What is superposition, really?

Light is a wave, and the wave wobbles in a direction across its path: up and down, side to side, or anything in between. That direction is its polarisation.

A polarising filter is a sheet with a built-in direction. Light wobbling along that direction passes. The filter blocks light wobbling across it. Ordinary light from the Sun or a bulb is a jumble of every direction, so a filter lets half of it through.

You use this every day. Glare off water and roads is mostly polarised side to side, so polarised sunglasses block that direction and pass the rest. Many phone screens use polarising layers too.

Now turn the light right down, until it arrives one photon at a time. Each photon is one particle of light, and it can't half pass a filter. It either passes or it doesn't.

So what happens when a photon arrives with its wobble tilted at an angle to the filter? The filter passes it some of the time and blocks it the rest. Which one happens to a given photon is random. How often each happens isn't random at all: the angle fixes it.

The pattern has a name. The share that passes equals the square of the cosine of the angle between the photon and the filter. Étienne-Louis Malus worked this out in 1809, using sunlight polarised by reflection, nearly a century before anyone knew light came in photons. For single photons it becomes the odds of passing.

Take a photon polarised at 45°, halfway between vertical and horizontal. Physicists say it's in a superposition of vertical and horizontal. That phrase gets misread more than almost anything in science.

Think of a compass. A north-east arrow isn't pointing north and east at the same time. It points one definite way: north-east. You can describe it as "some north plus some east", and that's useful, but the arrow itself is a single direction.

A 45° photon is the same. Its state is one definite direction. Send it to a 45° filter and it passes every single time: there's nothing left to settle. Ask a different question, "vertical or horizontal?", and the answer isn't settled until you measure. Nobody can predict it in advance, only the odds: 50 50.

So a superposition isn't a double life. It's one definite state, and the answer isn't settled until you measure. Whether there's anything left to settle depends on which question you ask.

Line the filter up with the state and nothing is random. Tilt it and each photon's result is random, settled only when it meets the filter, while the odds stay exact. Run a million photons at 45° and you get very close to half a million through, every time, in every lab in the world.

That's the honest summary of quantum randomness: individual results are random, and the statistics are exact. Why individual results are random, and whether something deeper decides them, is one of the big open questions. Chapter 6 lays out the main answers.

Here's what's going on. Each filter asks a question, and every photon that passes comes out lined up with that filter. Its old direction is gone.

With just two filters at right angles, every photon leaving the first filter is vertical, and a vertical photon never passes a horizontal filter.

Put a diagonal filter in the middle, and the photons that pass it come out diagonal. A diagonal photon has even odds at the last filter. So some get through. The middle filter doesn't add light. It changes which question the last filter is really answering.

This is the first big lesson of quantum measurement: asking a question changes the state to match the answer. Chapter 3 looks at what that means.

Go back to the arrow. Any direction can be split into two parts along the question's two answers, like north-east splitting into "north" and "east". Physicists call those parts amplitudes.

The rule for the odds is simple: square each part. For a 45° photon each part is about 0.71, and 0.71 squared is 0.5. Even odds. For a photon about 33° from vertical the parts are about 0.84 and 0.55. Squared, that's 70% and 30%.

Max Born set out this rule in 1926 and won the 1954 Nobel Prize in Physics for it. It has passed every test since. Every quantum technology, from a quantum computer's final answer to the security of quantum keys, rests on it.

Amplitudes can be negative as well as positive, and they can cancel each other out. Squaring turns them into odds that are never negative and always add up to one. That ability to cancel is the source of interference, and interference is what makes quantum computers useful. Chapter 4 shows it in action.

Electrons, protons and many atoms behave like tiny magnets. Physicists call this spin. Measure an electron's spin along any direction and you only ever get one of two answers: "up" or "down" along that direction. Nothing in between.

In 1922 Otto Stern and Walther Gerlach fired silver atoms through a shaped magnet in Frankfurt. They expected a smear. They got two separate spots. That was the first clear sight of a quantum two-answer question. Optional pages 2b and 2c tell the story, including a cheap cigar.

Spin follows the same pattern as polarisation: a definite state, a question, and odds set by the angle between them. The angles work slightly differently (for spin, "up" and "down" point in opposite directions rather than at right angles), but the logic is identical.

Any system with a two-answer question can carry a qubit: a photon's polarisation, an electron's spin, two energy levels of an atom. Chapter 7 builds on exactly this. MRI scanners already use the spin of protons in your body to make images. Optional page 2a shows how.