Categorified sl(N) invariants of colored rational tangles

We use categorical skew Howe duality to find recursion rules that compute categorified sl(N) invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress towards two conjectures about the colored HOMFLY homology of rational links.Comment: 45 pages, many figures, uses dcpic.sty, v2: minor changes and new example 5

Optimal Cloning of Pure States, Judging Single Clones

We consider quantum devices for turning a finite number N of d-level quantum systems in the same unknown pure state \sigma into M>N systems of the same kind, in an approximation of the M-fold tensor product of the state \sigma. In a previous paper it was shown that this problem has a unique optimal solution, when the quality of the output is judged by arbitrary measurements, involving also the correlations between the clones. We show in this paper, that if the quality judgement is based solely on measurements of single output clones, there is again a unique optimal cloning device, which coincides with the one found previously.Comment: 16 Pages, REVTe

Quantum information with continuous variables

Quantum information is a rapidly advancing area of interdisciplinary research. It may lead to real-world applications for communication and computation unavailable without the exploitation of quantum properties such as nonorthogonality or entanglement. We review the progress in quantum information based on continuous quantum variables, with emphasis on quantum optical implementations in terms of the quadrature amplitudes of the electromagnetic field.Comment: accepted for publication in Reviews of Modern Physic

Recognisable languages over monads

The principle behind algebraic language theory for various kinds of structures, such as words or trees, is to use a compositional function from the structures into a finite set. To talk about compositionality, one needs some way of composing structures into bigger structures. It so happens that category theory has an abstract concept for this, namely a monad. The goal of this paper is to propose monads as a unifying framework for discussing existing algebras and designing new algebras