On a product rule and quotient rule for certain quantum metrics on matrices

Quantum metrics in the sense of Rieffel were developed to address certain statements arising in the high-energy physics literature. Since quantum metrics are induced by a generalization of the Lipschitz constant seminorm, which measures the absolute maximum value of the derivative, these seminorms may exhibit a calculus-like structure. One way to identify a calculus-like structure in a non-commutative space-- that is, a quantum space-- is to look for results that resemble the form of the Leibniz product rule and quotient rule from calculus for quantum metrics. In this thesis, we focus on certain quantum metrics defined by conditional expectations. We provide computational evidence for an improved version of a product rule and an analytical derivation of a case of the quotient rule.