Up till now we have considered a two state quantum system,
specifically a spin-1/2 particle. However a quantum system is by no
means constrained to be a two state system. Much of the above
discussion for a 2 state quantum system is applicable to a general *n*
state quantum system.

In an *n* state system the Hilbert Space has *n* perpendicular axes,
or eigenstates, which represent the possible states that the system
can be measured in. As with the two state system, when we measure the
*n* state quantum system, we will always find it to be in exactly one
of the *n* states, and not a superposition of the *n* states. The
system is still allowed to exist in any superposition of the *n*
states while it is not being measured.

Mathematically if two state quantum system with coordinate axes
*x*_{0}, *x*_{1} can be fully described by:

| *X* > = *w*_{0}*| *x*_{0} > + *w*_{1}*| *x*_{1} > = = (*w*_{0}, *w*_{1})

Then an *n* state quantum system with coordinate axes
*x*_{0}, *x*_{1},..., *x*_{n-1} can be fully described by:

| *X* > = *w*_{k}*| *x*_{k} >

In general a quantum system with *n* base states can be represented by
the *n* complex numbers *w*_{0} to *w*_{n-1}. When this is done the
state may be written as:

| *X* > =

Where it is understood that *w*_{k} refers to the complex weighting
factor for the *k*'th eigenstate.

Using this information we can construct a quantum memory register out
of the qubits described in the previous section. Note that in general
a quantum register composed of *n* qubits requires 2^{n} complex
numbers to completely describe its state. A *n* qubit register can be
measured to be in one of 2^{n} states, and each state requires one
complex number to represent the projection of that total state onto
that state. In contrast a classical register composed of *n* bits
requires only *n* integers to fully describe its state.

This means that one can store an exponential amount of information in a quantum register relative to the number of qubits it contains. Here we see some of the first hints that a quantum computer can potentially be exponentially more efficient than a classical computer in some respects.