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.
The trace of the average mixing matrix of a quantum walk measures the "laziness" of the walk: the higher the trace, the more likely that the walker returns home in the long run. In this paper, we develop tools to study this graph invariant arising from Laplacian quantum walks.
It is known that the complete graph
Kn is the laziest connected graph on
n vertices. Using our machinery, we show that the star
Sn is the second laziest connected graph on
n vertices (and hence the laziest tree on
n vertices), the complete multipartite graph
Kn−2,1,1 is the third laziest connected graph on
n vertices, and the double star
DS(n−3,1) is the second laziest tree on
n vertices. We also show that on the same number of vertices, more unbalanced double stars are lazier.
We develop a general theory for a quantum-walk search on a star graph. A star graph has N edges each of which is attached to a central vertex. A graph G is attached to one of these edges, and we would like to find out to which edge it is attached. This is done by means of a quantum walk, a quantum version of a random walk. This walk contains O(sqrt[N]) steps, which represents a speedup over a classical search, which would require O(N) steps. The overall graph, star plus G, is divided into two parts, and we find that for a quantum speedup to occur, the eigenvalues associated with these two parts in the N→∞ limit must be the same. Our theory tells us how the initial state of the walk should be chosen, and how many steps the walk must make in order to find G.
The density matrix of a graph is the combinatorial laplacian matrix of a graph normalized to have unit trace. In this paper we generalize the entanglement properties of mixed density matrices from combinatorial laplacian matrices of graphs discussed in Braunstein et al. [Annals of Combinatorics, 10 (2006) 291] to tripartite states. Then we prove that the degree condition defined in Braunstein et al. [Phys. Rev. A, 73 (2006) 012320] is sufficient and necessary for the tripartite separability of the density matrix of a nearest point graph.
Consider the model of bipartite entanglement for a random pure state emerging in quantum information and quantum chaos, corresponding to the fixed trace Laguerre unitary ensemble (LUE) in Random Matrix Theory. We focus on correlation functions of Schmidt eigenvalues for the model and prove universal limits of the correlation functions in the bulk and also at the soft and hard edges of the spectrum, as these for the LUE. Further we consider the bounded trace LUE and obtain the same universal limits.
We recall the solvable [Formula: see text]-symmetric quantum square well on an interval of x ∈ (–L, L) := [Formula: see text] (with an α-dependent non-Hermiticity given by Robin boundary conditions) and generalize it. In essence, we just replace the support interval [Formula: see text] (reinterpreted as an equilateral two-pointed star graph with Kirchhoff matching at the vertex x = 0) with a q-pointed equilateral star graph [Formula: see text] endowed with the simplest complex-rotation-symmetric external α-dependent Robin boundary conditions. The remarkably compact form of the secular determinant is then deduced. Its analysis reveals that (i) at any integer q = 2, 3, …, there exists the same q-independent and infinite subfamily of the real energies, and (ii) at any special q = 2, 6, 10, …, there exists another, additional, q-dependent infinite subfamily of the real energies. In the spirit of the recently proposed dynamical construction of the Hilbert space of a quantum system, the physical bound-state interpretation of these eigenvalues is finally proposed.