In this thesis we present results and techniques from random matrix theory, the theory which studies matrices whose entries are random variables,and their connection to quantum information theory.More precisely, first we define the concept of the empirical spectral distribution of a matrix, which is the uniform discrete measure which is induced by the eigenvalues of the matrix. Next we study the limiting behaviour, in the sense of weak convergence of random variables, of the empirical spectral distribution of random matrices as their size grows. The cases we study, under certain conditions such as i.i.d. entries, finite moments and more, are the following:
1. The case of square symmetric random matrices.
2. The case of the product of a random matrix (not necessary square) with its conjugate transpose matrix, when the dimensions of the ma-trix are proportional.
3. The case of the product of a random matrix (not necessary square)with its conjugate transpose matrix, when one dimension grows faster than the other.
In each of these cases we show that the limit is a probability measure which is absolutely continuous to the Lebesgue measure and has compact support. Moreover, in each of these cases, we prove that the extreme eigenvalues of the matrices converge to the extreme points of the support of the corresponding limit of the empirical spectral distribution of the matrices,when the entries follow the standard Gaussian distribution.Next we present tools from random matrix theory that are useful inquantum information theory. We define several concepts such as the ∞−Wasserstein distance and the random (quantum) induced states, and prove some of their properties. It is proven that the concept of random quantum state is strongly related with matrices with standard Gaussian entries. Taking advantage of this connection we apply the results of the previous chapter to the study of random quantum states.In the last chapter, using the results that we obtain for the random quantum states, we prove the existence of a threshold function which depends only on the dimension of the space and separates with high probability the states which are entangled from those that are not entangled, having as a criterion the dimension of the space from which the states have been induced.
Random matrices,quantum theory,quantum information theory,Probability Theory,Mathematical analysis