Quantum Transition Probabilities

TL;DR

This paper extends the concept of quantum transition probabilities to mixed states and quantum effects, providing a more general framework applicable to a broader range of quantum systems and operations.

Contribution

It introduces a generalized formulation of transition probabilities in quantum mechanics that includes mixed states and quantum effects, expanding their applicability beyond pure states.

Findings

Transition probabilities are extended to mixed states and effects.

They depend on measured operations or instruments.

Special cases reduce to traditional pure state transition probabilities.

Abstract

Transition probabilities are an important and useful tool in quantum mechanics. However, in their present form, they are limited in scope and only apply to pure quantum states. In this article we extend their applicability to mixed states and to transitions between quantum effects. We also present their dependence on a measured operation or instrument. We begin by defining our concepts on a general quantum effect algebra. These concepts are illustrated using Holevo operations and instruments. We then present transition probabilities in the special case of the Hilbert space formulation of quantum mechanics. We show that for pure states and particular types of operations the transition probabilities reduce to their usual form. We give examples in terms of L\"uders operations and instruments.