Basic

Superposition means a linear combination of (two or many) configurations with their coefficients as complex number, such as $${3 \over 5}i{\mid }{0 }\rangle +{4 \over 5}{\mid }{1 }\rangle$$

A qubit, a quantum version of two binary bits (0 and 1), also two configurations, has two basis states: $|0\rangle$ and $| 1 \rangle$. So a qubit is a linear combination $$| \phi \rangle = c_1 |0\rangle + c_2 |1\rangle$$ where $c_1$ and $c_2$ are both complex numbers and mean the amplitudes attached to the possibilities of two configurations. Amplitudes are also called coefficients.

Interpretation of coefficient as probability

The probability for a specified configuration is given by the square of the absolute value of the coefficient.

If $c$ is a complex number, its absolute value denotes its distance from the $c$ point in the complex plane to the origin.

$$|z|={\sqrt {\operatorname {Re} (z)^{2}+\operatorname {Im} (z)^{2}}}={\sqrt {x^{2}+y^{2}}}$$

and so its square (is the probability of a specific configuration) $$(|c|)^2 = ({\sqrt {x^{2}+y^{2}}})^2=x^2 + y^2$$.

For $| \psi \rangle = c_1 |0\rangle + c_2 |1\rangle$, its coefficients also denote the weighs in a probability distribution, and $$|c_1|^2 + |c_2|^2 = 1$$

Examples

A qubit: $$|\psi \rangle ={3 \over 5}i{\mid }{0 }\rangle +{4 \over 5}{\mid }{1 }\rangle$$ where the probability of ${\mid }{0 }\rangle$ is $\left({3 \over 5}\right)^2$, and the probability of ${\mid }{1 }\rangle$ is $\left({4 \over 5}\right)^2$.

And also $\left({3 \over 5}\right)^2 + \left({4 \over 5}\right)^2 = 1$

Evolution

The fundamental law of quantum mechanics is that the evolution is linear, that is, if a configuration $A$ evolves to $A’$: $A \to A’$, and another $B \to B’$, then its superposition $\psi \to \psi'$

provided $$\psi = c_1 {\mid }{A }\rangle + c_2 {\mid }{B }\rangle$$ then $\psi’$ has the same coefficients as $\psi$. $$\psi’ = c_1 {\mid }{A’ }\rangle + c_2 {\mid }{B’ }\rangle$$