What Is a Qubit?

A classical bit is a switch. It is up or it is down, 1 or 0, and there is nothing else to say about it. Every computer you have ever used is built out of billions of these switches, and the entire discipline of classical computing is the art of arranging them so that flipping them in the right order does something useful. The bit is a wonderfully austere object: to describe one completely, you need exactly one piece of information, the answer to a single yes-or-no question.

A qubit is the physical system that replaces that switch — a single electron spin, the polarization of one photon, a superconducting loop carrying current in two directions at once. And the first surprise is that describing one completely takes considerably more than a yes-or-no answer. A qubit still has two special states we agree to call |0⟩ and |1⟩, and if you measure it you will always get one of those two answers. But between measurements, the qubit is not obliged to be sitting in either one.

The picture worth carrying around is this: a bit is a switch with two positions, and a qubit is an arrow that can point anywhere on the surface of a sphere. |0⟩ is the north pole. |1⟩ is the south pole. Everything else on that surface — the equator, the mid-latitudes, every point in between — is a legitimate state of the qubit that has no classical counterpart at all. That sphere has a name, the Bloch sphere, and it is the single most useful mental image in the subject.

Note what has just happened to the accounting. To pin down a point on a sphere you need two continuous numbers, a latitude and a longitude. So one qubit, the smallest quantum system there is, already needs two real numbers to describe rather than one bit. This is where the sense that quantum systems are "bigger" than classical ones begins, and it is worth noticing that it starts at n = 1, before entanglement or interference or anything else exotic enters the story.

The standard notation for a qubit state is α|0⟩ + β|1⟩. The angled bracket thing, |0⟩, is called a ket, and it is nothing more exotic than a label — a way of writing "the state we call zero" that is visually distinct from the ordinary number 0. The interesting content is in α and β, the two numbers out front. They are called amplitudes.

Read the expression as a weighted sum, exactly the way you would read a recipe: this state is "α parts |0⟩ mixed with β parts |1⟩". The state 1·|0⟩ + 0·|1⟩ is just |0⟩ — pure north pole, and a measurement will return 0 every single time. The state 0·|0⟩ + 1·|1⟩ is |1⟩. And the state with α = β = 1/√2, which lands you on the equator, is the famous even superposition: measure it and you get 0 half the time and 1 half the time.

The one rule the amplitudes must obey is normalization: |α|² + |β|² = 1. This is not a deep physical postulate so much as bookkeeping — it says the probabilities of all the possible outcomes add to one, which they had better. Geometrically it is exactly the statement that the arrow has length 1, which is what keeps it on the surface of the sphere rather than floating around inside it.

So α and β control the two things that distinguish one qubit state from another: how the weight is split between |0⟩ and |1⟩, and the relative phase between them. The first of those determines the latitude on the Bloch sphere. The second determines the longitude, and it is the part with no classical analogue whatsoever.