358 research outputs found
On graphs whose Laplacian matrix's multipartite separability is invariant under graph isomorphism
Normalized Laplacian matrices of graphs have recently been studied in the
context of quantum mechanics as density matrices of quantum systems. Of
particular interest is the relationship between quantum physical properties of
the density matrix and the graph theoretical properties of the underlying
graph. One important aspect of density matrices is their entanglement
properties, which are responsible for many nonintuitive physical phenomena. The
entanglement property of normalized Laplacian matrices is in general not
invariant under graph isomorphism. In recent papers, graphs were identified
whose entanglement and separability properties are invariant under isomorphism.
The purpose of this note is to characterize the set of graphs whose
separability is invariant under graph isomorphism. In particular, we show that
this set consists of , and all complete graphs.Comment: 5 page
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
Package of facts and theorems for efficiently generating entanglement criteria for many qubits
We present a package of mathematical theorems, which allow to construct
multipartite entanglement criteria. Importantly, establishing bounds for
certain classes of entanglement does not take an optimization over continuous
sets of states. These bonds are found from the properties of commutativity
graphs of operators used in the criterion. We present two examples of criteria
constructed according to our method. One of them detects genuine 5-qubit
entanglement without ever referring to correlations between all five qubits.Comment: 5 pages, 4 figure
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