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You can only extract one bit of information from a single qubit, but in some applications the ability to store a large amount of information and only later decide which portion is needed would still be useful.

If I have a large bitstring, can I store the entire string in the state of a single qubit (or small O(1)-sized collection of qubits) and still be able to retrieve any single bit?

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  • $\begingroup$ a qubit has only two states, but a multi-qubit system could store more than one bitstring. $\endgroup$
    – ZeroTheHero
    Oct 28, 2017 at 2:30
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    $\begingroup$ Relevant terms: holevo bound and quantum advice. The short answer is that, for most intents and purposes, when it comes to storing retrievable information, qubits aren't better than bits. $\endgroup$
    – Craig Gidney
    Oct 28, 2017 at 20:20
  • $\begingroup$ Not exactly the same, but perhaps relevant: You can transfer two bits by sending one qubit and already sharing a pair of entangled qubits <en.wikipedia.org/wiki/Superdense_coding>. $\endgroup$
    – Sebastian Riese
    May 27, 2019 at 15:48

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You have plenty of ways to "encode" classical data in a quantum systems. Given any data, encoded as a string of bits $\sigma$, you can:

  • You can take the same $n$-length string, and create the state $$ \otimes_{i=1}^n |\sigma_i\rangle $$.

Obviously, you can recover your data with $n$ different measurements. There are cases where this is what you want, but it's not always the case.

  • As is often done in quantum algorithmics, you can create the state $$ \sum_{i=1}^n \sigma_i |i\rangle $$ That uses just $\lceil log(n) \rceil$ qubits in order to have $2^n$ basis. There are plenty of situation where you can use a thing like this.

  • You could also encode your data as a relative phase of $\sigma$ with $n$ bits of precision on a single qubit. (not much useful..) There is no chance to recover the data with just one state.

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  • $\begingroup$ Could you please provide a more concrete example for the second case (for n=2 for example)? How would you go about extracting the information? How many measurements would you need? It seems to me that you can then also only extract one bit of information from it (one $\sigma_i$). $\endgroup$
    – KIAaze
    Mar 26 at 16:29

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