The Leggett-Garg correlation
From the above correlations, using definition (25.3) we find
Now the parameters of an experiment are under the control of the experimentalist. Therefore, we shall assume that the parameters a1, b1, a2, and b2 can be chosen to be whatever we wish, subject to the unitarity constraints | a112 +| b1|2 =
| a212 + | b212 = 1, and the limits of what is possible in the laboratory. Assuming there is no practical barrier, consider the reparametrization
where Q1, Q2, ф1 and ф2 are real. Now take ф1 = ф2. Plotting K as Q1 and Q2 range from —n to n each gives the following figure:
Fig. 25.1 Values ofK greater than 1 demonstrate a quantum mechanical violation of the Leggett-Garg inequality.
It is clear that there are values of the parameters where K exceeds the classical limit +1. In fact, the maximum value of K over the region shown is 1.5 This is an entirely non-classical result. How can we explain it?
Understanding the Leggett-Garg prediction
There is a deep issue being demonstrated in the above analysis, to do with the way humans think about reality and time. It is intimately bound up with the motif running throughout this book, viz., the contextuality of truth.
Let us say right at this point we are not going to ‘explain’ the quantum breakdown of the LG inequality. Merely saying that there is some bizarre quantum superposition effect going on explains nothing. What we can do is to say what is wrong with the classical calculation that set up the LG inequality in the first place. It’s not quantum that is wrong, but our preconditioned classical way of thinking about the universe and time.
In fact, we are fortunate that our faulty conditioning works 99.99999 . . . per cent of the time. Indeed, it took very sophisticated technology for quantum phenomena to be detected: they tend to hide themselves very successfully.
So how about the present scenario. Can we see where our classical thinking went wrong?
The answer is given by John Archibold Wheeler’s philosophy of observation: if you haven’t actually done it, don’t assume anything about it.
Consider the LG correlation K. It is calculated from three separate correlations. Now classically, these correlations can be regarded as coming from the same experiment. First, a bit evolves from time t0 to time tj. Snap!—we observe its state at that time, compare it with its initial state, and hence work out the temporal correlation C01, cost free. Then, without further ado, we let it evolve to time t2 where again, snap! we observe its final state and work out not just C12 but also C02. We have, after all, obtained all the information required to do those calculations.
But, according to Wheeler, the correlation C02 does not involve any observation at time tj. The evolution operator U20 is applied on the strict understanding that no attempt is made to extract information between times t0 and t2. And that is the essential point. Classically, we have been conditioned to think it must be the case that the ‘observed’ state of the SUO at time tj should play a role in C02. But how can it? We did not observe it when the system ran from t0 to t2.
Essentially, there are three separate ‘sub-experiments’ involved in the determination of K; evolution from t0 to tj, evolution from tj to t2, and evolution to t0 to t2. Classically, we can believe there is just one experiment. Quantum mechanically, we have to recognize context and factor it in.
What appears to be going on with the C02 correlation is that our lack of information about the ‘actual’ classical state of the SUO at time tj in the C02 subexperiment allows some sort of quantum interference between the two possible classical states up and down to occur at that time. This reminds us of the doubleslit experiment, where an interference pattern on a screen is observed provided no attempt is made to determine through which slit the particle had gone.
The above is not an explanation, because we cannot account for the mysterious rules of quantum mechanics, such as the Born outcome probability rule, and so much more. But perhaps that is just as well. How could humans ever deceive themselves into thinking they could explain and understand the universe, time, and hence themselves?