The goal of this series of lectures is to present some recent results in QIT which make use of random matrices. After an introduction to random matrix theory, I will present the method of moments, one of the most successful methods used to study the spectra of large random matrices. This will be the occasion to discuss integration over Gaussian spaces and over unitary groups. On the QIT side, I will focus on two main topics, random quantum states and random quantum channels. I will then prove two recent results, one on the asymptotic eigenvalue distribution of the partial transposition of random quantum states, and another on the output set of random quantum channels. Both will require some terminology and results from free probability, which will also be discussed in detail.
Useful recent reference: B. Collins and I. Nechita - Random matrix techniques in quantum information theory, J. Math. Phys. 57, 015215 (2016); http://dx.doi.org/10.1063/1.4936880; http://arxiv.org/abs/1509.04689
Plan of lectures:
Lecture 1.
- Introduction to Random Matrices
- Gaussian random variables and integration. The Wick formula
- The Haar measure on the unitary group. The Weingarten formula
Lecture 2.
- Random density matrices. The induced measure
- The asymptotic distribution of eigenvalues
- The partial transposition of random quantum states. Free probability theory
Lecture 3.
- Random quantum channels obtained from random isometries
- The maximal output entropy of quantum channels. The additivity question
- Product of conjugate random quantum channels
- The asymptotic output set of a random quantum channel
