The convex cone $\mathbb{V}_1$ generated by separable states is contained in the cone $\mathbb{T}$ of all positive semi-definite block matrices whose block transposes are also positive semi-definite. We characterize faces of the cone $\mathbb{V}_1$ induced by faces of the cone $\mathbb{T}$ which are determined by pairs of subspaces of matrices.
The convex cone V1 generated by separable states is contained in the cone T of all positive semi-definite block matrices whose block transposes are also positive semi-definite. We characterize faces of the cone V1 induced by faces of the cone T which are determined by pairs of subspaces of matrices.
We investigate the geometric structure of the set of spectra of bipartite absolute separable states (ASEPm,n) and absolute positive partial transpose states (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 is a spectrahedron for all m≤n: in particular, all its faces are exposed. While ASEP2,n is a spectrahedron, too, for ASEPm,n we can in general only show that it is a semialgebraic set for all m,n≥3. Furthermore, we provide a complete characterization of the faces and extreme points of 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.
In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of APPTm,n via an inscribed polytope Pm,n and conjecture that the maximal purity of APPTm,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 APPTm,n and demonstrate numerically that the minimum entropy eventually coincides with the polytope Pm,n as the local system dimension n increases. Finally, we show that the relative spectral volume of APPTm,n decays exponentially in n by a constant multiplicative factor of the relative volume of the inscribed polytope Pm,n.
Let K, K' be convex cones residing in finite-dimensional real vector spaces E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K and x_l' \in K' for all l. Let S(n), H(n), Q(n) be the spaces of n x n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S_+(n), H_+(n), Q_+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix in H(mn) = H(m) \otimes H(n) is H_+(m) \otimes H_+(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m) \otimes S(n), H(m) \otimes S(n), and for m < 3 in the space Q(m) \otimes S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n) \otimes S(2) is Q_+(n) \otimes S_+(2)-separable if and only if it is positive semidefinite.
We analyze the facial structures of the convex set consisting of all two qubit separable states. One of faces is a four dimensional convex body generated by the trigonometric moment curve arising from polyhedral combinatorics. Another one is an eight dimensional convex body, which is the convex hull of a homeomorphic image of the two dimensional sphere. Extreme points consist of points on the surface, and any two of them determines an edge. We also reconstruct the trigonometric moment curve in any even dimensional affine space using the qubit-qudit systems, and characterize the facial structures of the convex hull.
A state over time (stote) pairs a state ρ with a channel E as the Hermitian, generally non-positive operator (ρ⊗1)⋆JE. Characterizing the convex cone Q these operators generate was posed as an open problem in Hoogsteder-Riera, Calsamiglia and Winter (2026). We exhibit a linear copy of the bipartite separable cone as an exposed face of Q: for output systems of the form B=A′R with A′ a copy of the input, there is an explicit witness W0∈Q∗ and an explicit injective linear map L with σ∈SEP+(A:R)⟺L(σ)∈Q(A:A′R). Composing with the NP-hardness of separability shows that exact membership, weak membership, and weak optimization are NP-hard for Q and its dual. A quantitative stability bound makes the weak-membership reduction valid at inverse-polynomial accuracy. Thus polynomial-time membership tests do not exist unless P=NP. Independently of this complexity assumption, Fawzi's theorem implies that Q(A:A′R) admits no finite semidefinite representation when d=dimA≥2, r=dimR≥2, and d+r>5.