1,892 research outputs found
Polynomial invariants for a semisimple and cosemisimple Hopf algebra of finite dimension
We introduce new polynomial invariants of a finite-dimensional semisimple and
cosemisimple Hopf algebra A over a field by using the braiding structures of A.
We investigate basic properties of the polynomial invariants including
stability under extension of the base field. Furthermore, we show that our
polynomial invariants are indeed tensor invariants of the representation
category of A, and recognize the difference of the representation category and
the representation ring of A. Actually, by computing and comparing polynomial
invariants, we find new examples of pairs of Hopf algebras whose representation
rings are isomorphic, but representation categories are distinct.Comment: 64 page
Effective squeezing enhancement via measurement-induced non-Gaussian operation and its application to dense coding scheme
We study the measurement-induced non-Gaussian operation on the single- and
two-mode \textit{Gaussian} squeezed vacuum states with beam splitters and
on-off type photon detectors, with which \textit{mixed non-Gaussian} states are
generally obtained in the conditional process. It is known that the
entanglement can be enhanced via this non-Gaussian operation on the two-mode
squeezed vacuum state. We show that, in the range of practical squeezing
parameters, the conditional outputs are still close to Gaussian states, but
their second order variances of quantum fluctuations and correlations are
effectively suppressed and enhanced, respectively. To investigate an
operational meaning of these states, especially entangled states, we also
evaluate the quantum dense coding scheme from the viewpoint of the mutual
information, and we show that non-Gaussian entangled state can be advantageous
compared with the original two-mode squeezed state.Comment: REVTeX4, 14 pages with 21 figure
q-deformed integers derived from pairs of coprime integers and its applications
In connection with cluster algebras, snake graphs and q-integers, Kyungyong
Lee and Ralf Schiffler recently found a formula for computing the (normalized)
Jones polynomials of rational links in terms of continued fraction expansion of
rational numbers. Sophie Morier-Genoud and Valentin Ovsienko introduced
q-deformed continued fractions, and showed that by using them each coefficient
of the normalized Jones polynomial counted quiver representations of type A_n.
In this paper we introduce q-deformed integers defined by pairs of coprime
integers, which are motivated by the denominators and the numerators of their
q-deformed continued fractions, and give an efficient algorithm for computing
the (normalized) Jones polynomials of rational links. Various properties of
q-integers defined by pairs of coprime integers are investigated and shown its
applications
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