Kitaev's quantum double model from a local quantum physics point of view

A prominent example of a topologically ordered system is Kitaev's quantum double model $\mathcal{D}(G)$ for finite groups $G$ (which in particular includes $G = \mathbb{Z}_2$, the toric code). We will look at these models from the point of view of local quantum physics. In particular, we will review how in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the different superselection sectors of the model. In this way one finds that the charges are in one-to-one correspondence with the representations of $\mathcal{D}(G)$, and that they are in fact anyons. Interchanging two of such anyons gives a non-trivial phase, not just a possible sign change. The case of non-abelian groups $G$ is more complicated. We outline how one could use amplimorphisms, that is, morphisms $A \to M_n(A)$ to study the superselection structure in that case. Finally, we give a brief overview of applications of topologically ordered systems to the field of quantum computation.Comment: Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J. Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015). Mainly revie

Haag duality and the distal split property for cones in the toric code

We prove that Haag duality holds for cones in the toric code model. That is, for a cone Lambda, the algebra R_Lambda of observables localized in Lambda and the algebra R_{Lambda^c} of observables localized in the complement Lambda^c generate each other's commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if Lambda_1 \subset Lambda_2 are two cones whose boundaries are well separated, there is a Type I factor N such that R_{Lambda_1} \subset N \subset R_{Lambda_2}. We demonstrate this by explicitly constructing N.Comment: 15 pages, 2 figures, v2: extended introductio

Notions of Infinity in Quantum Physics

In this article we will review some notions of infiniteness that appear in Hilbert space operators and operator algebras. These include proper infiniteness, Murray von Neumann's classification into type I and type III factors and the class of F{/o} lner C*-algebras that capture some aspects of amenability. We will also mention how these notions reappear in the description of certain mathematical aspects of quantum mechanics, quantum field theory and the theory of superselection sectors. We also show that the algebra of the canonical anti-commutation relations (CAR-algebra) is in the class of F{/o} lner C*-algebras.Comment: 11 page

Human platelet lysate as a fetal bovine serum substitute improves human adipose-derived stromal cell culture for future cardiac repair applications

Adipose-derived stromal cells (ASC) are promising candidates for cell therapy, for example to treat myocardial infarction. Commonly, fetal bovine serum (FBS) is used in ASC culturing. However, FBS has several disadvantages. Its effects differ between batches and, when applied clinically, transmission of pathogens and antibody development against FBS are possible. In this study, we investigated whether FBS can be substituted by human platelet lysate (PL) in ASC culture, without affecting functional capacities particularly important for cardiac repair application of ASC. We found that PL-cultured ASC had a significant 3-fold increased proliferation rate and a significantly higher attachment to tissue culture plastic as well as to endothelial cells compared with FBS-cultured ASC. PL-cultured ASC remained a significant 25% smaller than FBS-cultured ASC. Both showed a comparable surface marker profile, with the exception of significantly higher levels of CD73, CD90, and CD166 on PL-cultured ASC. PL-cultured ASC showed a significantly higher migration rate compared with FBS-cultured ASC in a transwell assay. Finally, FBS- and PL-cultured ASC had a similar high capacity to differentiate towards cardiomyocytes. In conclusion, this study showed that culturing ASC is more favorable in PL-supplemented medium compared with FBS-supplemented medium

Topologische kwantumcomputers: rekenen met vlechten

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Anyons in infinite quantum systems : QFT in d=2+1 and the Toric Code

Contains fulltext : 92737.pdf (publisher's version ) (Open Access)Anyons, comprising a special type of particles that exhibit braid statistics, have received considerable attention in the past decade, from physicists, mathematicians, and computer scientists. It turns out that so-called modular tensor categories naturally appear in the description of the key properties of anyons, which partly explains the interest of mathematicians in the subject. In this thesis, we study the way such modular tensor categories emerge in two classes of models, viz. quantum field theory in low-dimensional Minkowski space-time, and quantum spin systems on an infinite spatial lattice. Although the nature of these two classes of models is quite different, we show in this thesis that they may both be discussed in a similar (mathematically rigorous) framework. This approach is inspired by the work of Dpplicher, Haag and Roberts (DHR) in algebraic quantum field theory. The DHR approach leads to braided tensor categories, which describe the charges in the theory (in our case, these are anyons). These categories need not be modular. In this thesis we present a method to remove an obstruction for modularity, in the case of a space-time of dimension three. We then study (a variant of) Kitaev's toric code, regarded as a quantum spin model on an infinite lattice. Using techniques inspired by the DHR programme, we show that one can obtain a modular tensor category describing all relevant properties of the anyons in this system. Finally, some generalisations to non-abelian models are discussed.Radboud Universiteit Nijmegen, 15 mei 2012Promotor : Landsman, N.P. Co-promotor : Müger, M.H.A.H.222 p