2.1States and complete tests

Part I left us with a vocabulary, preparations and tests, and one hard fact: outcomes are random, but their probabilities are fixed by the preparation. This lesson uses those to say precisely what a “quantum state” is, and to pick out the states about which the most, and the least, can be known.

A state is a list of odds

People speak of a photon being “in” a state as though the state were inside it. After the last two lessons that way of talking should make you uneasy. Here is a definition that uses nothing but laboratory procedure.

Take a preparation. Follow it with a test, many times over, and record how often each outcome occurs. Do the same for another test, and another: every test anyone could devise. Each outcome of each test settles down to a definite probability. That complete catalogue of probabilities is the state.

For example, to say that a photon is in the state “right circularly polarised” means this: there is a certain test (a quarter-wave plate followed by a crystal at the right angle) which it is sure to pass, and for any other arrangement of plates and crystals the odds of each exit are whatever classical optics gives for the brightness of a beam of right-circular light.

Three remarks about this definition.

• The state belongs to the preparation. Two different sets of hardware that lead to the same catalogue of odds prepare the same state, and nothing else about them matters.
• The catalogue is hugely repetitive. An arrow is fixed by its shadows on just a few axes; you do not need its shadow on every possible line. In the same way a small number of well-chosen tests fixes all the rest. For polarisation it takes three, as lesson 3.2 will show.
• Probability here means what it has meant all along: the fraction of trials, in the limit of very many repetitions of the same preparation. The imagined endless set of repetitions is called an ensemble. It exists only in our reasoning; in practice we repeat as often as patience allows.

And what is the system that the state describes? Whatever our tests address, and no more. A crystal of calcite asks about a photon’s polarisation and nothing else, so for these experiments the system is the polarisation: where the photon is, or what colour it is, does not come into it. In the Stern–Gerlach experiment the system was the magnetic moment of the atom, not the whole atom.

Tests that can be repeated

Put two identical crystals one after the other, set the same way. The first sends a photon straight on or shifts it sideways. The second does the same again. So there are three places a photon might end up: shifted twice, shifted once, or not at all.

A test that gives the same answer when it is done twice in quick succession is called repeatable. Do not confuse this with predictable: nobody could say which way the photon would go at the first crystal. Repeatable means only that, having gone one way, it will go that way again.

Plenty of real tests are not repeatable. A photographic plate absorbs the photon it detects, so there is nothing left to test again. Repeatable tests are an idealisation, like the frictionless slopes of mechanics, and like those they are what the theory is built on. From here on, “test” means a repeatable one.

A repeatable test has a second use. Block all of its exits but one, and what comes out of the remaining exit is sure to pass the same test again. So a repeatable test with one exit left open is a preparation. That is precisely what a polariser is.

Complete tests

A polarisation test has two outcomes, never more. Whatever combination of plates and crystals you assemble, you cannot sort photons by polarisation into three bins such that a photon from each bin stays in its bin when tested again. The Stern–Gerlach magnet also gave two outcomes for silver. For other atoms it gives three, or four, depending on the atom, but for a given kind of atom the number is always the same, however the magnet is turned.

Call the largest number of different outcomes any test of a system can have $N$. A test that has all $N$ is called complete, or maximal. For polarisation $N = 2$. For an atom of spin $s$ (a number such as $\tfrac12$, $1$, $\tfrac32$ that labels the kind of atom) it is $N = 2s + 1$.

The word “complete” here means something much weaker than it would in classical physics. A complete classical description of a spinning top gives all three components of its rotation at once. A complete quantum test of an atom’s spin gives one component, the one along the magnet, and that is all there is to be had: testing one component rules out testing the others at the same time, and scrambles what they would have given.

So in what sense is such a test complete? In the sense of the following rule, the first of the eleven that Peres letters A to K.

A. Statistical determinism. If a system is prepared so that it is certain to give a particular outcome in some complete test, then the probabilities of all the outcomes of every other test are fixed. They do not depend on any further detail of how the preparation was done.

A system prepared like that is said to be in a pure state. The easy way to make one is the way a polariser does it: perform the complete test and keep only the systems that come out of one exit.

Rule A says that a pure state is the most that can be known. If you know a photon will certainly go up at a horizontal-against-vertical crystal, you know its odds at every other crystal and every plate, and no extra information about the lamp or its history could improve on them. The odds for other tests are generally not certainties. A pure state is not one where everything is known; it is one where nothing more can be.

B. Equivalent tests. Two complete tests are the same test, for all purposes, if every preparation that gives a certain outcome in one also gives a certain outcome in the other. They then agree in their probabilities for every preparation.

This is mostly housekeeping: it says that a test is defined by the pure states it sorts, not by its hardware or by what its exits are called. But notice what it does not say. Two tests that happen to agree for some preparations need not be the same test. Circularly polarised light splits half and half at any crystal, at whatever angle, so all those crystals agree about circular light. They disagree about nearly everything else.

Knowing nothing

At the other extreme from a pure state is light straight from a lamp. Test its polarisation in any way you like, linear at any angle, circular, elliptical: half the photons go each way. Nothing is known about its polarisation, and no test finds anything.

C. Random mixtures. A system with $N$ distinguishable states can be prepared in such a way that every complete test gives each of its $N$ outcomes with probability $1/N$.

Such a preparation is called a random mixture. Two consequences follow from the definition of a state as a catalogue of odds.

There is only one of it. A state is its catalogue, and here every entry is $1/N$; there is no room for two different random mixtures.

And it stays put. Leave a random mixture alone for a while and then test it. “Wait, then test” is itself a complete test, so it too must give $1/N$ for every outcome. A system about which nothing is known does not, left to itself, turn into one about which something is known. Peres calls this the conservation of ignorance.

Disorder is not randomness

It is easy to slide from “random mixture” to “any jumble of photons”. The following game is there to stop you. It is the first exercise of the book’s second chapter.

Box A contains equal numbers of horizontally and vertically polarised photons, exactly equal, in no particular order. Box B contains equal numbers of right- and left-circular photons. Pull one photon from either box and test it any way you like: the odds are a half. One photon cannot tell you which box you have. Can all of them together?

The trick is in the word “exactly”. Test box A for horizontal against vertical and every photon answers according to how it was made: the counts come out exactly equal. Give box B the same test and each of its photons is a coin toss, so the counts are equal only on average and usually miss by a few. An exact tie points to box A; anything else proves box B.

With a million photons of each kind, tossed coins tie only about once in two thousand tries, so you would name the box wrongly less than once in three thousand games.

The boxes can be told apart because neither is a random mixture. Someone counted the photons into them, and that count is knowledge about the preparation. A box in which each photon had been made horizontal or vertical by an honest coin toss would be a different thing, and as lesson 3.4 will show, no test of any kind could tell it from one filled by tossing a coin between right and left.

This lesson follows §2-1 to §2-3. The book adds a careful discussion of why the instruments must be described in ordinary classical terms even though they are made of atoms; we touched on it at the end of lesson 1.4.