Dr. Tiju Cherian John (UA, Optical Sciences), The Quantum Central Limit Theorem and Monotonicity Conjectures Related to Entropy: A Bird's Eye View
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Abstract: In the context of the classical central limit theorem, let Y_n denote the n'th normalized sums of IID 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 paper 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 this story, Cushen and Hudson in 1971 proved a quantum probability analogue 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. Saikat Guha (UA, Optical Sciences) showed in 2008 that certain quantum analogues 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 give an overview of this area of research.