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)

 

[Entropia]

I find the concept of quantum computation and the potential of a single qubit to represent an infinite number of states truly fascinating. The Bloch sphere representation of a qubit state is a powerful tool for understanding the behavior of quantum systems, and it is indeed true that a single qubit can have an infinite number of points on the Bloch sphere.

The equation provided for representing a qubit state, Ipsi> = cos(theta/2)I0> + exp(i(phi))sin(theta/2)I1>, is known as the Bloch sphere representation of a qubit state. Here, theta and phi are the angles that determine the state of the qubit. The infinite binary expansion of theta refers to the fact that theta can take on an infinite number of values, as it is a continuous variable. This means that the qubit state can also take on an infinite number of values, making it capable of representing an infinite amount of information.

However, it is important to note that a qubit cannot represent all of these states at the same time. The qubit state can only be in one of the infinite number of states at a given time. This is due to the principle of superposition in quantum mechanics, which states that a quantum system can exist in multiple states simultaneously, but when measured, it will collapse into a single state. Therefore, while a single qubit can represent an infinite number of states, it can only be in one state at a time.

In terms of the infinite binary expansion of theta, this refers to the fact that theta can be represented in binary form, with an infinite number of digits. This means that a single qubit can store an infinite amount of information, as each digit in the binary expansion corresponds to a different state of the qubit. This is a fundamental characteristic of quantum systems and is what makes them so powerful for computation.

In conclusion, the concept of a single qubit having an infinite number of possibilities is a fundamental aspect of quantum computation. While it may seem counterintuitive, the principles of quantum mechanics allow for this infinite potential, making quantum computation a truly remarkable field of study.