Quantum Computation: A Single Qubit's Infinite Possibilities

[Amith2006]

A qubit state can be represented geometrically by Bloch sphere. A Bloch sphere has infinite points and so a single qubit can represent infinite number of states. Qubit state is represented by,
Ipsi> = cos(theta/2)I0> + exp(i(phi))sin(theta/2)I1>
It is said in the Book on Quantum Computation by Nielsen & Chuang that u can store an infinite text on a single qubit in the infinite binary expansion of theta. How is that possible? Though a single qubit can be in anyone of the infinite number of states, how can it represent all the states at the same time? I can understand infinite expansion of theta but what do they mean by infinite binary expansion of theta?

 

[nfelddav]

You know the "infinite number of states" are just superpositions of two states, yes?

how can it represent all the states at the same time?

It can't, it is in TWO states at the same time, it is ONE of an infinite number of superpositions.

How is that possible?

Just imagine theta written out in binary. It is an infinitely long binary string. You can store whatever you want on that string, and infinitely long text for example.
(the downside being that there is absolutely no way to recover it)