Simone Severini

The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States

Abstract.We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that the density matrix of a graph can be always written as a uniform mixture of pure density matrices of graphs. We consider the von Neumann entropy of these matrices and we characterize the graphs for which the minimum and maximum values are attained. We then discuss the problem of separability by pointing out that separability of density matrices of graphs does not always depend on the labelling of the vertices. We consider graphs with a tensor product structure and simple cases for which combinatorial properties are linked to the entanglement of the state. We calculate the concurrence of all graphs on four vertices representing entangled states. It turns out that for these graphs the value of the concurrence is exactly fractional.

Graph-Theoretic Approach to Quantum Correlations

Correlations in Bell and noncontextuality inequalities can be expressed as a positive linear combination of probabilities of events. Exclusive events can be represented as adjacent vertices of a graph, so correlations can be associated to a subgraph. We show that the maximum value of the correlations for classical, quantum, and more general theories is the independence number, the Lovász number, and the fractional packing number of this subgraph, respectively. We also show that, for any graph, there is always a correlation experiment such that the set of quantum probabilities is exactly the Grötschel-Lovász-Schrijver theta body. This identifies these combinatorial notions as fundamental physical objects and provides a method for singling out experiments with quantum correlations on demand.

On the Digraph of a Unitary Matrix

Given a matrix M of size n, the digraph D on n vertices is said to be the digraph of M, when

Number-theoretic nature of communication in quantum spin systems.

M_{ij}neq 0

Nondiscriminatory propagation on trees

if and only if (v,sub>i,v,sub>j) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph

COMMUNICATION IN XYZ ALL-TO-ALL QUANTUM NETWORKS WITH A MISSING LINK

overrightarrow{L}D

QUANTUM STATE TRANSFER THROUGH A QUBIT NETWORK WITH ENERGY SHIFTS AND FLUCTUATIONS

is the pattern of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and

Bounds on Entanglement Assisted Source-channel Coding Via the Lovász Theta Number and Its Variants

overrightarrow{L}D

Enumeration of (k,2)-noncrossing partitions

is the digraph of a unitary matrix, then

Some families of density matrices for which separability is easily tested

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