62,753 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
Higher Lorentzian Polynomials, Higher Hessians, and the Hodge-Riemann Property for Graded Artinian Gorenstein Algebras in Codimension Two
A homogeneous bivariate polynomial of degree with coefficients
in is the Macaulay dual generator of a codimension two standard
graded oriented Artinian Gorenstein -algebra of socle degree
. We show that the total positivity of a certain Toeplitz matrix determined
by the coefficients of is equivalent to a certain mixed Hodge-Riemann
relation on the algebra ; polynomials satisfying these conditions are
called higher Lorentzian polynomials. In degree , our conditions amount to
nonnegative ultra log concavity with no internal zeros, and we recover results
of Br\"and\'en-Huh in variables. A corollary of our results is that the
closure of the set of totally positive Toeplitz matrices is the set of totally
nonnegative Toeplitz matrices, which seems to be new.Comment: Rewritten with new results, authorship change, comments still
welcome!
- …