This paper is devoted to the study of the separability problem in the field of Quantum information theory. We deal mainly with the bipartite finite dimensional case and with two types of matrices, one of them being the PPT matrices. We proved that many results holds for both types. If these matrices have specific Hermitian Schmidt decompositions then the matrices are separable in a very strong sense. We proved that both types have what we call split decompositions. We defined the notion of weak irreducible matrix, based on the concept of irreducible state defined recently. These split decomposition theorems together with the notion of weak irreducible matrix, imply that these matrices are weak irreducible or a sum of weak irreducible matrices of the same type. The separability problem for these types of matrices can be reduced to the set of weak irreducible matrices of the same type. We also provided a complete description of weak irreducible matrices of both types. Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable, we found sharp inequalites providing separability for both types.
This paper is devoted to the study of the separability problem in the field of Quantum information theory. We deal mainly with the bipartite finite dimensional case and with two types of matrices, one of them being the PPT matrices. We proved that many results holds for both types. If these matrices have specific Hermitian Schmidt decompositions then the matrices are separable in a very strong sense. We proved that both types have what we call split decompositions. We defined the notion of weak irreducible matrix, based on the concept of irreducible state defined recently. These split decomposition theorems together with the notion of weak irreducible matrix, imply that these matrices are weak irreducible or a sum of weak irreducible matrices of the same type. The separability problem for these types of matrices can be reduced to the set of weak irreducible matrices of the same type. We also provided a complete description of weak irreducible matrices of both types. Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable, we found sharp inequalites providing separability for both types.
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).
Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e. it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect the separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem in quantum physics, which asks to tell if a given quantum state is entangled or not. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it. Moreover, we study psd decompositions of separable Hermitian tensors. When the psd rank is low, we first flatten them into cubic order tensors and then apply tensor decomposition methods to compute psd decompositions. We prove that this method works well if the psd rank is low. In computation, this flattening approach can detect the separability for much larger-sized Hermitian tensors. This method is a good starting point to determine psd ranks of separable Hermitian tensors.
Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e., it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem in quantum physics, which asks to tell if a given quantum state is entangled or not. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it. Moreover, we study psd decompositions of separable Hermitian tensors. When the psd rank is low, we first flatten them into cubic order tensors and then apply tensor decomposition methods to compute psd decompositions. We prove that this method works well if the psd rank is low. In computation, this flattening approach can detect separability for much larger sized Hermitian tensors. This method is a good start on determining psd ranks of separable Hermitian tensors.
The operator Schmidt rank is the minimum number of terms required to express a state as a sum of elementary tensor factors. Here we provide a new proof of the fact that any bipartite mixed state with operator Schmidt rank two is separable, and can be written as a sum of two positive semidefinite matrices per site. Our proof uses results from the theory of free spectrahedra and operator systems, and illustrates the use of a connection between decompositions of quantum states and decompositions of nonnegative matrices. In the multipartite case, we prove that any Hermitian Matrix Product Density Operator (MPDO) of bond dimension two is separable, and can be written as a sum of at most four positive semidefinite matrices per site. This implies that these states can only contain classical correlations, and very few of them, as measured by the entanglement of purification. In contrast, MPDOs of bond dimension three can contain an unbounded amount of classical correlations.
We investigate the problem of finding the optimal convex decomposition of a bipartite quantum state into a separable part and a positive remainder, in which the weight of the separable part is maximal. This weight is naturally identified with the degree of separability of the state. In a recent work, the problem was solved for two-qubit states using semidefinite programming. In this paper, we describe a procedure to obtain the optimal decomposition of a bipartite state of any finite dimension via a sequence of semidefinite relaxations. The sequence of decompositions thus obtained is shown to converge to the optimal one. This provides a systematic method to determine the so-called optimal Lewenstein-Sanpera decomposition of any bipartite state. Numerical results are provided to illustrate this procedure, and the special case of rank-2 states is also discussed.
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.