We present an alternative approach to unveil a different kind of entanglement in bipartite quantum states whose diagonal zero patterns in suitable matrix representations admit a nice description in terms of triangle-free graphs. Upon application of a local averaging operation, the separability of such states transforms into a simple matrix positivity condition, the violation of which implies the presence of entanglement. We completely characterize the class of triangle-free graphs, which allows for nontrivial entanglement detection using the above test. Moreover, we develop a recipe to construct a plethora of unique classes of positive partial transpose (PPT) entangled states in arbitrary dimensions. Finally, we link the task of entanglement detection in general states to the well-known graph-theoretic problem of finding triangle-free-induced subgraphs in a given graph.
Abstract Braunstein et al have started the study of entanglement properties of the quantum states through graph theoretical approach. Their idea was to start from a simple unweighted graph G and then they have defined the quantum state from the Laplacian of the graph G . A lot of research had already been done using the similar idea. We ask here the opposite one i.e can we generate a graph from the density matrix? To investigate this question, we have constructed a unital map ϕ such that ϕ ( ρ ) = L ρ + ρ , where the quantum state is described by the density operator ρ . The entries of L ρ depends on the entries of the quantum state ρ and the entries are taken in such a way that L ρ satisfies all the properties of the Laplacian. This make possible to design a simple connected weighted graph from the Laplacian L ρ . We show that the constructed unital map ϕ characterize the quantum state with respect to its purity by showing that if the determinant of the matrix ϕ ( ρ ) − I is positive then the quantum state ρ represent a mixed state. Moreover, we study the positive partial transpose (PPT) criterion in terms of the spectrum of the density matrix under investigation and the spectrum of the Laplacian associated with the given density matrix. Furthermore, we derive the inequality between the minimum eigenvalue of the density matrix and the weight of the edges of the connected subgraph of a simple weighted graph to detect the entanglement of d 1 ⨂ d 2 dimensional bipartite quantum states. Lastly, we have illustrated our results with few examples.
Graph-theoretic structures play a central role in the description and analysis of quantum systems. In this work, we introduce a new class of quantum states, called
Aα-graph states, which are constructed from either unweighted or weighted graphs by taking the normalised convex combination of the degree matrix
D and the adjacency matrix
AG of a graph
G. The constructed states are different from the standard graph states arising from stabiliser formalism. Our approach is also different from the approach used by Braunstein et al. This class of states depend on a tunable mixing parameter
α∈(0,1]. We first establish the conditions under which the associated operator
ραAG is positive semidefinite and hence represents a valid quantum state. We then derive a positive partial transposition (PPT) condition for
Aα-graph states in terms of graph parameters. This PPT condition involves only the Frobenius norm of the adjacency matrix of the graph, the degrees of the vertices and the total number of vertices. For simple graphs, we obtain the range of the parameter
α for which the
Aα-graph states represent a class of entangled states. We then develop a graph-theoretic formulation of a moments-based entanglement detection criterion, focusing on the recently proposed
p3-PPT criterion, which relies on the second and third moments of the partial transposition. Since the estimation of these moments is experimentally accessible via randomised measurements, swap operations, and machine-learning-based protocols, our approach provides a physically relevant framework for detecting entanglement in structured quantum states derived from graphs. This work bridges graph theory and moments-based entanglement detection, offering a new perspective on the role of combinatorial structures in quantum correlations.
In this paper we consider the separability problem for bipartite quantum states arising from graphs. Earlier it was proved that the degree criterion is the graph-theoretic counterpart of the familiar positive partial transpose criterion for separability, although there are entangled states with positive partial transpose for which the degree criterion fails. Here we introduce the concept of partially symmetric graphs and degree symmetric graphs by using the well-known concept of partial transposition of a graph and degree criteria, respectively. Thus, we provide classes of bipartite separable states of dimension $m\ifmmode\times\else\texttimes\fi{}n$ arising from partially symmetric graphs. We identify partially asymmetric graphs that lack the property of partial symmetry. We develop a combinatorial procedure to create a partially asymmetric graph from a given partially symmetric graph. We show that this combinatorial operation can act as an entanglement generator for mixed states arising from partially symmetric graphs.
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.
Motivated by the need for efficient local entanglement detection for applications in quantum information processing, sufficient conditions for arbitrary-dimensional local bipartite entanglement detection based on correlation matrices and joint probability distributions are investigated. In particular, their dependence on the nature of different classes of local measurements is explored for generalized measurements based on informationally complete
(N,M) positive-operator-valued measures (POVMs) [Siudzinska, Phys. Rev. A 105, 042209 (2022)]. It is shown that symmetry properties of
(N,M) POVMs necessarily imply that these sufficient conditions for bipartite entanglement detection exhibit characteristic scaling properties relating equivalent sufficient conditions. Based on these general scaling properties, the efficiency of different classes of local quantum measurement detecting typical bipartite entanglement is investigated quantitatively. For this purpose Euclidean volume ratios between locally detectable bipartite entangled states and all bipartite quantum states are determined numerically with the help of a Monte Carlo algorithm. Our results demonstrate that physically realizable
(N,M) POVMs are sufficient for optimal local entanglement detection. In particular, this implies that for this purpose the construction of optimal
(N,M) POVMs is not necessary. As questions concerning the existence and construction of optimal
(N,M) POVMs are still largely open, this may offer interesting perspectives for practical applications in quantum information processing.
We introduce and study bipartite quantum states that are invariant under the local action of the cyclic sign group. Due to symmetry, these states are sparse and can be parameterized by a triple of vectors. Their important semi-definite properties, such as positivity and positivity under partial transpose (PPT), can be simply characterized in terms of these vectors and their discrete Fourier transforms. We study in detail the entanglement properties of this family of symmetric states, showing that it contains PPT entangled states. For states that are diagonal in the Dicke basis, deciding separability is equivalent to a circulant version of the complete positivity problem. In local dimension d ≤ 5, we completely characterize these sets and construct entanglement witnesses; some partial results are also obtained for d = 6, 7. We construct a new family of states for which the properties of PPT and separability can be characterized for all dimensions, generalizing results from from the literature. Our results show that these states have a rich entanglement structure, even in the bosonic subspace.