8,606 research outputs found
On the degree conjecture for separability of multipartite quantum states
We settle the so-called degree conjecture for the separability of
multipartite quantum states, which are normalized graph Laplacians, first given
by Braunstein {\it et al.} [Phys. Rev. A \textbf{73}, 012320 (2006)]. The
conjecture states that a multipartite quantum state is separable if and only if
the degree matrix of the graph associated with the state is equal to the degree
matrix of the partial transpose of this graph. We call this statement to be the
strong form of the conjecture. In its weak version, the conjecture requires
only the necessity, that is, if the state is separable, the corresponding
degree matrices match. We prove the strong form of the conjecture for {\it
pure} multipartite quantum states, using the modified tensor product of graphs
defined in [J. Phys. A: Math. Theor. \textbf{40}, 10251 (2007)], as both
necessary and sufficient condition for separability. Based on this proof, we
give a polynomial-time algorithm for completely factorizing any pure
multipartite quantum state. By polynomial-time algorithm we mean that the
execution time of this algorithm increases as a polynomial in where is
the number of parts of the quantum system. We give a counter-example to show
that the conjecture fails, in general, even in its weak form, for multipartite
mixed states. Finally, we prove this conjecture, in its weak form, for a class
of multipartite mixed states, giving only a necessary condition for
separability.Comment: 17 pages, 3 figures. Comments are welcom
Hadwiger Number and the Cartesian Product Of Graphs
The Hadwiger number mr(G) of a graph G is the largest integer n for which the
complete graph K_n on n vertices is a minor of G. Hadwiger conjectured that for
every graph G, mr(G) >= chi(G), where chi(G) is the chromatic number of G. In
this paper, we study the Hadwiger number of the Cartesian product G [] H of
graphs.
As the main result of this paper, we prove that mr(G_1 [] G_2) >= h\sqrt{l}(1
- o(1)) for any two graphs G_1 and G_2 with mr(G_1) = h and mr(G_2) = l. We
show that the above lower bound is asymptotically best possible. This
asymptotically settles a question of Z. Miller (1978).
As consequences of our main result, we show the following:
1. Let G be a connected graph. Let the (unique) prime factorization of G be
given by G_1 [] G_2 [] ... [] G_k. Then G satisfies Hadwiger's conjecture if k
>= 2.log(log(chi(G))) + c', where c' is a constant. This improves the
2.log(chi(G))+3 bound of Chandran and Sivadasan.
2. Let G_1 and G_2 be two graphs such that chi(G_1) >= chi(G_2) >=
c.log^{1.5}(chi(G_1)), where c is a constant. Then G_1 [] G_2 satisfies
Hadwiger's conjecture.
3. Hadwiger's conjecture is true for G^d (Cartesian product of G taken d
times) for every graph G and every d >= 2. This settles a question by Chandran
and Sivadasan (They had shown that the Hadiwger's conjecture is true for G^d if
d >= 3.)Comment: 10 pages, 2 figures, major update: lower and upper bound proofs have
been revised. The bounds are now asymptotically tigh
On the strong partition dimension of graphs
We present a different way to obtain generators of metric spaces having the
property that the ``position'' of every element of the space is uniquely
determined by the distances from the elements of the generators. Specifically
we introduce a generator based on a partition of the metric space into sets of
elements. The sets of the partition will work as the new elements which will
uniquely determine the position of each single element of the space. A set
of vertices of a connected graph strongly resolves two different vertices
if either or
, where . An ordered vertex partition of
a graph is a strong resolving partition for if every two different
vertices of belonging to the same set of the partition are strongly
resolved by some set of . A strong resolving partition of minimum
cardinality is called a strong partition basis and its cardinality the strong
partition dimension. In this article we introduce the concepts of strong
resolving partition and strong partition dimension and we begin with the study
of its mathematical properties. We give some realizability results for this
parameter and we also obtain tight bounds and closed formulae for the strong
metric dimension of several graphs.Comment: 16 page
Controllability and observability of grid graphs via reduction and symmetries
In this paper we investigate the controllability and observability properties
of a family of linear dynamical systems, whose structure is induced by the
Laplacian of a grid graph. This analysis is motivated by several applications
in network control and estimation, quantum computation and discretization of
partial differential equations. Specifically, we characterize the structure of
the grid eigenvectors by means of suitable decompositions of the graph. For
each eigenvalue, based on its multiplicity and on suitable symmetries of the
corresponding eigenvectors, we provide necessary and sufficient conditions to
characterize all and only the nodes from which the induced dynamical system is
controllable (observable). We discuss the proposed criteria and show, through
suitable examples, how such criteria reduce the complexity of the
controllability (respectively observability) analysis of the grid
Perfect state transfer, graph products and equitable partitions
We describe new constructions of graphs which exhibit perfect state transfer
on continuous-time quantum walks. Our constructions are based on variants of
the double cones [BCMS09,ANOPRT10,ANOPRT09] and the Cartesian graph products
(which includes the n-cube) [CDDEKL05]. Some of our results include: (1) If
is a graph with perfect state transfer at time , where t_{G}\Spec(G)
\subseteq \ZZ\pi, and is a circulant with odd eigenvalues, their weak
product has perfect state transfer. Also, if is a regular
graph with perfect state transfer at time and is a graph where
t_{H}|V_{H}|\Spec(G) \subseteq 2\ZZ\pi, their lexicographic product
has perfect state transfer. (2) The double cone on any
connected graph , has perfect state transfer if the weights of the cone
edges are proportional to the Perron eigenvector of . This generalizes
results for double cone on regular graphs studied in
[BCMS09,ANOPRT10,ANOPRT09]. (3) For an infinite family \GG of regular graphs,
there is a circulant connection so the graph K_{1}+\GG\circ\GG+K_{1} has
perfect state transfer. In contrast, no perfect state transfer exists if a
complete bipartite connection is used (even in the presence of weights)
[ANOPRT09]. We also describe a generalization of the path collapsing argument
[CCDFGS03,CDDEKL05], which reduces questions about perfect state transfer to
simpler (weighted) multigraphs, for graphs with equitable distance partitions.Comment: 18 pages, 6 figure
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