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arXiv:2608.03390

The geometry of absolute separability and other convex matrix properties from spectrum

Jennifer Ahiable, Naga Bhavya Teja Kothakonda, Andreas Winter

quant-phmath-phSubmitted Aug 4, 2026• Announced Aug 5, 2026• 0 citations
v1 · Aug 4, 2026v2 · Sep 18, 2026
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Abstract

We investigate the geometric structure of the set of spectra of bipartite absolute separable states ($\mathrm{ASEP}_{m,n}$) and absolute positive partial transpose states ($\mathrm{APPT}_{m,n}$), i.e., bipartite quantum states that remain separable or PPT respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that $\mathrm{APPT}_{m,n}$ is a spectrahedron for all $m\leq n$: in particular, all its faces are exposed. In contrast, while $\mathrm{ASEP}_{2,n}$ is also a spectrahedron, we prove that in general $\mathrm{ASEP}_{m,n}$ is a semialgebraic set for all $m\leq n$. Furthermore, we provide a complete characterization of the faces and extreme points of $\mathrm{APPT}_{m,n}$ and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension $(mn-m-1)$. In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of $\mathrm{APPT}_{m,n}$ via an inscribed polytope $\mathcal{P}_{m,n}$ and conjecture that the maximal purity of $\mathrm{APPT}_{m,n}$ (along with its spectra) coincides with the polytope for arbitrary dimensions except when $m=n=2$. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of $\mathrm{APPT}_{m,n}$ and demonstrate numerically that the minimum entropy eventually coincides with the polytope $\mathcal{P}_{m,n}$ as the local system dimension $n$ increases. Finally, we show that the relative spectral volume of $\mathrm{APPT}_{m,n}$ decays exponentially in $n$ by a constant multiplicative factor of the relative volume of the inscribed polytope $\mathcal{P}_{m,n}$.

