4 research outputs found
Laplacian matrices of weighted digraphs represented as quantum states
Representing graphs as quantum states is becoming an increasingly important
approach to study entanglement of mixed states, alternate to the standard
linear algebraic density matrix-based approach of study. In this paper, we
propose a general weighted directed graph framework for investigating
properties of a large class of quantum states which are defined by three types
of Laplacian matrices associated with such graphs. We generalize the standard
framework of defining density matrices from simple connected graphs to density
matrices using both combinatorial and signless Laplacian matrices associated
with weighted directed graphs with complex edge weights and with/without
self-loops. We also introduce a new notion of Laplacian matrix, which we call
signed Laplacian matrix associated with such graphs. We produce necessary
and/or sufficient conditions for such graphs to correspond to pure and mixed
quantum states. Using these criteria, we finally determine the graphs whose
corresponding density matrices represent entangled pure states which are well
known and important for quantum computation applications. We observe that all
these entangled pure states share a common combinatorial structure.Comment: 19 pages, Modified version of quant-ph/1205.2747, title and abstract
has been changed, One author has been adde
Condition for zero and non-zero discord in graph Laplacian quantum states
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 non-zero 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 -states. We also consider graph Laplacian
states corresponding to simple graphs as a special case.Comment: 24 pages, this version is very similar to the one published in the
International Journal of Quantum Informatio
From Sharma-Mittal to von-Neumann Entropy of a Graph
In this article, we introduce the Sharma-Mittal entropy of a graph, which is
a generalization of the existing idea of the von-Neumann entropy. The
well-known R{\'e}nyi, Thallis, and von-Neumann entropies can be expressed as
limiting cases of Sharma-Mittal entropy. We have explicitly calculated them for
cycle, path, and complete graphs. Also, we have proposed a number of bounds for
these entropies. In addition, we have also discussed the entropy of product
graphs, such as Cartesian, Kronecker, Lexicographic, Strong, and Corona
products. The change in entropy can also be utilized in the analysis of growing
network models (Corona graphs), useful in generating complex networks
Phase Squeezing of Quantum Hypergraph States
Corresponding to a hypergraph with vertices, a quantum hypergraph
state is defined by , where is a -variable Boolean function depending
on the hypergraph , and denotes a binary vector of length
with at -th position for . The non-classical
properties of these states are studied. We consider annihilation and creation
operator on the Hilbert space of dimension acting on the number states
. The Hermitian number and phase
operators, in finite dimensions, are constructed. The number-phase uncertainty
for these states leads to the idea of phase squeezing. We establish that these
states are squeezed in the phase quadrature only and satisfy the Agarwal-Tara
criterion for non-classicality, which only depends on the number of vertices of
the hypergraphs. We also point out that coherence is observed in the phase
quadrature