Properties of Sequential Products

• URL: http://arxiv.org/abs/2307.16327v1
• Date: Sun, 30 Jul 2023 21:48:07 GMT
• Title: Properties of Sequential Products
• Authors: Stanley Gudder
• Abstract summary: We define the sequential product $a[mathcalI]b$ of $a$ then $b$. It is observed that $bmapsto a[mathcalI]b$ is an additive, convex morphism. We consider repeatable effects and conditions on $a[mathcalI]b$ that imply commutativity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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