Intuitionistic Quantum Logic of an <Emphasis Type="Italic">n</Emphasis>-level System

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra M _n (?) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σ _n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem.In our approach, the nondistributive lattice ?(M _n (?)) of projections in M _n (?) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice \(\mathcal{O}(\Sigma_{n})\) of functions from the poset \(\mathcal{C}(M_{n}(\mathbb{C}))\) of all unital commutative C*-subalgebras C of M _n (?) to ?(M _n (?)). The lattice \(\mathcal{O}(\Sigma_{n})\) is essentially the (pointfree) topology of the quantum phase space Σ _n , and as such defines a Heyting algebra. Each element of \(\mathcal{O}(\Sigma_{n})\) corresponds to a “Bohrified” proposition, in the sense that to each classical context \(C\in\mathcal{C}(M_{n}(\mathbb{C}))\) it associates a yes-no question (i.e. an element of the Boolean lattice ?(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann.     

Intensional dynamic programming. A Rosetta stone for structured dynamic programming

We present intensional dynamic programming (IDP), a generic framework for structured dynamic programming over atomic, propositional and relational representations of states and actions. We first develop set-based dynamic programming and show its equivalence with classical dynamic programming. We then show how to describe state sets intensionally using any form of structured knowledge representation and obtain a generic algorithm that can optimally solve large, even infinite, MDPs without explicit state space enumeration. We derive two new Bellman backup operators and algorithms. In order to support the view of IDP as a Rosetta stone for structured dynamic programming, we review many existing techniques that employ either propositional or relational knowledge representation frameworks.     

Hollow-core microstructured polymer optical fiber

We have fabricated microstructured polymer optical fibers that guide light in a hollow core using the photonic bandgap mechanism. The hollow core allows the use of polymer fibers to be extended to wavelength ranges where material absorption typically prohibits their use, with attenuation lower than the material loss observed in the infrared. The fabrication method is similar to other microstructured polymer optical fibers, which has favorable implications for the feasibility of manufacturing such bandgap fibers.     

Design and characterization of arrayed waveguide gratings using ultra-low loss Si3N4 waveguides

Planar waveguides with ultra-low propagation loss are necessary for integrating optoelectronic systems that require long optical time delay or narrowband optical filters. In this paper, we review an ultra-low loss planar waveguide platform that uses thin (<150 nm) Si₃N₄ cores and thick (>8 μm) SiO₂ cladding layers. In particular, we discuss the performance of arrayed waveguide gratings (AWGs) fabricated with the platform. We propose the use of a practical design method that takes the statistical nature of worst-case crosstalk into account. We also demonstrate the measurement of amplitude and phase error distributions in an AWG using an optical backscatter reflectometer. We show that the waveguides have phase errors small enough to achieve AWG crosstalk below ?30 dB, while crosstalk below ?40 dB should also be possible with optimization of the component design.     

Reaction Calorimetry for the Development of Ultrasound-Induced Polymerization Processes in CO2-Expanded Fluids

A strong viscosity increase upon polymerization hinders radical formation during an ultrasound-induced bulk polymerization. Since CO₂ acts as a strong anti-solvent for most polymers, it can be used to reduce the viscosity of the reaction mixture. In this work, a process for the ultrasound-induced polymerization in CO₂-expanded fluids has been developed. Temperature oscillation calorimetry has been applied to study the influence of CO₂ on the viscosity during the ultrasound-induced polymerization. In contrast to polymerizations in bulk, the results show that a low viscosity is maintained during polymerization reactions in CO₂-expanded methyl methacrylate (MMA). As a consequence, a constant or even increasing polymerization rate is observed when pressurized CO₂ is applied. Moreover, the ultrasound-induced polymer scission in CO₂-expanded MMA is demonstrated, which appears to be a highly controlled process. Finally, a preliminary sustainable process design is presented for the production of 10 kg/hour pure PMMA (specialty product) in CO₂-expanded MMA by ultrasound-induced initiation.     

Synthesis of thiol-modified peptide nucleic acids designed for post-assembly conjugation reactions

Two orthogonally protected PNA monomers were prepared having the mercaptomethyl moiety attached to the PNA backbone. These building blocks were employed in solid-phase PNA synthesis and it was shown that Boc/S-p-methoxybenzyl protection scheme was only satisfactory for the introduction of N-terminal thiol modification while the Fmoc/S-butylthio protected monomer proved to be amenable to elongation. The mercaptomethyl modification did not influence the thermal stability of a PNA/RNA duplex. The feasibility of PNA-PNA native ligation was demonstrated.