This study presents a foundational exploration of free probability theory, focusing on the key ideas of non-commutative probability space, free independence, and free convolution. The proposed method delves into the important relationship between random matrices and free probability theory. This study introduces an innovative approach for identifying quantum entangled states within the realm of quantum information theory, employing the Nica–Speicher semigroup construction proposition. The proposed method holds promise for improving our comprehension and ability to manage the detection of quantum entanglement efficiently.

The study of the behavior of spectrally negative Lévy processes has been a subject of interest for many years. In particular, around 2000, there was a major advancement in the understanding of scale functions for these processes. These scale functions are not only of theoretical interest but are also essential in practical fields such as ruin theory, optimal stopping problems, and optimal control problems. Moreover, recent research has explored the scale functions for other stochastic processes that lack positive jumps, aiming to apply these functions to similar challenges. In this study, we discuss the properties of scale functions for spectrally negative Lévy processes and discuss their applications. In addition, we provide an overview of the scale functions for several other stochastic processes without positive jumps.