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arXiv:2512.05560

A Hierarchy of Entanglement Cones via Rank-Constrained $C^*$-Convex Hulls

Mohsen Kian

math-phmath.FAquant-phSubmitted Dec 5, 2025• Announced Dec 8, 2025• 0 citations
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Abstract

This paper systematically investigates the geometry of fundamental quantum cones, the separable cone ($\mathscr{P}_+$) and the Positive Partial Transpose (PPT) cone ($\mathcal{P}_{\mathrm{PPT}}$), under generalized non-commutative convexity. We demonstrate a sharp stability dichotomy analyzing $C^*$-convex hulls of these cones: while $\mathscr{P}_+$ remains stable under local $C^*$-convex combinations, its global $C^*$-convex hull collapses entirely to the cone of all positive semidefinite matrices, $\operatorname{MCL}(\mathscr{P}_+) = \mathscr{P}_0$. To gain finer control and classify intermediate structures, we introduce the concept of ``$k$-$C^*$-convexity'', by using the operator Schmidt rank of $C^*$-coefficients. This constraint defines a new hierarchy of nested intermediate cones, $\operatorname{MCL}_k(\cdot)$. We prove that this hierarchy precisely recovers the known Schmidt number cones for the separable case, establishing a generalized convexity characterization: $\operatorname{MCL}_k(\mathscr{P}_+) = \mathcal{T}_k$. Applied to the PPT cone, this framework generates a family of conjectured non-trivial intermediate cones, $\mathcal{C}_{\mathrm{PPT}, k}$.

