by the prescribed diagonal state.

Definition 1.1 (Exact maximum and attainment).

Formalization. D5/S3/Quantum/Entanglement/AbsolutePPT/QutritQuditMaximumPurity.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, Conjecture 6.7, page 29: Let 𝒫_{m,n} ⊆ APPT_{m,n} be the inscribed absolute PPT polytope with 2 ≤ m ≤ n, n > 2. Then max_{λ∈APPT_{m,n}} ∑{i=1}^{mn} λ_i² = max{λ∈𝒫_{m,n}} ∑_{i=1}^{mn} λ_i² and occurs at the spectra given by Eq. (44). The encoding substitutes m=3 and Corollary 6.4’s two candidate values, quantifies every natural n≥3, uses density matrices on Fin 3 × Fin n, and expresses a maximum by the real sSup together with attainment at an explicit computational-basis diagonal state. For 3≤n≤8 the entry at (0,0) is 3/(3n+2) and all other entries are 1/(3n+2), giving the spectrum (3,1,…,1)/(3n+2). For n≥9 the entries with first-factor index 0 are 2/(4n) and the other entries are 1/(4n), giving n copies of 2/(4n) and 2n copies of 1/(4n). The second index test uses Fin.val, and the real diagonal entries are cast to complex numbers by Complex.ofReal. APPT is the literal predicate defined in QutritPerturbationAttainment; the purity is Complex.re of Matrix.trace (rho*rho). All displayed quotients are real division. Tran, remark after Theorem A, page 4: Determining the exact maximum APPT purity remains open.

Theorem 1.2 (Unconditional exact maximum).

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

Resolves. Problems/ahiable-kothakonda-winter-2026-appt-qutrit-maximum-purity (proved) by D5/S3/Quantum/Entanglement/AbsolutePPT/QutritQuditMaximumPurity.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 unconditional proof combines exact finite-sector dual certificates with a K1-only uniform cone estimate and the prescribed explicit diagonal density matrices. It proves this qutrit sector of the source conjecture and leaves m≥4 and APPT versus absolute separability open.

References