Comments
48 pages, 11 figures

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We investigate the geometric structure of the set of spectra of bipartite absolute separable states (ASEPm,n\mathrm{ASEP}_{m,n}ASEPm,n​) and absolute positive partial transpose states (APPTm,n\mathrm{APPT}_{m,n}APPTm,n​), i.e., bipartite quantum states that remain separable or PPT respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ is a spectrahedron for all m≤nm\leq nm≤n: in particular, all its faces are exposed. In contrast, while ASEP2,n\mathrm{ASEP}_{2,n}ASEP2,n​ is also a spectrahedron, we prove that in general ASEPm,n\mathrm{ASEP}_{m,n}ASEPm,n​ is a semialgebraic set for all m≤nm\leq nm≤n. Furthermore, we provide a complete characterization of the faces and extreme points of APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension (mn−m−1)(mn-m-1)(mn−m−1).
In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ via an inscribed polytope Pm,n\mathcal{P}_{m,n}Pm,n​ and conjecture that the maximal purity of APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ (along with its spectra) coincides with the polytope for arbitrary dimensions except when m=n=2m=n=2m=n=2. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ and demonstrate numerically that the minimum entropy eventually coincides with the polytope Pm,n\mathcal{P}_{m,n}Pm,n​ as the local system dimension nnn increases. Finally, we show that the relative spectral volume of APPTm,n\mathrm{APPT}_{m,n}APPTm,n​ decays exponentially in nnn by a constant multiplicative factor of the relative volume of the inscribed polytope Pm,n\mathcal{P}_{m,n}Pm,n​.
Abstract Quantum states that remain separable (i.e. not entangled) under any global unitary transformation are known as absolutely separable and form a convex set. Despite extensive efforts, the complete characterization of this set remains largely unknown. In this work, we employ linear maps and their inverses to derive new sufficient analytical conditions for absolute separability in arbitrary dimensions, providing extremal points of this set and improving its characterization. Additionally, we employ convex geometry optimization to refine the characterization of the set when multiple non-comparable criteria for absolute separability are available. We also address the closely related problem of characterizing the absolute PPT (positive partial transposition) set, which consists of quantum states that remain positive under partial transposition across all unitary transformations. Finally, we extend our results to multipartite states. We are proud to dedicate our work to Professor Ryszard Horodecki, whose pioneering contributions to the field of quantum entanglement continue to inspire us all. With deep gratitude and respect.
We study maximum purity and minimum von Neumann entropy of absolutely positive partial transpose (APPT) states, together with the relative volume of their spectral sets. A consequence of Hildebrand's criterion gives an explicit outer spectral polytope, whose vertices we classify. Optimizing purity over this polytope, together with the Song--Chen result for the 2⊗32\otimes32⊗3 system, yields, for every mn≥6mn\ge6mn≥6, an explicit upper bound on the purity of APPT states that is asymptotic to 4/(3mn)4/(3mn)4/(3mn). This improves the previous 2/(mn)2/(mn)2/(mn) upper bound for absolutely separable states. The new bound applies to every APPT state, a class containing all absolutely separable states, and is sharp for every 2⊗n2\otimes n2⊗n system with n≥3n\ge3n≥3. For every 3⊗n3\otimes n3⊗n system with n≥3n\ge3n≥3, however, the unique outer-polytope maximizer is not APPT, so the bound is strict and disproves the Dũng--Khôi qutrit--qudit conjecture. The polytope also gives an explicit entropy lower bound in arbitrary bipartite dimensions and, together with the Song--Chen extreme-point classification, the exact minimum entropy for every 2⊗n2\otimes n2⊗n system. Finally, exact formulas for the relative volumes of an inner polytope and the outer spectral polytope give explicit two-sided bounds on the qubit--qudit relative spectral volume ana_nan​ whose ratio is less than 444 and tends to 333. Consequently, an=Θ ⁣(n(4/27)n)a_n=Θ\!\left(\sqrt n(4/27)^n\right)an​=Θ(n​(4/27)n), and the relative volume of the qubit--qudit APPT spectral set (equivalently, the absolutely separable spectral set) has exact exponential decay rate ln⁡(27/4)\ln(27/4)ln(27/4).
Absolutely positive partial transpose (APPT) states, which remain PPT under every global unitary, provide a spectral relaxation of absolutely separable states, which remain separable under every such transformation. Despite substantial progress, whether these two notions coincide beyond qubit-qudit systems remains unresolved, motivating the study of structured state families in which the APPT spectral constraints and separability can be analyzed exactly. We study this problem for Bell-diagonal states in two settings: the standard two-qutrit Weyl-Heisenberg basis and the two-ququart lattice basis generated by two-qubit Pauli operators. In the two-qutrit case, we show that a linearized necessary condition for APPT defines a polytope of spectra in which every associated Weyl-Heisenberg Bell-diagonal state admits an exact decomposition into explicitly separable states. This yields a separability criterion that applies to a region even larger than the APPT set itself. In the two-ququart lattice Bell-diagonal setting, we show that a state remains PPT under every permutation of its Bell coefficients if and only if its six largest Bell coefficients sum to at most 1/2, and we prove that all such states are separable. Equivalently, every entangled lattice Bell-diagonal state becomes NPT after a suitable coefficient permutation and is therefore not APPT. The proofs combine convex-polytope analysis, affine and symplectic orbit reductions, and exact rational decompositions into separable states supported on affine Lagrangian subsets. Our results rule out entangled APPT states in both Bell-diagonal families, and provide simple spectral inequalities as sufficient criteria for separability in both cases. Being basis-specific, however, they do not by themselves settle whether APPT and absolute separability coincide.
A quantum state is called absolutely separable (resp. absolutely positive partial transpose (APPT)) if it remains separable (resp. positive partial transpose (PPT)) under any global unitary transformation. It is known that the set of all absolutely separable states is a subset of the set of all APPT states. Moreover, these two sets are identical for qubit-qudit systems. In this note, we give an upper bound for the purity of APPT bipartite states (and therefore for the purity of absolutely separable bipartite states). For qubit-qudit systems, this upper bound becomes the maximum purity of APPT (and absolutely separable) states.
Separability from the spectrum is an important and ongoing research topic in quantum entanglement. In this study, we investigate properties related to absolute separability from the spectrum in qudit-qudit states in the bipartite state space Hmn=Hm⊗Hn. First, we propose the necessary and sufficient conditions for absolute positive partial transpose states in the Hilbert space H4n. These conditions are equivalent to the positive semidefiniteness of twelve matrices resulting from the symmetric matricizations of eigenvalues. Furthermore, we demonstrate that these sufficient conditions can be extended to the general Hmn case, improving existing conclusions in the literature. These sufficient conditions depend only on the first few leading and last few leading eigenvalues, significantly reducing the complexity of determining absolute separable states. On the other hand, we also introduce additional sufficient conditions for determining that states in Hmn are not absolutely separable. These conditions only depend on 2m − 1 eigenvalues of the mixed states. Our sufficient conditions are not only simple and easy to implement, but also effective in practice. As applications, we derive distance bounds for eigenvalues and purity bounds for general absolutely separable states.
Absolutely stabilizer states are those that remain convex mixtures of stabilizer states after conjugation by any unitary. Here we give a characterization of such states for multiple qudits of all prime dimensions by introducing a polytope of their allowed spectra. We illustrate this through the examples of one qubit, two qubits, and one qutrit. In particular, the set of absolutely stabilizer states for a single qubit is a ball inscribed in the stabilizer octahedron, but for higher dimensions the geometry is more complicated. For odd-prime-dimensional qudits, we also give a complete characterization of absolutely Wigner-positive states, i.e., states whose Wigner function remains nonnegative after conjugation by any unitary. In so doing, we show there are absolutely Wigner-positive states that are not absolutely stabilizer, which can be seen as a unitarily invariant version of bound magic. We then study the radii of the largest balls contained in the sets of absolutely stabilizer states and absolutely Wigner-positive states. These radii respectively tell us the lowest possible purity of nonstabilizer and Wigner-negative states. Conversely, we also find the radius of the smallest ball containing the set of absolutely Wigner-positive states, giving a tight purity-based necessary condition thereof. Finally, we apply our methods to magic witnessing and find simple sufficient conditions for determining if an nnn-qubit state is reducible to an mmm-qubit magic state for any m<nm<nm<n.