Properties of Sequential Products

TL;DR

This paper investigates the properties of sequential products of effects in finite-dimensional quantum systems, exploring their mathematical structure, measurement implications, and extensions to observables, culminating in an uncertainty principle for conditioned measurements.

Contribution

It introduces a comprehensive analysis of sequential products, characterizes their properties for different measurement operations, and extends the concept to observables and instruments with new uncertainty relations.

Findings

Sequential product acts as an additive, convex morphism.

Measurement interference affects sequential product properties.

An uncertainty principle for conditioned observables is established.

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.