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.
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.
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.
This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph theoretic conditions for zero and nonzero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and [Formula: see text]-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.
In this paper we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and we call them graph Laplacian quantum states. Then we obtain structural properties of these graphs such that the corresponding graph Laplacian states have zero quantum discord by investigating structural properties of clustered graphs which provide a family of commuting normal matrices formed by the blocks of its Laplacian matrices. We apply these results on some important mixed quantum states, such as the Werner, Isotropic, and
X-states.
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.
Using the signed Laplacian matrix, and weighted and hybrid graphs, we present additional ways to interpret graphs as grid states. Hybrid graphs offer the most general interpretation. Existing graphical methods that characterize entanglement properties of grid states are adapted to these interpretations. These additional classes of grid states are shown to exhibit rich entanglement properties, including bound entanglement. Further, we introduce graphical techniques to construct bound-entangled states in a modular fashion. We also extend the grid states model to hypergraphs. Our work, on one hand, opens up possibilities for constructing additional families of mixed quantum states in the grid state model. On the other hand, it can serve as an instrument for investigating entanglement problems from a graph theory perspective.