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.
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.
This paper is devoted to the study of the separability problem in the field of Quantum information theory. We focus on the bipartite finite dimensional case and on two types of matrices: SPC and PPT matrices (see definitions 32 and 33). We prove that many results hold for both types. If these matrices have specific Hermitian Schmidt decompositions then they are separable in a very strong sense (see theorem 38 and corollary 39). We prove that both types have what we call \textbf{split decompositions} (see theorems 41 and 42). We also define the notion of weakly irreducible matrix (see definition 43), based on the concept of irreducible state defined recently in \cite{chen1}, \cite{chen} and \cite{chen2}.}{These split decomposition theorems imply that every SPC (PPT) matrix can be decomposed into a sum of s+1 SPC (PPT) matrices of which the first s are weakly irreducible, by theorem 48, and the last one has a further split decomposition of lower tensor rank, by corollary 49. Thus the SPC (PPT) matrix is decomposed in a finite number of steps into a sum of weakly irreducible matrices. Different components of this sum have support on orthogonal local Hilbert spaces, therefore the matrix is separable if and only if each component is separable. This reduces the separability problem for SPC (PPT) matrices to the case of weakly irreducible SPC (PPT) matrices. We also provide a complete description of weakly irreducible matrices of both types (see theorem 46).}{Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable (see theorem 58), we found sharp inequalites providing separability for both types (see theorems 61 and 62).
Abstract Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform thek-fold maximal tensor product of a cone C1 C1 into thek-fold minimal tensor product of a cone C2 C2 , and the pair (C1,C2) (C1,C2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that (C1,C2) (C1,C2) is resilient if either C1 C1 or C2 C2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.
A linear map between real symmetric matrix spaces is positive if all positive semidefinite matrices are mapped to positive semidefinite ones. A real symmetric matrix is separable if it can be written as a summation of Kronecker products of positive semidefinite matrices. This paper studies how to check if a linear map is positive or not and how to check if a matrix is separable or not. We propose numerical algorithms, based on Lasserre's type of semidefinite relaxations, for solving such questions. To check the positivity of a linear map, we construct a hierarchy of semidefinite relaxations for minimizing the associated biquadratic form over the unit spheres. We show that the positivity can be detected by solving a finite number of such semidefinite relaxations. To check the separability of a matrix, we construct a hierarchy of semidefinite relaxations. If it is not separable, we can get a mathematical certificate for that; if it is, we can get a decomposition for the separability.
A Hierarchy of Entanglement Cones via Rank-Constrained C∗-Convex Hulls
This paper systematically investigates the geometry of fundamental quantum cones, the separable cone (P+) and the Positive Partial Transpose (PPT) cone (PPPT), under generalized non-commutative convexity. We demonstrate a sharp stability dichotomy analyzing C∗-convex hulls of these cones: while P+ remains stable under local C∗-convex combinations, its global C∗-convex hull collapses entirely to the cone of all positive semidefinite matrices, MCL(P+)=P0. 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, MCLk(⋅). We prove that this hierarchy precisely recovers the known Schmidt number cones for the separable case, establishing a generalized convexity characterization: MCLk(P+)=Tk. Applied to the PPT cone, this framework generates a family of conjectured non-trivial intermediate cones, CPPT,k.
A pair of proper cones (C1,C2) is said to have the Lorentz factorization property (LFP) if every (C1,C2)-positive map factors through a direct sum of Lorentzian cones, i.e., cones over Euclidean balls. Clearly, (C1,C2) has the LFP if either C1 or C2 is a direct sum of Lorentzian cones, and our main goal is to find other examples. We show that such examples cannot be found for pairs (C1,C2) where C1=C2, or in the case where both C1 and C2 are polyhedral. We also focus on the case where C1=C□ is the square-based cone in R3. Here, we show that (C□,C) has the LFP whenever C is a symmetric cone, i.e., a direct sum of Lorentz cones, cones of positive semidefinite matrices over the real numbers, complex numbers or quaternions, and the cone of 3×3 positive semidefinite matrices over the octonions. We leave open the question whether there are more examples, but we show that this list cannot be extended by any strictly convex cone C or for a cone C with dim(C)≤5. Finally, we discuss an application to a problem in quantum information theory.
A real symmetric matrix is separable if it can be written as a sum of Kronecker products of positive semidefinite matrices. This talk concerns how to check if a matrix is separable or not. We propose a numerical algorithm, based on Lasserre type semidefinite relaxations, for solving the question. To check the separability of a matrix, we construct a hierarchy of semidefinite relaxations. If it is not separable, we can get a mathematical certificate for that; if it is, we can get a decomposition for the separability. This is a joint work with Xinzhen Zhang.