Free entropy

Philippe Biane
Université Paris-Est Marne-la-Vallée
April 22, 2014
Free entropy is a quantity introduced 20 years ago by D. Voiculescu in order to investigate noncommutative probability spaces (e.g. von Neumann algebras). It is based on approximation by finite size matrices. I will describe the definition and main properties of this quantity as well as applications to von Neumann algebras. I will also explain a new approach based on work with Y. Dabrowski, using random matrices, which leads to the solution of some problems concerning this quantity.

Results and open problems in theory of quantum complexity

Andris Ambainis
University of Latvia; Member, School of Mathematics
April 22, 2014
I will survey recent results and open problems in several areas of quantum complexity theory, with emphasis on open problems which can be phrased in terms of classical complexity theory or mathematics but have implications for quantum computing:

1. Quantum vs. classical query complexity
2. Quantum vs. classical query complexity for almost all inputs
3. Quantum counterparts of Valiant-Vazirani theorem (reducing NP to unique-NP)

A Riemann-Roch theorem in Bott-Chern cohomology

Jean-Michel Bismut
Université Paris-Sud
April 21, 2014
If \(M\) is a complex manifold, the Bott-Chern cohomology \(H_{\mathrm{BC}}^{(\cdot,\cdot)}\left(M,\mathbf{C}\right)\) of \(M\) is a refinement of de Rham cohomology, that takes into account the \((p,q)\) grading of smooth differential forms. By results of Bott and Chern, vector bundles have characteristic classes in Bott-Chern cohomology, which will be denoted with the subscript \(\mathrm{BC}\). Let \(p:M\to S\) be a proper holomorphic submersion of complex manifolds. Let \(F\) be a holomorphic vector bundle on \(M\) and let \(Rp_{*}F\) be its direct image.

True Randomness: Its Origin and Expansion

Yaoyun Shi
University of Michigan
April 21, 2014
How can we produce randomness of almost perfect quality, in large quantities, and under minimal assumptions? This question is fundamental not only to modern day information processing but also to physics. Yet a satisfactory answer is still elusive to both the practice and the theory of randomness extraction. Here we propose a solution through a new paradigm of extracting randomness from physical systems and basing security on the validity of physical theories, such as quantum mechanics and special relativity.

Limiting Eigenvalue Distribution of Random Matrices Involving Tensor Product

Leonid Pastur
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
April 16, 2014
We consider two classes of \(n \times n\) sample covariance matrices arising in quantum informatics. The first class consists of matrices whose data matrix has \(m\) independent columns each of which is the tensor product of \(k\) independent \(d\)-dimensional vectors, thus \(n=d^k\). The matrices of the second class belong to \(\mathcal{M}_n(\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}), \ n=d_1 d_2\) and are obtained from the standard sample covariance matrices by the partial transposition in \(\mathbb{C}^{d_2}\).

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