The boundary rigidity problem in the presence of a magnetic field

For a compact Riemannian manifold with boundary, endowed with a magnetic potential $\alpha$, we consider the problem of restoring the metric $g$ and the magnetic potential $\alpha$ from the values of the Ma\~n\'e action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this case, knowledge of the Ma\~n\'e action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction. This problem can only be solved up to an isometry and a gauge transformation of $\alpha$. For the linearized problem, we show injectivity, up to the natural obstruction, under explicit bounds on the curvature and on $\alpha$. We also show injectivity and stability for $g$ and $\alpha$ in a generic class $\mathcal{G}$ including real analytic ones. For the nonlinear problem, we show rigidity for real analytic simple $g$, $\alpha$. Also, rigidity holds for metrics in a given conformal class, and locally, near any $(g,\alpha)\in \mathcal{G}$.Comment: This revised version contains a proof that 2D simple magnetic systems are boundary rigid. Some references have been adde

Tangles of superpositions and the convex-roof extension

We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is, SL(2,\CC) invariant tangles. We highlight two key concepts that contain valuable information about the tangle of a density matrix: the {\em zero-polytope} is a convex set of density matrices with vanishing tangle whereas the {\em convex characteristic curve} readily provides a non-trivial lower bound for the convex roof and serves as a tool for constructing the convex roof outside the zero-polytope. Both concepts are derived from the tangle for superpositions of the eigenstates of the density matrix. We illustrate their application by considering examples of density matrices for two-qubit and three-qubit states of rank 2, thereby pointing out both the power and the limitations of the concepts.Comment: 7 pages, 5 figures, revtex