What is a qubit? 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 as you might guess correspond to the states 0 and 1 for a classical bit. Notation like ‘|>’ is called the Dirac notation, and it’s the standard notation for states in quantum mechanics.
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, often called superpositions: |ψ> = α|0> + β|1> with the numbers α and β being complex 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.
Another difference is that the value of the bit can be determined at any time: computers do this all the time when they retrieve the contents of their memory, to check whether it is in state 0 or 1. On the contrary, 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, with |α|2 +|β|2 = 1, since the probabilities must sum to one.