227 research outputs found
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 of order and apply to it
the Gram-Schmidt orthonormalization procedure by columns to obtain a Haar
distributed orthogonal matrix . If denotes the vector formed by
the first -coordinates of the th row of and
, our main result shows that the euclidean norm of
converges exponentially fast to , up to negligible terms.
To show the extent of this result, we use it to study the convergence of the
supremum norm and we find a coupling that improves by a factor
the recently proved best known upper bound of . 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 factor is a subfactor of
the ultrapower of the hyperfinite II 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
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