Law of the Iterated Logarithm for some Markov operators

The Law of the Iterated Logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell cycle model examined by A. Lasota and M.C. Mackey, J. Math. Biol. (1999).Comment: 23 page

Coupling capacity in C*-algebras

Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states, and establish a duality formula that shows the equality of two of the parameters. Restricting to abelian C*-algebras we recover instances of Monge-Kantorovich duality and establish a connection with quantitative versions of Arveson's Null Set Theorem. On the other hand, restricting to matrix algebras we recover and generalise quantum versions of Strassen's Theorem. We show that in the latter case our parameters can detect maximal entanglement and separability.Comment: 19 page

Implementing Shor's algorithm on Josephson Charge Qubits

We investigate the physical implementation of Shor's factorization algorithm on a Josephson charge qubit register. While we pursue a universal method to factor a composite integer of any size, the scheme is demonstrated for the number 21. We consider both the physical and algorithmic requirements for an optimal implementation when only a small number of qubits is available. These aspects of quantum computation are usually the topics of separate research communities; we present a unifying discussion of both of these fundamental features bridging Shor's algorithm to its physical realization using Josephson junction qubits. In order to meet the stringent requirements set by a short decoherence time, we accelerate the algorithm by decomposing the quantum circuit into tailored two- and three-qubit gates and we find their physical realizations through numerical optimization.Comment: 12 pages, submitted to Phys. Rev.