7,944 research outputs found
Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization
In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton’s method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers’ equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a FE space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity).Peer ReviewedPostprint (author's final draft
Entanglement verification with realistic measurement devices via squashing operations
Many protocols and experiments in quantum information science are described
in terms of simple measurements on qubits. However, in a real implementation,
the exact description is more difficult, and more complicated observables are
used. The question arises whether a claim of entanglement in the simplified
description still holds, if the difference between the realistic and simplified
models is taken into account. We show that a positive entanglement statement
remains valid if a certain positive linear map connecting the two
descriptions--a so-called squashing operation--exists; then lower bounds on the
amount of entanglement are also possible. We apply our results to polarization
measurements of photons using only threshold detectors, and derive procedures
under which multi-photon events can be neglected.Comment: 12 pages, 2 figure
An introduction to operational quantum dynamics
In the summer of 2016, physicists gathered in Torun, Poland for the 48th
annual Symposium on Mathematical Physics. This Symposium was special; it
celebrated the 40th anniversary of the discovery of the
Gorini-Kossakowski-Sudarshan-Lindblad master equation, which is widely used in
quantum physics and quantum chemistry. This article forms part of a Special
Volume of the journal Open Systems & Information Dynamics arising from that
conference; and it aims to celebrate a related discovery -- also by Sudarshan
-- that of Quantum Maps (which had their 55th anniversary in the same year).
Nowadays, much like the master equation, quantum maps are ubiquitous in physics
and chemistry. Their importance in quantum information and related fields
cannot be overstated. In this manuscript, we motivate quantum maps from a
tomographic perspective, and derive their well-known representations. We then
dive into the murky world beyond these maps, where recent research has yielded
their generalisation to non-Markovian quantum processes.Comment: Submitted to Special OSID volume "40 years of GKLS
Testing complete positivity
We study the modified dynamical evolution of the neutral kaon system under
the condition of complete positivity. The accuracy of the data from planned
future experiments is expected to be sufficiently precise to test such a
hypothesis.Comment: 12 pages, latex, no figures, to appear in Mod. Phys. Lett.
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