In the context of the classical central limit theorem, let Y_n denote the n-th normalized sums of  i.i.d copies of a random variable X with mean 0 and variance 1. Following Shannon's work in information theory, Lieb conjectured in 1978 that the differential entropy of Y_n increases monotonically in n. This conjecture was finally settled by Artstein, Ball,  Barthe  and Naor (ABBN) in 2004. In fact, the ABBN article proved more general results and tied the so-called entropy power inequalities into this framework. These inequalities are extremely useful in proving several coding theorems in information theory. 

On the non-commutative side of the story, Cushen and Hudson, in 1971, proved a quantum probability analog of the classical central limit theorem. The monotonicity of von Neumann entropy under the Cushen-Hudson central limit theorem remains an open problem in this area. Guha in 2008 showed that certain quantum analogs of entropy power inequalities, if proved, will produce several coding theorems in quantum information theory, but these problems also remain open to this day. In this talk, we discuss an overview of this area of research.