# Fine's Theorem for Leggett-Garg tests with an arbitrary number of measurement times

## TL;DR

This paper generalizes Fine's theorem for Leggett-Garg tests to an arbitrary number of measurement times, providing necessary and sufficient conditions for classicality in systems with multiple temporal correlations.

## Contribution

It extends Fine's theorem to multiple measurement times and all two-time correlations, broadening the applicability of Leggett-Garg tests.

## Key findings

- Proved Fine's theorem for any number of measurement times.
- Extended LG framework to include all two-time correlations.
- Analyzed the large number of measurements limit.

## Abstract

If the time evolution of a system can be understood classically, then there must exist an underlying probability distribution for the variables describing the system at all times. It is well known that for systems described by a single time-evolving dichotomic variable $Q$ and for which a given set of temporal correlation functions are specified, a necessary set of conditions for the existence of such a probability are provided by the Leggett-Garg (LG) inequalities. Fine's theorem in this context is the non-trivial result that a suitably augmented set of LG inequalities are both necessary and sufficient conditions for the existence of an underlying probability. We present a proof of Fine's theorem for the case of measurements on a dichotomic variable at an abitrary number of times, thereby generalizing the familiar proofs for three and four times. We demonstrate how the LG framework and Fine's theorem can be extended to the case in which all possible two-time correlation functions are measured (instead of the partial set of two-time correlators normally studied). We examine the limit of a large number of measurements for both of the above cases.