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A Hierarchy of Entanglement Cones via Rank-Constrained C∗C^*C∗-Convex Hulls
This paper systematically investigates the geometry of fundamental quantum cones, the separable cone (P+\mathscr{P}_+P+​) and the Positive Partial Transpose (PPT) cone (PPPT\mathcal{P}_{\mathrm{PPT}}PPPT​), under generalized non-commutative convexity. We demonstrate a sharp stability dichotomy analyzing C∗C^*C∗-convex hulls of these cones: while P+\mathscr{P}_+P+​ remains stable under local C∗C^*C∗-convex combinations, its global C∗C^*C∗-convex hull collapses entirely to the cone of all positive semidefinite matrices, MCL⁡(P+)=P0\operatorname{MCL}(\mathscr{P}_+) = \mathscr{P}_0MCL(P+​)=P0​. To gain finer control and classify intermediate structures, we introduce the concept of ``kkk-C∗C^*C∗-convexity'', by using the operator Schmidt rank of C∗C^*C∗-coefficients. This constraint defines a new hierarchy of nested intermediate cones, MCL⁡k(⋅)\operatorname{MCL}_k(\cdot)MCLk​(⋅). We prove that this hierarchy precisely recovers the known Schmidt number cones for the separable case, establishing a generalized convexity characterization: MCL⁡k(P+)=Tk\operatorname{MCL}_k(\mathscr{P}_+) = \mathcal{T}_kMCLk​(P+​)=Tk​. Applied to the PPT cone, this framework generates a family of conjectured non-trivial intermediate cones, CPPT,k\mathcal{C}_{\mathrm{PPT}, k}CPPT,k​.
Motivated by the importance of the purification principle in quantum theory and generalized probabilistic theories, we study purifications using only the geometry of a finite-dimensional proper convex cone. We prove an existence theorem for indecomposable cones and intermediate tensor cones containing the maximally entangled state; in particular, every interior point of an indecomposable homogeneous cone admits a purification. This applies to Lorentz cones, for example. We also give a criterion for uniqueness up to local automorphisms. On the boundary, we show that if every proper face of CCC is simplicial, then only pure points can admit purifications, and we demonstrate that this conclusion fails in the presence of non-simplicial faces. Examples involving positive semidefinite cones, Lorentz cones, kkk-positive maps, PPT tensors, and polyhedral cones illustrate both existence and non-uniqueness phenomena.
We develop a new method for entanglement detection in bipartite quantum states by using the violation of the rank-1-generated property of matrices. The positive-semidefinite matrices form a convex cone that has extremal elements of rank 1. But, convex conic subsets resulting from the presence of linear constraints allow extremal elements of rank >= 2. The problem of deciding when a matrix is rank-1 generated, i.e, a sum of rank-1 positive-semidefinite (PSD) matrices, has been studied extensively in optimization theory. This rank-1 generated property acts as an entanglement criterion, and we use this property to find novel classes of PPT (Positive under partial transposition) entangled states. We do this by mapping some faces of PPT density matrices to convex cones that are not rank-1 generated. We show that all separable states map to a rank-1 generated matrix. In general, the same is not true for the corresponding mapping of PPT entangled states. We also extend this approach to construct PPT entangled edge states. Finally, we provide witnesses that detect violations of the rank-1 generated property.
Deciding whether a mixed quantum state is separable or entangled is a difficult problem in general. Separable states are positive under partial transposition [they are positive-partial-transpose (PPT) states], but this simple test does not exclude all entangled states. In order to understand the entangled PPT states, having so-called bound entanglement, we want to study the extremal PPT states. An extremal PPT state is either a pure product state, then it is separable, or it is entangled, in which case the state and its partial transpose must still be rather low-rank density matrices, although it is known that the rank must be at least 4. In a previous paper we presented a complete classification of the rank-4 entangled PPT states in dimension $3\ifmmode\times\else\texttimes\fi{}3$, generalizing the construction by Bennett et al. from unextendible sets of orthogonal product vectors. In the present paper we continue our investigations of the low-rank entangled PPT states, mostly in dimension $3\ifmmode\times\else\texttimes\fi{}3$, using a combination of analytical and numerical methods. We use perturbation theory in order to construct rank-5 entangled PPT states close to the known rank-4 states, and in order to compute dimensions and study the geometry of surfaces of low-rank PPT states. We exploit the close connection between low-rank PPT states and product vectors. In particular, we show how to reconstruct a low-rank PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.
Randomized measurement protocols give experimental access to low-degree polynomial invariants of a quantum state rather than to the state itself, which raises the question of what the best entanglement criterion is that can be built from finitely many such invariants. In this work we answer this question through convex duality. For each degree we introduce the cone of local unitary trace inequalities that are valid for separable bipartite states in all local dimensions, together with its one-matrix counterparts for Hermitian and positive semidefinite matrices. Each cone is the polar dual of a moment cone; we also determine the minimal combinatorial parametrization of the separable cone. The framework organizes the known moment criteria: the PPT criterion using the first three moments of the partial transposition is the image under an embedding of a Hankel determinant from the positive semidefinite cone to the bipartite cone, which we show to be an extremal ray, and we determine the first nontrivial separable cone completely, finding exactly four extremal rays. The framework allows us to analyze the realignment criterion and its centered (or enhanced) variant. We prove that the even singular value moments of the (centered) realignment matrix are trace invariants, associated with explicit fixed-point-free involutions. Extracting the optimal trace-norm bound from mmm such moments thus becomes a dimension-free truncated moment problem, which we solve in closed form for m=2m = 2m=2 and relax to a semidefinite program of size O(m)O(m)O(m) for arbitrary mmm. The dual solutions are polynomial minorants of x\sqrt{x}x​ on [0,1][0,1][0,1], and two explicit families, binomial and L1L^1L1-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.
Bound entangled (BE) states are strange in nature: a nonzero amount of free entanglement is required to create them but no free entanglement can be distilled from them under local operations and classical communication (LOCC). Even though the usefulness of such states has been shown in several information processing tasks, there exists no simple method to characterize them for an arbitrary composite quantum system. Here we present a $(d\ensuremath{-}3)/2$-parameter family of BE states each with positive partial transpose (PPT). This family of PPT-BE states is introduced by constructing an unextendible product basis (UPB) in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ with ddd odd and $d\ensuremath{\ge}5$. The range of each such PPT-BE state is contained in a $2(d\ensuremath{-}1)$-dimensional entangled subspace, whereas the associated UPB subspace is of dimension ${(d\ensuremath{-}1)}^{2}+1$. We further show that each of these PPT-BE states can be written as a convex combination of $(d\ensuremath{-}1)/2$ rank-4 PPT-BE states. Moreover, we prove that these rank-4 PPT-BE states are extreme points of the convex compact set P\mathcal{P}P of all PPT states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, namely, the Peres set. An interesting geometric implication of our result is that the convex hull of these rank-4 PPT-BE extreme points---the $(d\ensuremath{-}3)/2$ simplex---is sitting on the boundary between the set P\mathcal{P}P and the set of non-PPT states. We also discuss consequences of our construction in the context of quantum state discrimination by LOCC.
Bound entangled (BE) states are strange in nature: non-zero amount of free entanglement is required to create them but no free entanglement can be distilled from them under local operations and classical communication (LOCC). Even though usefulness of such states has been shown in several information processing tasks, there exists no simple method to characterize all these states for an arbitrary composite quantum system. Here we present a (d−3)/2(d-3)/2(d−3)/2-parameter family of BE states each with positive partial transpose (PPT). This family of PPT-BE states is introduced by constructing an unextendible product basis (UPB) in Cd⊗Cd\mathbb{C}^d\otimes\mathbb{C}^dCd⊗Cd with ddd odd and d≥5d\ge 5d≥5. We also provide the `tile structure' corresponding to these UPBs. The range of each such PPT-BE state is contained in a 2(d−1)2(d-1)2(d−1) dimensional entangled subspace whereas the associated UPB-subspace is of dimension (d−1)2+1(d-1)^2+1(d−1)2+1. We further show that each of these PPT-BE states can be written as a convex combination of (d−1)/2(d-1)/2(d−1)/2 number of rank-4 PPT-BE states. Moreover, we prove that these rank-4 PPT-BE states are extreme points of the convex compact set P\mathcal{P}P of all PPT states in Cd⊗Cd\mathbb{C}^d\otimes\mathbb{C}^dCd⊗Cd, namely the {\it Peres} set. An interesting geometric implication of our result is that the convex hull of these rank-4 PPT-BE extreme points -- the (d−3)/2(d-3)/2(d−3)/2-simplex -- is sitting on the boundary between the set P\mathcal{P}P and the set of non-PPT states. We also discuss consequences of our construction in the context of quantum state discrimination by LOCC.