Dual Instruments and Sequential Products of Observables

TL;DR

This paper develops a mathematical framework for dual operations and sequential products of quantum observables and instruments, extending the theory with new definitions and properties, especially in the context of L"uders and Holevo operations.

Contribution

It introduces the concept of a dual operation for each quantum operation and defines sequential products of effects and observables relative to specific instruments.

Findings

Every operation has a unique dual operation and effect.

Properties of the sequential product are established and illustrated.

Sequential products of finite observables are analyzed in the context of L"uders and Holevo instruments.

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 L\"uders 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 L\"uders and Holevo instruments are studied.