Quantum bit explained in a technical manner with animation
Single qubit
The general expression of a single qubit in computational basis is given by
where and it must follow that the expression is normalised, (i.e. ). Computational basis refers to the fact that and were used to described the qubit state . In literature, people often instead convey the message in the following manner: “The qubit state is written using the following basis ”.
Now, you may be wondering, “what on earth is or ?”. Well, they are referred to what is known as ”ket 0 / ket 1” in Dirac notation and they are basically shorthand for vectors or column matrices. Below, we write the relationship explicitly.
It may seem that we need 4 degrees of freedom (which are real numbers in this case) to describe the qubit state since each of the 2 complex constants are described using two real numbers. However, by using Hopf coordinates, we can rewrite the expression in eqn () like so
where we have omitted the global phase factor because it is irrelevant in the sense that it has no physically observable meaning. As we can see from eqn (), the number of degrees of freedom needed to describe the qubit state has reduced from 4 to 2.
From eqn (), we see that the possible states that our single qubit can take on is a continuum, as opposed to a classical bit, which can take on only state or . It is this property which gives qubits advantage over the classical bits.
Besides the computational basis, there are also the -basis () and the -basis (), where each of these can be expressed in terms of and as shown below
Animation
The computational basis, also known as the -basis, together with the - and -basis can be plotted as along the , , and axis in the Bloch sphere representation, respectively. An animation showing how the qubit state can be represented as a Bloch vector along the Bloch sphere (which is a unit-sphere) in 3D space can be seen below.
