So I want to conclude this chapter on measurements by telling you something about some foundational

aspects of measurements.

And the thing we will start with is called the Leggett-Garg inequality, which is a sort

of balance inequality in time.

And so the basic idea you already know, namely in quantum mechanics the results of measurements,

if you do several measurements also depend on what previous measurements you did.

For example, if you have the sequence of measurements in a spin one half, where you first measure

the x component and then at some later time again measure the x component and there is

no time evolution in between and you don't do any other measurement, then maybe the first

result will be random because the initial state might have been different than an eigenstate

of Sx, but once you know the first measurement outcome then you are guaranteed, if your measurement

apparatus is ideal, that the second outcome will be exactly the same.

However if you were to try and measure Sy in between, then of course according to what

we know this would protect an eigenstate of the Sy operator and so it completely scrambles

the state and the next Sx measurement can then again give one of two outcomes randomly.

So everything somehow depends on the order of how you do the measurements.

And this can be contrasted to the classical idea of a physical system where we would think

that every observable like position or momentum or any function of position and momentum has

some definite time evolution, whether you look at it or you don't look at it and whenever

you choose to do a measurement then at least in principle ideally there is a way of doing

the measurement such that you almost don't perturb the system and so then the subsequent

evolution will not have been changed.

So you just can look at different points in time what your system is doing.

So the story I have just been telling about the classical idea of observing a system can

be summarized into assumptions.

The first one would be if you take any observable like position or momentum or anything else

then at each point in time it will have some definite value even if you don't look at it.

And for all purposes let's call this observable Q.

And the second assumption would be that at least ideally there are measurements where

you can measure Q at an arbitrary point in time without changing its subsequent evolution.

And of course in quantum mechanics we know this in general not true.

Now the idea of Legate and Garg was to derive inequalities a la bell in a sense that you

would take an arbitrary system and some cleverly chosen observable you would try to measure

it and then from the measurement outcomes you would show that they are incompatible

with these assumptions.

So you take the point of view I don't know whether quantum mechanics holds or not but

I want to test whether my particular physical system can be described by assuming these

two assumptions or not.

And the particular observables they want to look at for reasons of simplicity are those

that only have two values plus or minus one like the two values of a spin but you could

also imagine you measure the position of a particle and then if the coordinate is positive

you assign it the value plus one and if it's negative you assign the value minus one to

this capital Q.

So there's always a way of assigning this.

So now we will just consider capital Q that can just have two values plus minus one.

And more specifically we are interested in a measurement sequence that invokes measurements

at certain points in time like t1, t2, t3 and I will call the results Qj.

So Qj is just plus or minus one and it has to be observed at time tj.

So if these assumptions are true then you can just apply the rules of classical statistics

say to correlators of Q1 and Q2 and so on.