How many classical bits are needed to represent a qubit?

Question

I have two question concerning information content of qubit.

Question 1: How many classical bits are needed to represent a qubit:

A qubit can be represented by a vector $q = \begin{pmatrix}\alpha \\\beta \end{pmatrix}, ~~ \alpha, \beta \in \mathbb{C}$. So, we need four real numbers to represent it. But when facts that (i) $|\alpha|^2+|\beta|^2 = 1$ and (ii) two qubits which differ in global phase only are indistinguishable, are taken into account, only two real numbers are necessary (angles on Bloch sphere). Moreover, we can choose a precission of the qubit representation (i.e. number of decimal places in $\alpha$ and $\beta$ or Bloch sphere angles) which influence number of classical bits needed to described the qubit.

So, it seems to me that the qubit representation cannot be used for measuring qubit information content but only memory consumption in simulation. Am I right?

Question 2: What is an (effective) information content of qubit:

A superdense coding enable us to send two classical bits via one qubit. But on the other hand you need two entangled qubits prepared in advance.

Given these facts, what is an information content of qubit? One or two classical bits? Or do I need to use another angle of view given the fact that qubits are "continuous" (i.e. described by complex numbers $\alpha$ and $\beta$)?

Answer 1

There are two types of information in physics:

• Classical information
• Quantum information

Physics doesn't answer the question "What is (classical or quantum) information?". This is philosophic question, and physics never answers questions of this kind. Instead, physics answers another question "How (classical or quantum) information is measured?". The answer is trivial:

• Classical information is measured in bits
• Quantum information is measured in qubits

Then people hear this first time, they think this is just a tautology, but this is not; this is mathematically correct way of thinking about information. We don't use vague terms like "effective information" and don't ask vague questions like "What is an (effective) information content of qubit".

Now we come to the question: How these two types of information are related?

First, how many bits are needed to represent a qubit? We need infinite number of bits to represent a qubit with infinite accuracy; but infinite accuracy is never needed; any practical computational problem requires finite accuracy. You can run quantum algorithms on a classical computer, given the classical computer has enough memory (memory requirements grow exponentially with the number of qubits).

Another question: how many bits can be transmitted by transmitting one qubit?. Here we have 2 fundamental results: Holevo theorem and Superdense coding.

Answer 2

It takes an infinite amount of information to specify the state of a qubit. That's for the reason you said: those two angles to specify the point on the Bloch sphere are continuous, and that means you need an infinite amount of precision.

That being said, only one bit of accessible information is gained by measuring a qubit. If you had enough time and money, and many copies of the same state, you could keep doing measurements on the qubit. With each measurement you would narrow down uncertainty on what the exact state of the qubit is.

As for superdense coding, it does turn out that transferring one qubit from a system of two can transfer both classical bits, but it's important to remember qubits act a bit nonintuitively in larger systems. It's incorrect to say that one qubit is here and the other is there. They together constitute a two-qubit state, which in general is not divisible into two one-qubit states.