Moments have been intensively studied in probability theory, quantum information theory or matrix analysis. During the last years, non-commutative moment estimates have received considerable attention in the scientific community. Firstly, this phenomenon originates in studies on the extreme properties of the standard deviation in quantum information theory and its recent applications in commutator estimates. Secondly, there has been an urgent need driven by particle physics to develop a non-commutative version of the classical theory of compact metric spaces and the Gromov-Hausdorff distance. One building block on this road is the concept of strongly Leibniz seminorms which appears in the work of Rieffel on the problem of convergence of finitely generated modules over quantum metric spaces. Interestingly, the simplest example of such seminorms is the non-commutative standard deviation. On the other hand, we need to mention that one can find Leibniz-type inequalities in the general theory of Dirichlet forms and non-linear PDEs as well where they are known as the Kato-Ponce inequalities.
The primary goal of the research project is to provide a better understanding of non-commutative moment inequalities in terms of their commutative counterparts and the strongly Leibniz seminorms through new examples. Specifically, we addressed the following fundamental questions: How large the higher-order non-commutative moments can be? Do all central moments have the strong Leibniz property?
These questions are sitting at an exciting intersection of several mathematical disciplines, including linear algebra, singular value inequalities, matrix and functional analysis and rearrangement inequalities.
We completely solved the first question in case of the fourth central moment and proved that the corresponding commutative bound is, in fact, an upper bound in the general matrix case. However, examples show that the sharp upper bound can be strictly smaller for special matrices. Furthermore, we have provided several particular answers in the general case as well.
In regards to the second question, first, we studied the Leibniz inequality of central moments in ordinary probability spaces. We demonstrated that all central moments satisfy the Leibniz property, but they are not necessarily strongly Leibniz. We have been presenting several proofs to this problem that reveal a strong connection to rearrangement inequalities and open up the way to much more general theorems. Numerical simulations support the conjecture of the Leibniz property for singular values of Hermitian products as well. Our convexity approach to the problem has led to new proofs of several singular value inequalities in the literature.
