Qubit (or quantum bit) is the fundamental unit of quantum information — the quantum analog of a classical bit. Unlike a classical bit, which is strictly 0 or 1, a qubit can exist in a superposition of both basis states simultaneously, with complex-valued probability amplitudes determining the likelihood of each outcome upon measurement.

A classical bit can be 0 or 1. There are no probabilities involved at all, but we can assign them anyway. If a bit is 1, then it's 1 with 100% probability and 0 with 0% probability. Conversely, if a bit is 0, then it's 1 with 0% probability and 0 with 100% probability.

A qubit works in a similar way. It also has two basis states: $\lvert 0\rangle$ and $\lvert 1\rangle$, except the total probability is distributed between them. A qubit $\lvert\psi\rangle$ is written in the following form, using Dirac notation:

$$\lvert\psi\rangle = a\lvert 0\rangle + b\lvert 1\rangle$$

The coefficients $a$ and $b$ are not percentages — they are complex numbers called probability amplitudes. For example, a qubit might have amplitude $a = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}$ for state $\lvert 0\rangle$ and amplitude $b = \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}$ for state $\lvert 1\rangle$, all at the same time. The probability amplitudes evolve over time according to the Schrödinger equation; for example, $a$ and $b$ can rotate or exchange magnitudes over time.

A qubit state continues to evolve as this object of two complex probability amplitudes — until it is measured. When a qubit is measured, it collapses to either $\lvert 0\rangle$ or $\lvert 1\rangle$ with 100% probability. At that point, there are no longer any probability amplitudes involved. The qubit behaves exactly like a classical bit and we might as well refer to $\lvert 0\rangle$ as 0 and $\lvert 1\rangle$ as 1.

But how do we know whether a qubit is going to collapse to 0 or 1 if $a$ and $b$ are complex numbers rather than percentages? The answer is given by the Born rule. The Born rule states that the real probability of a measurement yielding a certain result equals the modulus squared of the probability amplitude associated with that result.

$$P_0 = |a|^2 \qquad P_1 = |b|^2$$

We do, however, require the qubit always exists as at least something with 100% probability when we measure it — a qubit can't half-exist or exist with 150% probability. Therefore the squared moduli must add up to 1. This is also called the second axiom of probability:

$$|a|^2 + |b|^2 = 1$$

The state of a qubit can be visualised geometrically as a point on the surface of the Bloch sphere, where the north pole is $\lvert 0\rangle$, the south pole is $\lvert 1\rangle$, and all other points correspond to superposition states.