Indirect acquisition of information in quantum mechanics: states associated with tail events

TL;DR

This paper explores how quantum systems encode information about their long-term behavior through tail events, linking asymptotic measurement operators to histories that depend only on infinite future data.

Contribution

It establishes a mathematical framework connecting tail events in quantum histories with asymptotic observables, advancing understanding of indirect quantum measurements.

Findings

Quantum states determine probability measures on history spaces.

Functions depending on tail events correspond to asymptotic observables.

The framework applies to non-demolition measurements in quantum mechanics.

Abstract

The problem of reconstructing information on a physical system from data acquired in long sequences of direct (projective) measurements of some simple physical quantities - histories - is analyzed within quantum mechanics; that is, the quantum theory of indirect measurements, and, in particular, of non-demolition measurements is studied. It is shown that indirect measurements of time-independent features of physical systems can be described in terms of quantum-mechanical operators belonging to an algebra of asymptotic observables. Our proof involves associating a natural measure space with certain sets of histories of a system and showing that quantum-mechanical states of the system determine probability measures on this space. Our main result then says that functions on that space of histories measurable at infinity (i.e., functions that only depend on the tails of histories) correspond to operators in the algebra of asymptotic observables.