A real is computable if its left cut, L; is computable. If q i i is a computable sequence of rationals computably converging to ; then fq i g; the corresponding set, is always computable. A computably enumerable c.e. real is a real which is the limit of an increasing computable sequence of rationals, and has a left cut which is c.e. We study the Turing… (More)
We show there is a computable linear order with a 0 2 initial segment that is not isomorphic to any computable linear order.
Given a computably enumerable set B; there is a Turing degree which is the least jump of any set in which B is computably enumer-able, namely 0 0 : Remarkably, this is not a phenomenon of computably enumerable sets. We show that for every subset A of N; there is a Tur-ing degree, c 0
0 1 classes are important to the logical analysis of many parts of mathematics. The 0 1 classes form a lattice. As with the lattice of computable enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality namely the notion of a thin class. We prove a number of results relating… (More)
We present a proof of the following result due to Solovay: There exists a noncomputable 0 2 real x such that H x n 6 H n + O1:
Scalable quantum technologies may be achieved by faithful conversion between matter qubits and photonic qubits in integrated circuit geometries. Within this context, quantum dots possess well-defined spin states (matter qubits), which couple efficiently to photons. By embedding them in nanophotonic waveguides, they provide a promising platform for quantum… (More)