On the dimension of subspaces with bounded Schmidt rank

We consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states. This is motivated in part by results of Hayden et al., which show that in large d x d--dimensional systems there exist random subspaces of dimension almost d^2, all of whose states have entropy of entanglement at least log d - O(1). It is also related to results due to Parthasarathy on the dimension of completely entangled subspaces, which have connections with the construction of unextendible product bases. Here we take as entanglement measure the Schmidt rank, and determine, for every pair of local dimensions dA and dB, and every r, the largest dimension of a subspace consisting only of entangled states of Schmidt rank r or larger. This exact answer is a significant improvement on the best bounds that can be obtained using random subspace techniques. We also determine the converse: the largest dimension of a subspace with an upper bound on the Schmidt rank. Finally, we discuss the question of subspaces containing only states with Schmidt equal to r.Comment: 4 pages, REVTeX4 forma

Evolution of Massive Haloes in non-Gaussian Scenarios

We have performed high-resolution cosmological N-body simulations of a concordance LCDM model to study the evolution of virialized, dark matter haloes in the presence of primordial non-Gaussianity. Following a standard procedure, departures from Gaussianity are modeled through a quadratic Gaussian term in the primordial gravitational potential, characterized by a dimensionless non-linearity strength parameter f_NL. We find that the halo mass function and its redshift evolution closely follow the analytic predictions of Matarrese et al.(2000). The existence of precise analytic predictions makes the observation of rare, massive objects at large redshift an even more attractive test to detect primordial non-Gaussian features in the large scale structure of the universe.Comment: 7 pages,3 figures, submitted to MNRA