How many bits can a qubit hold? I know there are probabilities involved in a qubit, but there must be some limit to how many bits a qubit can take. If different qubits can have different amounts of information then we can think of these as different sizes of processor-registers: $8$-bit, $16$-bit, $32$-bit or $64$-bit, can't we?

But my question is really, is there a limit to how much information content (discreete-wise) a qubit can store?


The answer to your question is "1, but with some qualifications". I start by saying "1" because there is a bound called Holevo bound which says the following.

If you have $N$ classical bits of information, and wish to send them using qubits from $A$ to $B$, in such a way that $B$ can confidently reconstruct what the classical information was, say by writing it down in a book, then you will need $N$ qubits.

So that's the first part of the answer: each qubit only transmits one classical bit, when used for the purpose of classical communication.

The second part is that qubits are nevertheless much more subtle than classical bits and can be used to perform a greater variety of information tasks. There are examples of information processing tasks where we have good reason to think that, to get a given outcome, if $n$ qubits are required then of order $2^n$ classical bits are required. It is not possible to prove this (because of a difficulty in computer science: we don't know how to rule out the possibility of very clever classical algorithms that might be discovered one day), but there is good reason to think that it is a true statement.

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