A qubit or quantum bit is the basic unit of quantum information.
Just as a classical bit has a state – either 0 or 1 – a qubit also has a state. Two possible states for a qubit are the states |0⟩ and |1⟩, which correspond to the states 0 and 1 for a classical bit. Notation like ‘| ⟩’ is called the Dirac notation. The difference between bits and qubits is that a qubit can be in a state other than |0⟩ or |1⟩. It is also possible to form linear combinations of states |ψ⟩ = α |0⟩ + β |1⟩. The numbers α and β are complex numbers, although for many purposes not much is lost by thinking of them as real numbers. Put another way, the state of a qubit is a vector in a two-dimensional complex vector space. The special states |0⟩ and |1⟩ are known as computational basis states, and form an orthonormal basis for this vector space.
We can examine a bit to determine whether it is in the state 0 or 1. For example, computers do this all the time when they retrieve the contents of their memory. Rather remarkably, we cannot examine a qubit to determine its quantum state, that is, the values of α and β. Instead, quantum mechanics tells us that we can only acquire much more restricted information about the quantum state. When we measure a qubit we get either the result 0, with probability |α|^2 , or the result 1, with probability |β|^2 . Naturally, |α|^2 + |β|^2 = 1, since the probabilities must sum to one. Thus, in general a qubit’s state is a unit vector in a two-dimensional complex vector space.