Case Study: The Leggett-Garg Inequalities

\chapter{The Leggett-Garg Inequalities}
Consider a system characterized by a dichotomous observable, which assumes values $\pm1$.
Leggett-Garg inequalities (from now on LGI) set constrains on the value accessible to the two-times correlations functions $C_{ij}= \langle Q_{i}Q_{j} \rangle$, obtain measuring it at $t_{i}$ and $t_{j}$. The simplest of them is:
\begin{equation}\label{LGI}
-3 \leq K \leq 1
\end{equation}
\begin{equation}\label{K}
K=C_{12}+C_{23}-C_{31}
\end{equation}
This inequality is the focus of this chapter. Sections 2.1 is dedicated to the two assumptions required to obtain the inequality, a proof of (\ref{LGI}) is given.
In section 2.2 I examine under which conditions a violation of (\ref{LGI}) can be observed, particular

This observation leads to the clumsiness loophole (see section 2.2).

\subsection{Correlation Functions}
These assumptions lead to some properties of correlation functions $C_{ij}$, essential for the proof of (\ref{LGI}). Consider firstly their definition:

Where $Q_{i}$ is the value of a dichotomous observable measured at time $t_{i}$, and the joint probabilities are obtained repeating the experiment.

Postulate 1 provides that Q(t) is a well defined quantity for every time t, even for times different from $t_{i}$ and $t_{j}$. Thus it is possible to define a three-time probability and to obtain $P_{ij}(Q_{i},Q_{j})$ as its marginal, for notation simplicity set $i=1$ and $j=2$.
Where the subscript $12$ specifies at which times the measurements are actually carried out.
Because of what I have just said $P_{12}(Q_{1},Q_{2},Q_{3})$ is well defined and the other joint probabilities can be found as marginals of similar distributions.

Postulate 2 provides that all these three-times probabilities are the same, the choice of (i,j) is irrelevant since the measurements don't influence the

Kofler and Brukner have proven that this is indeed the case for any nontrivial time independent Hamiltonian \cite{kofler2008conditions}, in the next section I report the explicit proof.

\section{LGI violation}
Firstly note that in eq. (\ref{LGI}) just the upper bound is of interest, considering that we are summing three quantities bound in module to unity:

Consider a general quantum system characterized by a set of energy eigenstate ${ \{ \ket{u_{i}}\}}$. The proof is effective for any initial state different from an eigenstate of \textbf{H}, for simplicity consider: This\footnote{Note that the ensemble of eigenvectors don't need to be restricted to these two values.} is assumed to be the result of the first measure (at time $t_{0}$), whom is assigned the eigenvalue +1, for every other state the eigenvalue is set to -1. The dichotomic observable is then