We present a 3D topological picture-language for quantum information. Our approach combines charged excitations carried by strings, with topological properties that arise from embedding the strings in the interior of a 3D manifold with boundary. A quon is a composite that acts as a particle. Specifically, a quon is a hemisphere containing a neutral pair of open strings with opposite charge. We interpret multiquons and their transformations in a natural way. We obtain a type of relation, a string–genus “joint relation,” involving both a string and the 3D manifold. We use the joint relation to obtain a topological interpretation of the C* Hopf algebra relations, which are widely used in tensor networks. We obtain a 3D representation of the controlled NOT (CNOT) gate that is considerably simpler than earlier work, and a 3D topological protocol for teleportation.

Let $ \gamma = \frac{1}{2}\left( {1 + \sqrt {5} } \right) $ denote the golden ratio. H. Davenport and W. M. Schmidt showed in 1969 that, for each non-quadratic irrational real number ξ, there exists a constant$c$> 0 with the property that, for arbitrarily large values of$X$, the inequalities $ \left| {{x_0}} \right| \leqslant X,\,\,\,\left| {{x_0}\xi - {x_1}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}}\,\,\,{\text{and}}\,\,\,\left| {{x_0}{\xi^2} - {x_2}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}} $ admit no non-zero solution $ \left( {{x_0},{x_1},{x_2}} \right) \in {\mathbb{Z}^3} $ . Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers ξ such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each$X$≥ 1. Such$extremal$numbers are transcendental and their set is stable under the action of $ {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) $ on $ \mathbb{R}\backslash \mathbb{Q} $ by linear fractional transformations. In this paper, it is shown that there exist extremal numbers ξ for which the Lagrange constant ν(ξ) = liminf_{$q$→∞}$q$||$q$ξ|| is $ \frac{1}{3} $ , the largest possible value for a non-quadratic number, and that there is a natural bijection between the $ {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) $ -equivalence classes of such numbers and the non-trivial solutions of Markoff’s equation.

We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries (mn−−−√log(mn))−1 for m,n≥2. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with [0,π]. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on spectral perturbation theory and scattering theory. Finding an explicit diagonalisation of the multiplicative Hilbert matrix remains an interesting open problem.

The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the HausdorffYoung inequality, Young's inequality, the HirschmanBeckner uncertainty principle, the DonohoStark uncertainty principle. We characterize the minimizers of the uncertainty principles and then we prove Hardy's uncertainty principle by using minimizers. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's <i></i>-lattices, modular tensor categories, etc.

We characterize mappings$S$_{$i$}and$T$_{$i$}, not necessarily linear, from sets $\mathcal {J}_{i}$ ,$i$=1,2, onto multiplicative subsets of function algebras, subject to the following conditions on the peripheral spectra of their products:$σ$_{$π$}($S$_{1}($a$)$S$_{2}($b$))⊂$σ$_{$π$}($T$_{1}($a$)$T$_{2}($b$)) and$σ$_{$π$}($S$_{1}($a$)$S$_{2}($b$))∩$σ$_{$π$}($T$_{1}($a$)$T$_{2}($b$))≠∅, $a\in \mathcal {J}_{1}$ , $b\in \mathcal {J}_{2}$ . As a direct consequence we obtain a large number of previous results about mappings subject to various spectral conditions.