Quantum Systems on Linear Groups

Hendra I. Nurdin, Naoki Yamamoto

Ho Yiu Chung, Cihan Okay, Igor Sikora

Abstract A linear constraint system is specified by linear equations over the group Zd of integers modulo d. Their operator solutions play an important role in the study of quantum contextuality and non-local games. In this paper, we use the theory of simplicial sets to develop a framework for studying operator solutions of linear systems. Our approach refines the well-known group-theoretical approach based on solution groups by identifying these groups as algebraic invariants closely related to the fundamental group of a space. In this respect, our approach also makes a connection to the earlier homotopical approach based on cell complexes. Within our framework, we introduce a new class of linear systems that come from simplicial sets and show that any linear system can be reduced to one of that form. Then we specialize in linear systems that are associated with groups. We provide significant evidence for a conjecture stating that for odd d every linear system admitting a solution in a group admits a solution in Zd.

Jesús Clemente-Gallardo, Giuseppe Marmo

We review the geometrical formulation of quantum mechanics to identify, according to Klein's programme, the corresponding group of transformations. For closed systems, it is the unitary group. For open quantum systems, the semigroup of Kraus maps contains, as a maximal subgroup, the general linear group. The same group emerges as the exponentiation of the C*-algebra associated with the quantum system, when thought of as a Lie algebra. Thus, open quantum systems seem to identify the general linear group as associated with quantum mechanics and moreover suggest to extend the Klein programme also to groupoids. The usual unitary group emerges as a maximal compact subgroup of the general linear group.

L. E. Reichl

Jincheng Zhang

This paper introduces a novel approach to quantum control theory by leveraging the principles of non-linear topological groups. Traditional quantum control methods often struggle to accurately model and control systems exhibiting significant non-linearities. This research proposes a framework for mapping quantum control systems onto non-linear topological groups, capitalizing on the inherent stability and symmetry properties of these groups to achieve enhanced control precision and robustness. We demonstrate how topological invariants can be utilized to characterize the system's dynamics and derive optimal control strategies. The core of this approach lies in the ability to represent complex quantum systems within a simplified topological space, enabling efficient analysis and design of control protocols. This work provides a theoretical foundation for a new generation of quantum control techniques, particularly relevant for systems with intricate non-linear interactions.

Brian Jongwon Choi