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Journal of Mathematical Physics

Spectral criteria for absolute positive partial transpose (PPT) in qudit-qudit state systems

Liang Xiong, Nung-Sing Sze

Published Jan 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1063/5.0273779

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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.
Separability from the spectrum is a significant and ongoing research topic in quantum entanglement. In this study, we investigate properties related to absolute separability from the spectrum in qudits-qudits states in the bipartite states space Hmn=Hm⊗Hn\mathcal{H}_{mn}=\mathcal{H}_m \otimes \mathcal{H}_nHmn​=Hm​⊗Hn​. Firstly, we propose the necessary and sufficient conditions for absolute separable states in the Hilbert space H4n\mathcal{H}_{4n}H4n​. These conditions are equivalent to the positive semidefiniteness of twelve matrices resulting from the symmetric matricizations of eigenvalues. Furthermore, we demonstrate that this sufficient condition can be extended to the general Hmn\mathcal{H}_{mn}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\mathcal{H}_{mn}Hmn​ are not absolutely separable. These conditions only depend on 2m−12m-12m−1 eigenvalues of the mixed states. Our sufficient conditions are not only simple and easy to implement. As applications, we derive distance bounds for eigenvalues and purity bounds for general absolutely separable states.
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.
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.
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​), \emph{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. While ASEP2,n\mathrm{ASEP}_{2,n}ASEP2,n​ is a spectrahedron, too, for ASEPm,n\mathrm{ASEP}_{m,n}ASEPm,n​ we can in general only show that it is a semialgebraic set for all m,n≥3m,n\geq 3m,n≥3. 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−1mn-m-1mn−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​.
The absolutely separable (resp. PPT) states remain separable (resp. positive partial transpose) under any global unitary operation. We present a compact form of the extreme points in the sets of absolutely separable states and PPT states in two-qubit and qubit-qudit systems. The results imply that each extreme point has at most three distinct eigenvalues. We establish a necessary and sufficient condition for determining extreme points of the set of absolutely PPT states in two-qutrit and qutrit-qudit systems, expressed as solvable linear equations. We also demonstrate that any extreme point in qutrit-qudit system has at most seven distinct eigenvalues. We introduce the concept of robustness of nonabsolute separability. It quantifies the minimal amount by which a state needs to mix with other states such that the overall state is absolutely separable. We show that the robustness satisfies positivity, invariance under unitary transformation, monotonicity and convexity, so it is a good measure within the resource theory of nonabsolute separability. Analytical expressions for this measure are given for pure states in arbitrary system and rank-two mixed states in two-qubit system.
The absolute separability problem asks for a characterization of the quantum states ρ∈Mm⊗Mnρ\in M_m\otimes M_nρ∈Mm​⊗Mn​ with the property that UρU†UρU^\daggerUρU† is separable for all unitary matrices UUU. We investigate whether or not it is the case that ρρρ is absolutely separable if and only if UρU†UρU^\daggerUρU† has positive partial transpose for all unitary matrices UUU. In particular, we develop an easy-to-use method for showing that an entanglement witness or positive map is unable to detect entanglement in any such state, and we apply our method to many well-known separability criteria, including the range criterion, the realignment criterion, the Choi map and its generalizations, and the Breuer-Hall map. We also show that these two properties coincide for the family of isotropic states, and several eigenvalue results for entanglement witnesses are proved along the way that are of independent interest.