Dual Instruments and Sequential Products of Observables

Abstract

We first show that every operation possesses an unique dual operation and measures an unique effect. If $a$ and $b$ are effects and $J$ is an operation that measures $a$, we define the sequential product of $a$ then $b$ relative to $J$. Properties of the sequential product are derived and are illustrated in terms of Lüders and Holevo operations. We next extend this work to the theory of instruments and observables. We also define the concept of an instrument (observable) conditioned by another instrument (observable). Identity, state-constant and repeatable instruments are considered. Sequential products of finite observables relative to Lüders and Holevo instruments are studied.

27 pages, comments are welcome

References in corpus (2)

• Translation of Lueders' "Uber die Zustandsanderung durch den Messprozess"
• Quantum Instruments and Conditioned Observables

Cited by in corpus (3)

• Causal potency of consciousness in the physical world
• A Theory of Quantum Instruments
• Can QBism exist without Q? Morphophoric measurements in generalised probabilistic theories