Quantum systems have infinitely more information than the corresponding classical system. An example of a two-state classical system is a coin which can be either heads-up or heads-down. This corresponds to a single "bit" of information; we could code it as "0" or "1".

The quantum two-state system or "qubit" is usually described by a normalized complex vector with two elements such as the "ket":
$\left(\begin{array}{c} \alpha \\ \beta \end{array}\right)$
where $\alpha$ and $\beta$ are two complex numbers subject to $|\alpha|^2+|\beta|^2 =1$. An alternative is the corresponding "bra", which has the same information content:
$\left(\begin{array}{cc} \alpha^* & \beta^* \end{array}\right)$
A less common representation of a qubit is the (pure) density matrix form:
$\left(\begin{array}{cc} |\alpha|^2 & \alpha\beta^* \\ \alpha^*\beta & |\beta|^2 \end{array} \right)$,
in which the extraneous complex phase information has been removed. From this we see that there are only two real informational degrees of freedom in a qubit, for example, $|\alpha|^2$ and the complex phase of $\alpha\beta^*$. Another way of representing these two degrees of freedom is with the Bloch sphere.

Probably the most popular textbook for quantum information is Nielsen and Chuang's Quantum Computation and Quantum Information.

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