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.
