Rigidity for pp-Laplacian type equations on compact Riemannian manifolds

In this paper, we obtain two rigidity results for $p$-Laplacian type equations on compact Riemannian manifolds by using of the carr\'e du champ and nonlinear flow methods, respectively, where rigidity means that the PDE has only constant solution when a parameter is in a certain range. Moreover, an interpolation inequality is derived as an application.Comment: 14 page

Apply current exponential de Finetti theorem to realistic quantum key distribution

In the realistic quantum key distribution (QKD), Alice and Bob respectively get a quantum state from an unknown channel, whose dimension may be unknown. However, while discussing the security, sometime we need to know exact dimension, since current exponential de Finetti theorem, crucial to the information-theoretical security proof, is deeply related with the dimension and can only be applied to finite dimensional case. Here we address this problem in detail. We show that if POVM elements corresponding to Alice and Bob's measured results can be well described in a finite dimensional subspace with sufficiently small error, then dimensions of Alice and Bob's states can be almost regarded as finite. Since the security is well defined by the smooth entropy, which is continuous with the density matrix, the small error of state actually means small change of security. Then the security of unknown-dimensional system can be solved. Finally we prove that for heterodyne detection continuous variable QKD and differential phase shift QKD, the collective attack is optimal under the infinite key size case.Comment: 11 pages, 2 figures, detailed version, applications adde