No abstract

The evolution of quantum light through linear optical devices can be described by the scattering matrix S of the system. For linear optical systems with m possible modes, the evolution of n input photons is given by a unitary matrix U = ϕ m,M (S), derived from a known homomorphism, ϕ m,M , which depends on the size of the resulting Hilbert space of the possible photon states, M. We present a method to decide whether a given unitary evolution U for n photons in m modes can be achieved with linear optics or not and the inverse transformation ϕ −1 m,M when the transformation can be implemented. Together with previous results, the method can be used to find a simple optical system which implements any quantum operation within the reach of linear optics. The results come from studying the adjoint map between the Lie algebras corresponding to the Lie groups of the relevant unitary matrices.

Abstract. Given a complete isometric immersion ϕ : P m −→ N n in an ambient Riemannian manifold N n with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space M n w , we determine a set of conditions on the extrinsic curvatures of P that guarantees that the immersion is proper and that P has finite topology in the line of the results in [24] and [25]. When the ambient manifold is a radially symmetric space, it is shown an inequality between the (extrinsic) volume growth of a complete and minimal submanifold and its number of ends which generalizes the classical inequality stated in [1] for complete and minimal submanifolds in R n . We obtain as a corollary the corresponding inequality between the (extrinsic) volume growth and the number of ends of a complete and minimal submanifold in the Hyperbolic space together with Bernstein type results for such submanifolds in Euclidean and Hyperbolic spaces, in the vein of the work [12].

Abstract. We obtain an estimate of the Cheeger isoperimetric constant in terms of the volume growth for a properly immersed submanifold in a Riemannian manifold which possesses at least one pole and sectional curvature bounded from above.

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