Seminar on Free Probability
and Random Matrices

Winter 2019

In this talk we are interested in finding an optimal upper bound for sums of product of matrices entries with some restrictions. These kinds of sums appear canonically in many problems such as the trace of the product of matrices. In order to find this upper bound we will associate a graph to every sum. This translate the problem to finding an upper bound to some operator associated to a graph. We will use some traffic and functional analysis techniques to solve our problem.

Classical/quantum capacity of quantum channels will be discussed in the first half. Then I will introduce a class of channels that can preserve SU(2)-symmetries.

We covered the most of the background knowledge that we need to construct the operator-valued infinitesimal free convolution (OVIC ) last talk. In this talk, we will define the C^*-operator-valued infinitesimal probability spaces and start to construct the OVIC.

In this talk, minimum output entropy (MOE) and Holevo/quantum capacity of quantum channels will be discussed. And I will sketch how the additivity conjecture of MOE was disproved by random unitary channels.

In this talk, we will recall some basic knowledge of operator-valued probability spaces, and some analytical properties of the operator-valued Cauchy transforms. Then, we will introduce the C^*-operator-valued infinitesimal probability spaces, and the operator-valued infinitesimal Cauchy transform. We will also construct the operator-valued infinitesimal R-transform if time permits.

As a cornerstone for discussing quantum information theory, I will explain what quantum states and quantum channels are in this talk. And I will introduce some important properites of quantum channels such as the entanglement-breaking property and classical/quantum capacity.

I will continue from last week.

In this talk, we will introduce the operator-valued infinitesimal probability space and we will give an characterization of operator-valued infinitesimal freeness.

In the month of March the seminar will move to the Centre de Recherches Mathématiques at l'Université de Montréal for a one month focus program.

I will discuss § 2.3 Difference-Differential Calculus and § 2.4 Taylor-Taylor Expansion.

I will continue the discussion of the difference-differential calculus.

I will discuss the difference-differential calculus.

When one extends the Cauchy transform to the operator valued case one cannot recover the law of the random variable without extending to the matricial case. This requires one to develop the theory of fully matricial functions. These are functions defined on open sets in C*-algebras that extend to matrices of all orders in a regular way. I will present the example of the Cauchy transform and introduce fully matricial functions.

This talk will review the construction of the tensor product of modules and its universal property. This property can be used to establish associativity, commutativity over direct sums, and a check for linear independence when our modules are vector spaces. I also hope to give a brief introduction to tensor norms.