29,987 research outputs found
Subspaces with a common complement in a Banach space
We study the problem of the existence of a common algebraic complement for a
pair of closed subspaces of a Banach space. We prove the following two
characterizations: (1) The pairs of subspaces of a Banach space with a common
complement coincide with those pairs which are isomorphic to a pair of graphs
of bounded linear operators between two other Banach spaces. (2) The pairs of
subspaces of a Banach space X with a common complement coincide with those
pairs for which there exists an involution S on X exchanging the two subspaces,
such that I+S is bounded from below on their union. Moreover we show that, in a
separable Hilbert space, the only pairs of subspaces with a common complement
are those which are either equivalently positioned or not completely asymptotic
to one another. We also obtain characterizations for the existence of a common
complement for subspaces with closed sum
Matroid and Tutte-connectivity in infinite graphs
We relate matroid connectivity to Tutte-connectivity in an infinite graph.
Moreover, we show that the two cycle matroids, the finite-cycle matroid and the
cycle matroid, in which also infinite cycles are taken into account, have the
same connectivity function. As an application we re-prove that, also for
infinite graphs, Tutte-connectivity is invariant under taking dual graphs.Comment: 11 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
The laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states
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
graph on four vertices representing entangled states. It turns out that for
some of these graphs the value of the concurrence is exactly fractional.Comment: 20 pages, 11 figure
Energy as an Entanglement Witness for Quantum Many-Body Systems
We investigate quantum many-body systems where all low-energy states are
entangled. As a tool for quantifying such systems, we introduce the concept of
the entanglement gap, which is the difference in energy between the
ground-state energy and the minimum energy that a separable (unentangled) state
may attain. If the energy of the system lies within the entanglement gap, the
state of the system is guaranteed to be entangled. We find Hamiltonians that
have the largest possible entanglement gap; for a system consisting of two
interacting spin-1/2 subsystems, the Heisenberg antiferromagnet is one such
example. We also introduce a related concept, the entanglement-gap temperature:
the temperature below which the thermal state is certainly entangled, as
witnessed by its energy. We give an example of a bipartite Hamiltonian with an
arbitrarily high entanglement-gap temperature for fixed total energy range. For
bipartite spin lattices we prove a theorem demonstrating that the entanglement
gap necessarily decreases as the coordination number is increased. We
investigate frustrated lattices and quantum phase transitions as physical
phenomena that affect the entanglement gap.Comment: 16 pages, 3 figures, published versio
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