Senior Honors Thesis Presentation: "On the Study of Commutator Indicators for Finite Matrices"

Abstract: In his paper, Quantum Unsharpness and Symplectic Rigidity, Leonid Polterovich talks about the importance of the “intrinsic noise operator”, a measurement which analyzes the probabilities of an observable in a quantum system (mathematically, these observables are functions in a vector space). Polterovich proves that the maximum value the intrinsic noise operator can take is related to the “degree of non-commutativity” of the observable. In my thesis, I analyze this degree for a simple space and treat operators as matrices, creating my own value called the “commutator indicator”. Finding the commutator indicator turns into the following problem: we are given n matrices A_1, A_2, . . . , A_n that sum up to the identity matrix. We must take the matrix norm of the sum of the adjacent pairwise commutators of these matrices, and the commutator indicator is the maximum possible value of this norm. In my paper, I present the results for maximizing the commutator indicator for three 2x2 matrices, along with the concrete maximum value itself and a set of matrices that yield this value. Then with the help of programming, I generalize to four 2x2 matrices and make some conjectures for that case and cases beyond.

Host: Xiang Tang