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