Properties of Sequential Products

Abstract: Our basic concept is the set $\mathcal{E}(H)$ of effects on a finite dimensional complex Hilbert space $H$. If $a,b\in\mathcal{E}(H)$, we define the sequential product $a[\mathcal{I}]b$ of $a$ then $b$. The sequential product depends on the operation $\mathcal{I}$ used to measure $a$. We begin by studying the properties of this sequential product. It is observed that $b\mapsto a[\mathcal{I}]b$ is an additive, convex morphism and we show by examples that $a\mapsto a[\mathcal{I}]b$ enjoys very few conditions. This is because a measurement of $a$ can interfere with a later measurement of $b$. We study sequential products relative to Kraus, L\"uders and Holevo operations and find properties that characterize these operations. We consider repeatable effects and conditions on $a[\mathcal{I}]b$ that imply commutativity. We introduce the concept of an effect $b$ given an effect $a$ and study its properties. We next extend the sequential product to observables and instruments and develop statistical properties of real-valued observables. This is accomplished by employing corresponding stochastic operators. Finally, we introduce an uncertainty principle for conditioned observables.