Euclidean distance between Haar orthogonal and gaussian matrices

In this work we study a version of the general question of how well a Haar distributed orthogonal matrix can be approximated by a random gaussian matrix. Here, we consider a gaussian random matrix $Y_n$ of order $n$ and apply to it the Gram-Schmidt orthonormalization procedure by columns to obtain a Haar distributed orthogonal matrix $U_n$. If $F_i^m$ denotes the vector formed by the first $m$-coordinates of the $i$th row of $Y_n-\sqrt{n}U_n$ and $\alpha=\frac{m}{n}$, our main result shows that the euclidean norm of $F_i^m$ converges exponentially fast to $\sqrt{ \left(2-\frac{4}{3} \frac{(1-(1 -\alpha)^{3/2})}{\alpha}\right)m}$, up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm $\epsilon_n(m)=\sup_{1\leq i \leq n, 1\leq j \leq m} |y_{i,j}- \sqrt{n}u_{i,j}|$ and we find a coupling that improves by a factor $\sqrt{2}$ the recently proved best known upper bound of $\epsilon_n(m)$. Applications of our results to Quantum Information Theory are also explained.Comment: v2: minor modifications to match journal version, 26 pages, 0 figures, J Theor Probab (2016

Comparison of delensing methodologies and assessment of the delensing capabilities of future experiments

Most of the CMB experiments proposed for the next generation aim to detect the Primordial Gravitational Wave Background (PGWB). The fulfillment of this objective depends on our capacity to separate Galactic foreground emissions and to \emph{delens} the secondary B-mode component induced by weak gravitational lensing. Focusing on the latter of these efforts, in this work we briefly review the basic aspects of lensing, and exhaustively compare the performance of current delensing methodologies and implementations within the Born approximation as a preparation for the analysis of the data to come in the following years. Two of the main conclusions that can be drawn from our study are that, for next-generation experiments, delensing efficiency will still be limited by the quality of the data itself rather than by the limitations of current delensing methodologies, and that template delensing within the antilensing approximation will be the optimal (balancing accuracy and computational cost) technique to employ. We then evaluate the delensing capabilities of future experiments (like the Simons Observatory, the CMB Stage-IV, or the LiteBIRD and PICO satellites) by applying that methodology onto numerical simulations of the typical CMB and lensing potential reconstructions that they are expected to produce, and quantify how internal and external delensing will help them to improve their sensitivity to detect the PGWB. We also consider the benefits that a joint analysis of their data would provide.Comment: 37 pages, 12 figures, submitted to JCA

Connes' embedding problem and Tsirelson's problem

We show that Tsirelson's problem concerning the set of quantum correlations and Connes' embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchberg's QWEP conjecture) are essentially equivalent. Specifically, Tsirelson's problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the set of correlations between commuting C*-algebras. Connes' embedding problem asks whether any separable II$_1$ factor is a subfactor of the ultrapower of the hyperfinite II$_1$ factor. We show that an affirmative answer to Connes' question implies a positive answer to Tsirelson's. Conversely, a positve answer to a matrix valued version of Tsirelson's problem implies a positive one to Connes' problem