How can infinite information be theoretically encoded or stored in a single qubit?

Question

I've just gotten started with Nielsen and Chuang's text, and I'm a little stuck. They mention that theoretically, it would be possible to store an infinite amount of information in the state of a single qubit. I'm not sure I completely comprehend this.

Here's how I rationalized it: You take all the information you want to store, put it in binary form, and make it the real component of $\alpha $ or $\beta$ (the coefficients of the computational basis states).

Now I'm not sure if I've understood it right, but since it's still fuzzy in my head it would be great to get some kind of ELI5 explanation or possibly a more detailed picture of how this would, even theoretically, be possible.

Apologies if the question doesn't meet standards. I'm new to the forum and would be open to feedback regarding asking questions or answering them.

Answer 1

I'm not sure what passage in Nielsen and Chuang you have in mind and I see all of this differently. I don't see any need to believe that it is "theoretically" possible store an infinite amount of information in a qubit. My answer to the paradox is that amplitudes aren't stored information. A qubit does not know its amplitudes any more than a randomized bit knows the chance that it is a 1. If the bit has 0.637 chance of being 1, that does not mean that 0.637 has been stored anywhere. The size of the bit's brain is exactly one bit; it can only tell you 0 or 1 if you ask it the one question that it can answer. Now a qubit can answer any one out of a continuous family of binary questions; but it can still only answer one such question in the sense that that completely determines its posterior state for future questions. A qubit is still too small to give a clean answer to any question with more than two answers, and it certainly does not have room to store decimal expansions of numbers.

To reiterate, quantum amplitudes are similar to classical probabilities. They are statistical features that are not directly stored by the systems that obey the statistics.

The Holevo-Nayak theorem says that n qubits cannot store any more than n classical bits. That's the real answer to the question of how a qubit can encode or store infinite information, "theoretically" or otherwise. Answer: It can't.

Answer 2

Highly Relevant: (Physics SE) Informational capacity of qubits and photons

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Here's how I rationalized it: You take all the information you want to store, put it in binary form, and make it the real component of $\alpha $ or $\beta$ (the coefficients of the computational basis states).

Yes, and then if you could then prepare a qubit precisely in the $\alpha|0\rangle + \beta|1\rangle$, in some sense you would be storing infinite information in a single qubit.

Though, the drawback is, firstly it's not possible to prepare quantum states that precisely in practice due to noise and other engineering limitations. Secondly, even if you managed to do that, you wouldn't be able to recover that information by measuring the qubit, as qubits immediately collapse to one of their basis states ($|0\rangle$ and $|1\rangle$ being the standard basis states).

The "encoding infinite information" idea is funny because it's certainly possible to claim that hypothetically if you can produce radio waves with frequency an integer multiple of $\pi$ or any other non-recurring, non-terminating irrational number for that matter that has unlimited decimal places, you are storing infinite information in that radio wave. It doesn't mean that the information is useful or practically retrievable!

Answer 3

Here is another way to think about it. You can, in principle, store an infinite amount of information into a qubit, in the sense that you might need arbitrarily many bits to exactly pinpoint its state.

However, this is not as weird or surprising as one could think. You can make the same argument about a (classical) probability distributions. Given any amount of information, I can always find a way to encode it into a probability distribution over a bit. For example, given $N$ bits of classical information in the form of a bitsring $\equiv(x_1,...,x_N)$, just define $x$ as the number having that bitstring as binary decomposition, and then use a probability distribution with $p_0=x 2^{-M}$ for a big enough $M$.

About the matter of retrieving the information "stored" this way, you find in both classical and quantum case that there is no way to do that with a single measurement. In other words, the more information you want to retrieve from a probability distribution, the more you need to sample from it. Holevo's theorem essentially tells you that quantum mechanics doesn't give any advantages over the classical case in this task.