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bibkey: hildebrand2007pptspectra authors: Roland Hildebrand year: 2007 title: “Positive partial transpose from spectra” doi: 10.1103/PhysRevA.76.052325 url: https://arxiv.org/abs/quant-ph/0502170 claim: “For 3-by-n dimensions, absolute PPT is equivalent to two explicitly displayed 3-by-3 spectral linear matrix inequalities.” strata_touched:

• D5/S3/Quantum/Entanglement/AbsolutePPT/QutritSpectralReduction license: citation-only triage: anchor

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DOI: https://doi.org/10.1103/PhysRevA.76.052325

Source: https://arxiv.org/abs/quant-ph/0502170

Qutrit spectral criterion

Pages 5–6; the two LMIs are equation (5), and the following statement is Corollary 4. The source wording and matrices are:

Let now $m=3$, $n\ge 3$. Then $p=3$, $p_{+}=6$, $p_{-}=3$. We shall now determine the set ${\Sigma }_{\pm }(3)$. If $x_1>x_2>x_3>0$, then we have $$x_1^2>x_1{x}_2>\max (x_2^2,x_1{x}_3)\ge \min (x_2^2,x_1{x}_3)>x_2{x}_3>x_3^2.$$ However, we can have both $x_2^2\ge x_1{x}_3$ and $x_1{x}_3\ge x_2^2$. Hence ${\Sigma }_{\pm }(3)$ consists of two elements, and the corresponding matrices $\Lambda$ are given by $${\Lambda }_1=\left(\begin{array}{ccc}{\lambda }_{3n}& {\lambda }_{3n-1}& {\lambda }_{3n-3}\\ -{\lambda }_1& {\lambda }_{3n-2}& {\lambda }_{3n-4}\\ -{\lambda }_2& -{\lambda }_3& {\lambda }_{3n-5}\end{array}\right),{\Lambda }_2=\left(\begin{array}{ccc}{\lambda }_{3n}& {\lambda }_{3n-1}& {\lambda }_{3n-2}\\ -{\lambda }_1& {\lambda }_{3n-3}& {\lambda }_{3n-4}\\ -{\lambda }_2& -{\lambda }_3& {\lambda }_{3n-5}\end{array}\right).$$ Thus we obtain the two LMIs $$\left(\begin{array}{ccc}2{\lambda }_{3n}& {\lambda }_{3n-1}-{\lambda }_1& {\lambda }_{3n-3}-{\lambda }_2\\ {\lambda }_{3n-1}-{\lambda }_1& 2{\lambda }_{3n-2}& {\lambda }_{3n-4}-{\lambda }_3\\ {\lambda }_{3n-3}-{\lambda }_2& {\lambda }_{3n-4}-{\lambda }_3& 2{\lambda }_{3n-5}\end{array}\right)⪰0,\left(\begin{array}{ccc}2{\lambda }_{3n}& {\lambda }_{3n-1}-{\lambda }_1& {\lambda }_{3n-2}-{\lambda }_2\\ {\lambda }_{3n-1}-{\lambda }_1& 2{\lambda }_{3n-3}& {\lambda }_{3n-4}-{\lambda }_3\\ {\lambda }_{3n-2}-{\lambda }_2& {\lambda }_{3n-4}-{\lambda }_3& 2{\lambda }_{3n-5}\end{array}\right)⪰0.$$

Corollary 4. Let $A$ be a self-adjoint PSD operator on the Hilbert space $H_{3n}$. Let ${\lambda }_1,\dots ,{\lambda }_{3n}$ be the eigenvalues of $A$ in decreasing order. Then $A$ is PPT with respect to any decomposition of $H_{3n}$ as a tensor product $H_3\otimes H_n$ if and only if linear matrix inequalities (5) hold.

The delivery uses the necessary direction only. The Lean nine-coordinate boundary vector contains the top three eigenvalues followed by the bottom six, all in decreasing order; source indices start at one and Lean Fin indices start at zero.