The Leggett-Garg Inequalities and Non-Invasive Measurability

TL;DR

This paper discusses the Leggett-Garg inequalities, their assumptions, and the implications of their violation, emphasizing the role of non-invasive measurability in quantum versus classical systems.

Contribution

It provides a clear derivation of the inequalities, analyzes the assumptions involved, and examines the limitations of undetectable measurements in testing non-invasiveness.

Findings

Violations of Leggett-Garg inequalities challenge classical realism.

Undetectable measurements are not always non-invasive.

A realist hidden variable theory can violate the inequalities.

Abstract

The Leggett-Garg inequalities are a set of inequalities obeyed by classical systems but violated in quantum theory. Their violation has been taken as evidence that quantum theory lacks a `realistic' formulation. However in addition to realism the derivation of the Leggett-Garg inequalities relies on another, more obscure assumption of `non-invasive measurability.' The significance of this assumption and the consequences for the interpretation of violations of the Leggett-Garg inequalities have been hotly debated. In this paper we present a pedagogical introduction to the issues, focussing on the significance of non-invasive measurability. We give a simple derivation of the Leggett-Garg inequalities paying particular attention to where and when the two assumptions are used, and we give an example of a realist but not non-invasively measurable hidden variable theory which violates them. We also discuss recent attempts to experimentally implement non-invasive measurements via so called `undetectable measurements' and we show that even undetectable measurements are not necessarily non-invasive.