6,695 research outputs found
Categorification of Hopf algebras of rooted trees
We exhibit a monoidal structure on the category of finite sets indexed by
P-trees for a finitary polynomial endofunctor P. This structure categorifies
the monoid scheme (over Spec N) whose semiring of functions is (a P-version of)
the Connes--Kreimer bialgebra H of rooted trees (a Hopf algebra after base
change to Z and collapsing H_0). The monoidal structure is itself given by a
polynomial functor, represented by three easily described set maps; we show
that these maps are the same as those occurring in the polynomial
representation of the free monad on P.Comment: 29 pages. Does not compile with pdflatex due to dependency on the
texdraw package. v2: expository improvements, following suggestions from the
referees; final version to appear in Centr. Eur. J. Mat
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
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