1.4The question you cannot ask

Peres sets his readers a design problem and tells them not to give up on it easily. Someone hands you a single photon. It is linearly polarised, but you are not told along which direction. Build an instrument that finds out. Before reading any further, try it.

Why it cannot be done

However you played, your average error will have come out somewhere around 27° or worse. Nobody does better, and the reason is not a lack of ingenuity.

The analyser has two exits, so one photon gives one of two answers: a single yes or no. An angle is a number that can take any value from 0° to 180°. One yes or no cannot carry that much.

Then test the photon again, you might say, at a different angle. But recall the end of lesson 1.2: a photon that comes out of an analyser is polarised along that analyser’s axis, whatever it was before. The first test wiped out the thing you were trying to find. A second test would only tell you about the first analyser.

So the question “what is the polarisation of this photon?” has no experiment that answers it. Peres’s position, which this course takes too, is that a question no experiment can answer is not a deep mystery about the photon. It is a badly formed question. The well-formed one is different:

Is this photon polarised along such-and-such a direction?

That question names the direction first and asks for a yes or a no. An analyser set to that direction answers it every time.

Overlapping answers

The difference between the two questions is easier to see in a picture that has nothing to do with physics. Scatter squares over a sheet of paper so that they overlap, and mark a point. “Which square is the point in?” has no answer: it may be in several. “Is the point in this square?” always has one.

Polarisations overlap in just this way. A photon prepared at 30° will pass a test for polarisation along 0° three times in four, because $\cos^2 30^\circ = 0.75$. Passing the test for 0° does not show that it was prepared at 0°. Almost any preparation could have passed.

Here is the same point as a challenge. I prepare a long series of photons, each at an angle of my choosing, and keep a list. I send them to you without the list. There is no instrument with which you could sort them into bins, 0° to 10°, 10° to 20° and so on, in a way that matches my records.

Many copies are a different matter

Yet the polarisation of a beam of light is measured every day, as precisely as anyone likes. There is no contradiction. A beam is an enormous number of photons all prepared the same way, and each one can be given its own test.

Between one photon and a whole beam there is a sliding scale, and it is worth seeing how knowledge builds up along it.

The curve is drawn by reasoning backwards. If the hidden angle were $\theta$ and the analyser is at 0°, each photon goes “+” with probability $\cos^2\theta$. The chance of getting some particular run with $n_+$ pluses and $n_-$ minuses is then proportional to

Read that as a function of $\theta$ with the counts fixed, and it tells you how well each candidate angle accounts for what was seen. Angles that make the observed run likely are favoured; angles that make it nearly impossible are ruled out. Peres’s own example is seven photons, four up and three down, which gives the lumpy curve $\cos^8\theta\,\sin^6\theta$.

Two things to notice in the figure.

• With one analyser setting there are two peaks. A photon at 30° and one at 150° pass a 0° analyser equally often, so no number of photons at that setting will separate them. A second setting does it at once. One question, however often repeated, is not enough; you need different questions.
• The peak narrows slowly. Its width shrinks as one over the square root of the number of photons: a hundred times the photons for ten times the precision.

Working out the odds of a result from a known preparation is called prediction. Working back from results to the preparation, as here, is retrodiction. Laboratories do far more of the second.

Preparations and tests

It is time to fix some vocabulary, because ordinary words like “measure” have just let us down. Watch a physicist at the bench and you see two kinds of activity.

A preparation is a procedure that is completely specified, like a recipe: this lamp, this polariser at this angle, this pinhole. Anyone can follow it and produce the same thing.

A test begins like a preparation, but ends with something the experimenter did not know in advance: which detector fired. That piece of news is the outcome.

These two are the starting-points of the theory. They are not defined in terms of anything simpler, in the way that Euclid’s geometry does not define points and lines but only says how they relate. We are free to choose which preparation to make and which test to perform. We are not free to choose the outcome.

With that, the whole scope of quantum theory can be stated in a sentence.

What quantum theory is. A set of rules for calculating the probabilities of the outcomes of tests that follow specified preparations.

Nothing in that sentence mentions what a photon is, or where it is between the lamp and the detector. It speaks only of what was done and what was then seen. Most working physicists would describe their subject more generously, as the theory of atoms and nuclei and solids. But when the meaning of a calculation is in doubt, this is the description to fall back on, and the rest of the course is built on it.

Two ways of describing

One feature of this set-up deserves a flag, because it will come up again. The lamp, the crystal and the counters are described in everyday terms: where they are, how they are set, which one clicked. They have to be, since an experiment is something we must be able to report to each other in plain language. The photon is described differently, by an abstract mathematical object that we shall build in Part III.

This does not mean that large things and small things obey different laws. It means that the same world is being described in two ways, and that somewhere between photon and printout the description changes. Where exactly to draw that line, and why the result does not depend on where it is drawn, is the subject of the last chapter of the book.

The same caution applies here as in geometry. A drawn line is not Euclid’s line; it only comes close. The lamps and crystals in a real laboratory come close to the ideal preparations and tests of the theory, and no closer.