Well-posedness and exponential equilibration of a volume-surface reaction-diffusion system with nonlinear boundary coupling

We consider a model system consisting of two reaction-diffusion equations, where one species diffuses in a volume while the other species diffuses on the surface which surrounds the volume. The two equations are coupled via a nonlinear reversible Robin-type boundary condition for the volume species and a matching reversible source term for the boundary species. As a consequence of the coupling, the total mass of the two species is conserved. The considered system is motivated for instance by models for asymmetric stem cell division. Firstly we prove the existence of a unique weak solution via an iterative method of converging upper and lower solutions to overcome the difficulties of the nonlinear boundary terms. Secondly, our main result shows explicit exponential convergence to equilibrium via an entropy method after deriving a suitable entropy entropy-dissipation estimate for the considered nonlinear volume-surface reaction-diffusion system.Comment: 31 page

Mixed State Entanglement: Manipulating Polarisation-Entangled Photons

There has been much discussion recently regarding entanglement transformations in terms of local filtering operations and whether the optimal entanglement for an arbitrary two-qubit state could be realised. We introduce an experimentally realisable scheme for manipulating the entanglement of an arbitrary state of two polarisation entangled qubits. This scheme is then used to provide some perspective to the mathematical concepts inherent in this field with respect to a laboratory environment. Specifically, we look at how to extract enhanced entanglement from systems with a fixed rank and in the case where the rank of the density operator for the state can be reduced, show how the state can be made arbitrarily close to a maximally entangled pure state. In this context we also discuss bounds on entanglement in mixed states.Comment: 12 pages, 10 figure

A survey of uncertainty principles and some signal processing applications

The goal of this paper is to review the main trends in the domain of uncertainty principles and localization, emphasize their mutual connections and investigate practical consequences. The discussion is strongly oriented towards, and motivated by signal processing problems, from which significant advances have been made recently. Relations with sparse approximation and coding problems are emphasized

Optimal Uncertainty Quantification

We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call Optimal Uncertainty Quantification (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as extreme values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions, they have finite-dimensional reductions. As an application, we develop Optimal Concentration Inequalities (OCI) of Hoeffding and McDiarmid type. Surprisingly, contrary to the classical sensitivity analysis paradigm, these results show that uncertainties in input parameters do not necessarily propagate to output uncertainties. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact, suggesting the feasibility of the framework for important complex systems

Efficient Approximation of Quantum Channel Capacities

We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite dimensional output, possibly under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To provide an $\varepsilon$-close estimate to the capacity, the presented algorithm requires $O(\tfrac{(N \vee M) M^3 \log(N)^{1/2}}{\varepsilon})$, where $N$ denotes the input alphabet size and $M$ the output dimension. We then generalize the method for the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite dimensional output. For channels with a finite dimensional quantum mechanical input and output, the idea of a universal encoder allows us to approximate the Holevo capacity using the same method. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multidimensional integration problem. For families of quantum channels fulfilling a certain assumption we show that the complexity to derive an $\varepsilon$-close solution to the Holevo capacity is subexponential or even polynomial in the problem size. We provide several examples to illustrate the performance of the approximation scheme in practice.Comment: 36 pages, 1 figur