8,458 research outputs found
A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles
We use convex polyhedral cones to study a large class of multivariate
meromorphic germs, namely those with linear poles, which naturally arise in
various contexts in mathematics and physics. We express such a germ as a sum of
a holomorphic germ and a linear combination of special non-holomorphic germs
called polar germs. In analyzing the supporting cones -- cones that reflect the
pole structure of the polar germs -- we obtain a geometric criterion for the
non-holomorphicity of linear combinations of polar germs. This yields the
uniqueness of the above sum when required to be supported on a suitable family
of cones and assigns a Laurent expansion to the germ. Laurent expansions
provide various decompositions of such germs and thereby a uniformized proof of
known results on decompositions of rational fractions. These Laurent expansions
also yield new concepts on the space of such germs, all of which are
independent of the choice of the specific Laurent expansion. These include a
generalization of Jeffrey-Kirwan's residue, a filtered residue and a coproduct
in the space of such germs. When applied to exponential sums on rational convex
polyhedral cones, the filtered residue yields back exponential integrals.Comment: 30 page
Quantum resource studied from the perspective of quantum state superposition
Quantum resources,such as discord and entanglement, are crucial in quantum
information processing. In this paper, quantum resources are studied from the
aspect of quantum state superposition. We define the local superposition (LS)
as the superposition between basis of single part, and nonlocal superposition
(NLS) as the superposition between product basis of multiple parts. For quantum
resource with nonzero LS, quantum operation must be introduced to prepare it,
and for quantum resource with nonzero NLS, nonlocal quantum operation must be
introduced to prepare it. We prove that LS vanishes if and only if the state is
classical and NLS vanishes if and only if the state is separable. From this
superposition aspect, quantum resources are categorized as superpositions
existing in different parts. These results are helpful to study quantum
resources from a unified frame.Comment: 9 pages, 4 figure
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