On 24 January 2026, at VNUHCM–University of Science (HCMUS), researcher Nguyễn Duy Ái Nhân (specialising in Algebra and Number Theory) successfully defended his doctoral dissertation at the institutional level. The thesis, titled “Quantum Distance Functions, Non-linear Matrix Equations, and Related Issues”, was conducted under the academic supervision of Associate Professor Đinh Trung Hòa and Dr Nguyễn Anh Thi.

 The dissertation focuses on theoretical problems within modern mathematics, specifically concepts relating to quantum distance, non-linear matrix equations, and matrix mean theory. These research directions hold fundamental significance in operator theory and quantum mathematics. A primary contribution of the thesis is the construction of a novel quantum distance function of the Hellinger type. Upon determining the structure of this function, the researcher proved several vital mathematical properties, including the midpoint property, data processing inequalities, and the existence of solutions for the corresponding least squares problem. These findings contribute to the system of mathematical tools used to describe and analyse the distance between quantum states.

 In addition, the thesis extends two forms of non-linear matrix equations from the weighted geometric mean case to weighted means corresponding to any symmetric Kubo-Ando mean. Within this research framework, the candidate proved the existence of solutions for non-linear matrix equations under general conditions, where the involved matrices are invertible and the parameters satisfy appropriate hypotheses.

 The dissertation also devotes a significant portion to the study of systems of non-linear matrix equations related to the Wasserstein mean. By representing the Wasserstein mean as a weighted geometric mean of order 2 and exploiting the connection between the Wasserstein mean and the arithmetic mean, the researcher established and proved the existence of solutions for several non-linear matrix equation systems. This group of results carries substantial theoretical weight, as the Wasserstein mean does not belong to the class of Kubo-Ando matrix means, thus requiring bespoke analytical approaches.

The results achieved in the dissertation help clarify the relationships between various forms of matrix means, while simultaneously opening possibilities for constructing new quantum distance functions based on those relationships. From a scientific perspective, the research provides a further theoretical basis for the study of operator monotone functions and problems involving non-linear matrix equations.