trureturing — Discovering truth

bibkey: ahiablekothakondawinter2026geometry authors: Jennifer Ahiable; Naga Bhavya Teja Kothakonda; Andreas Winter year: 2026 title: “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 claim: “Conjecture 6.7 states equality of APPT maximum purity and the inscribed polytope maximum, with the spectra of equation (44). Theorem 6.2 proves that the linear condition (31) implies absolute PPT; Section 7 asks to show that (31) is sufficient for absolute separability.” strata_touched:

• D5/S3/Quantum/Entanglement/AbsolutePPT/QutritPerturbationAttainment
• D5/S3/Quantum/Entanglement/AbsolutePPT/QutritQuditMaximumPurity
• D5/S3/Quantum/Entanglement/AbsoluteSeparability/AhiableKothakondaWinterEq31 license: citation-only triage: anchor

Verified locator

DOI: https://doi.org/10.48550/arXiv.2608.03390

Source: https://arxiv.org/abs/2608.03390v2

Conjecture 6.7

Page 29, verbatim statement (equation reference rendered as its printed number):

Let ${\mathcal{P}}_{m,n}\subseteq {\mathrm{APPT}}_{m,n}$ be the inscribed absolute PPT polytope with $2\le m\le n,n>2$. Then $$\underset{\lambda \in {\mathrm{APPT}}_{m,n}}{\max }\sum _{i=1}^{mn}{\lambda }_i^2=\underset{\lambda \in {\mathcal{P}}_{m,n}}{\max }\sum _{i=1}^{mn}{\lambda }_i^2$$ and occurs at the spectra given by Eq. (44).

Inscribed-polytope maximum

Corollary 6.4, pages 26–27, verbatim source statement:

Let ${\mathcal{P}}_{m,n}\subseteq {\mathrm{APPT}}_{m,n}$ be the inscribed absolute PPT polytope with $2\le m\le n,n>2$. The maximum achievable purity of a quantum state $\rho \in {\mathcal{M}}_m\otimes {\mathcal{M}}_n$ with spectrum $\lambda \in {\mathcal{P}}_{m,n}$ is given by $$\underset{\lambda \in {\mathcal{P}}_{m,n}}{\max }\sum _{i=1}^{mn}{\lambda }_i^2=\{\begin{array}{ll}\frac{mn+8}{(mn+2{)}^2},& \text{if }n<n^{\ast },\\ \frac{m}{(m^2-1)n},& n\ge n^{\ast }\text{ and }\left(m\text{ is odd or }n\text{ is even}\right),\\ \frac{mn(m^2-1)+2m}{((m^2-1)n+1{)}^2},& n\ge n^{\ast },m\text{ is even and }n\text{ is odd }\end{array}$$ where $$n^{\ast }=\frac{2(m^2-2)+2\sqrt{(m^2-1)(m^2-4)}}{m}.$$ Furthermore, the spectrum of the maximal purity state with ${\lambda }_1\ge \cdots \ge {\lambda }_{mn}\ge 0$ is given by $$\{\begin{array}{ll}{\lambda }_1=\frac{3}{mn+2},{\lambda }_j=\frac{1}{mn+2}& \text{for }j=2,\dots ,mn,\text{ if }n<n^{\ast }\\ {\lambda }_i=a,{\lambda }_j=b& \text{for }i=1,\dots ,t\text{ and }j=t+1,\dots ,mn,\text{ if }n\ge n^{\ast }\end{array}$$ where the block size $t=\lceil \frac{(m-1)n}{2}\rceil$, and the eigenvalues $a$ and $b$ evaluate exactly to $$a=\frac{m+1}{2t+mn(m-1)}\text{and}b=\frac{m-1}{2t+mn(m-1)}.$$

Theorem 6.2 and the Section 7 question

Theorem 6.2, page 23, verbatim:

Given a mixed state $\rho \in {\mathcal{M}}_m\otimes {\mathcal{M}}_n$ with a decreasingly ordered eigenvalue spectrum ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_{mn}\ge 0$, if the following linear inequality holds: $$2{\lambda }_{mn}+\sum _{k=1}^{m-1}{\lambda }_{mn-k}\ge \sum _{k=1}^{m-1}{\lambda }_k(31)$$ then the spectrum lies within ${\mathcal{P}}_{m,n}$, and $\rho$ is absolutely PPT.

Section 7 (Conclusion), page 39:

Thus, another matter of interest is to resolve Conjecture 6.7 in the positive, and to show that the linear condition in Eq. (31) is sufficient for absolute separability: because then the maximum purity over all three sets must coincide.

Absolute separability convention

Page 2:

Within this class of states lies a convex and compact subset of separable states that remain separable after the transformation $U\rho U^{†}$ under all global unitaries $U$ in the unitary group $\mathcal{U}(mn)$, widely referred to as absolutely separable states.

Partial transpose convention

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$\Gamma (X\otimes Y)=X\otimes Y^T$, where $X\in {\mathcal{M}}_m$ and $Y\in {\mathcal{M}}_n$, and $T$ denotes the standard matrix transpose.

Absolute PPT convention

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Defined analogously to absolutely separable states, absolute PPT states are those states which remain PPT after global unitary rotations and have been completely characterized across all dimensions [17].

Page 11:

Suppose $\lambda$ is the spectrum of a bipartite state. Then $\lambda \in {\mathrm{APPT}}_{m,n}$ if and only if for all unitaries $U\in \mathcal{U}(mn)$, (U\diag(\lambda)U^\dagger)^\Gamma \geq 0.

The matrix encoding uses the second-factor partial transpose on Fin 3 × Fin n. Its trace-squared purity is the real part of Matrix.trace (rho * rho). The qutrit sector substitutes m = 3 and t = n in Corollary 6.4. The two candidate purities become (3n + 8)/(3n + 2)^2 and 3/(8n).