On the number of real eigenvalues of products of random matrices and an application to quantum entanglement

The probability that there are $k$ real eigenvalues for an $n$ dimensional real random matrix is known. Here we study this for the case of products of independent random matrices. Relating the problem of the probability that the product of two real 2 dimensional random matrices has real eigenvalues to an issue of optimal quantum entanglement, this is fully analytically solved. It is shown that in $\pi/4$ fraction of such products the eigenvalues are real. Being greater than the corresponding known probability ($1/\sqrt{2}$) for a single matrix, it is shown numerically that the probability that {\it all} eigenvalues of a product of random matrices are real tends to unity as the number of matrices in the product increases indefinitely. Some other numerical explorations, including the expected number of real eigenvalues is also presented, where an exponential approach of the expected number to the dimension of the matrix seems to hold.