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Separability from spectrum
Characterize the spectra that force every bipartite state with those eigenvalues to be separable, and decide whether a PPT test suffices.
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| 1 | +# Separability from spectrum | |
| 2 | + | |
| 3 | +Some mixed states are so noisy that no global unitary, however entangling, can make them entangled. Whether a state has this property depends only on its eigenvalues. The problem is to describe exactly which eigenvalue lists guarantee it, and in particular whether a simple, computable test based on the partial transpose already gives the full answer. | |
| 4 | + | |
| 5 | +## Problem | |
| 6 | + | |
| 7 | +Let $2\le m\le n$ and let $\lambda=(\lambda_1\ge\dots\ge\lambda_{mn}\ge0)$ be a probability vector. Call $\lambda$ absolutely separable if every state on $\mathbb{C}^m\otimes\mathbb{C}^n$ with spectrum $\lambda$ is separable, that is, | |
| 8 | + | |
| 9 | +$$U\operatorname{diag}(\lambda)\,U^\dagger\ \text{is separable for every unitary } U\in\mathrm{U}(mn).$$ | |
| 10 | + | |
| 11 | +Call $\lambda$ absolutely PPT if $U\operatorname{diag}(\lambda)U^\dagger$ has a positive partial transpose for every $U$. Write $\mathrm{ASEP}_{m,n}$ and $\mathrm{APPT}_{m,n}$ for these two sets of spectra. Since separable states are PPT, $\mathrm{ASEP}_{m,n}\subseteq\mathrm{APPT}_{m,n}$. | |
| 12 | + | |
| 13 | +**Question.** Characterize $\mathrm{ASEP}_{m,n}$. In particular, is $\mathrm{ASEP}_{m,n}=\mathrm{APPT}_{m,n}$ when $m,n\ge3$? The first open case is two qutrits, $m=n=3$. | |
| 14 | + | |
| 15 | +## Background | |
| 16 | + | |
| 17 | +The question arises when someone holds a highly mixed state of two systems and may apply any joint unitary: can entanglement be created at all? It is No. 15 in the Open Quantum Problems collection of IQOQI Vienna, posed by E. Knill in 2003 [6]. States of this kind were named absolutely separable by Kuś and Życzkowski in 2001. The absolutely PPT spectra are fully described by finitely many linear matrix inequalities [2], so a positive answer would turn an intractable-looking property into a semidefinite test. | |
| 18 | + | |
| 19 | +## What is known | |
| 20 | + | |
| 21 | +- Every state with purity $\operatorname{Tr}\rho^2\le 1/(mn-1)$ is separable, and this purity bound is tight; it misses many absolutely separable spectra (Gurvits and Barnum) [6]. | |
| 22 | +- For two qubits, $\lambda$ is absolutely separable exactly when $\lambda_1-\lambda_3-2\sqrt{\lambda_2\lambda_4}\le0$ [1]. | |
| 23 | +- Hildebrand characterized $\mathrm{APPT}_{m,n}$ in all dimensions by linear matrix inequalities on the eigenvalues [2]. | |
| 24 | +- Johnston proved $\mathrm{ASEP}_{2,n}=\mathrm{APPT}_{2,n}$ for every $n$, so all qubit–qudit cases are solved [3]. | |
| 25 | +- Arunachalam, Johnston and Russo showed that the two sets agree on isotropic states, and that many standard separability criteria (range, realignment, the Choi and Breuer–Hall maps) cannot detect entanglement in any absolutely PPT state [4]. | |
| 26 | +- Ahiable, Kothakonda and Winter (2026) showed that $\mathrm{APPT}_{m,n}$ is a spectrahedron whose faces are all exposed, and described its faces and extreme points; for $m,n\ge3$ they could only show that $\mathrm{ASEP}_{m,n}$ is semialgebraic [5]. | |
| 27 | +- Within Bell-diagonal families, absolutely PPT implies separable for two qutrits in the Weyl–Heisenberg basis and for two ququarts in the two-qubit Pauli basis ([Patra and Hiesmayr, 2026 preprint](https://arxiv.org/abs/2610.12000)). General states remain open. | |
| 28 | + | |
| 29 | +## What would count as progress | |
| 30 | + | |
| 31 | +- A separating example: a spectrum $\lambda\in\mathrm{APPT}_{3,3}$, a unitary $U$ and an entanglement witness $W$ with $\operatorname{Tr}\big(W\,U\operatorname{diag}(\lambda)U^\dagger\big)<0$. This is checkable in exact arithmetic and would answer the question in the negative. | |
| 32 | +- A proof that $\mathrm{ASEP}_{3,3}=\mathrm{APPT}_{3,3}$, for instance by giving certified separable decompositions for every extreme point of $\mathrm{APPT}_{3,3}$ in the description of [5]. | |
| 33 | +- New explicit families of absolutely separable spectra for $m,n\ge3$ that reach the boundary of $\mathrm{APPT}_{m,n}$. | |
| 34 | + | |
| 35 | +## References | |
| 36 | + | |
| 37 | +1. F. Verstraete, K. Audenaert and B. De Moor, "Maximally entangled mixed states of two qubits", *Physical Review A* 64, 012316 (2001). [doi:10.1103/PhysRevA.64.012316](https://doi.org/10.1103/PhysRevA.64.012316) | |
| 38 | +2. R. Hildebrand, "Positive partial transpose from spectra", *Physical Review A* 76, 052325 (2007). [doi:10.1103/PhysRevA.76.052325](https://doi.org/10.1103/PhysRevA.76.052325) | |
| 39 | +3. N. Johnston, "Separability from spectrum for qubit-qudit states", *Physical Review A* 88, 062330 (2013). [doi:10.1103/PhysRevA.88.062330](https://doi.org/10.1103/PhysRevA.88.062330) | |
| 40 | +4. S. Arunachalam, N. Johnston and V. Russo, "Is absolute separability determined by the partial transpose?", *Quantum Information and Computation* 15, 694–720 (2015). [arXiv:1405.5853](https://arxiv.org/abs/1405.5853) | |
| 41 | +5. J. Ahiable, N. B. T. Kothakonda and A. Winter, "The geometry of absolute separability and other convex matrix properties from spectrum", preprint (2026). [arXiv:2608.03390](https://arxiv.org/abs/2608.03390) | |
| 42 | +6. E. Knill (contact), "Separability from spectrum", Open Quantum Problems, IQOQI Vienna, problem 15 (posted 2003). [Archived page](https://web.archive.org/web/20231029013533/https://oqp.iqoqi.oeaw.ac.at/separability-from-spectrum) | |
| 43 | + | |
| 44 | +## Source | |
| 45 | + | |
| 46 | +Imported from [QIQCOP Zoo op_b063bdae4363cda8](https://qiqc-op.com/problem/op_b063bdae4363cda8/), curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion. | |