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In Nielsen and Chuang, it's said that to describe a known quantum state precisely takes an infinite amount of classical information since $|\psi\rangle$ takes values in a continuous space (from the 3rd paragraph in chapter 1.37 "Example: quantum teleportation").

I'm confused about how classical information will be used to describe quantum states in the first place. Is it that we find the strings of 0's and 1's that are equal to the numerical value of the amplitudes of our quantum state ($\alpha$ and $\beta$) and these 2 strings of 0's and 1's are what describe this known quantum state?

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    $\begingroup$ Can you reference the precise statement in N&C so that one might look up the relevant discussion? $\endgroup$
    – Dvij D.C.
    Aug 4 at 18:43
  • $\begingroup$ @DvijD.C. added :) $\endgroup$
    – Alexia.
    Aug 4 at 19:20
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    $\begingroup$ related on physics: physics.stackexchange.com/a/381580/58382 $\endgroup$
    – glS
    Aug 9 at 23:45
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Generally speaking, in order to describe elements of a set $A$ using classical information we need two ingredients: a non-empty finite alphabet $\Sigma$ and an encoding $E: A\to\Sigma^\omega$ which injectively maps objects in $A$ to the set $\Sigma^\omega$ of sequences of symbols in $\Sigma$.

Infinite encodings

If the set $A$ is uncountable, as is the case for the set of all pure quantum states, then $E$ necessarily maps most elements in $A$ to infinite sequences of symbols.

This is what Nielsen & Chuang mean when they say that a generic quantum state requires an infinite amount of information to be described exactly. Another example of this phenomenon occurs for the decimal encoding of real numbers where a generic real number requires an infinite number of digits to be specified exactly.

In practice, we generally impose a restriction where $E:A'\to\Sigma^*$ maps a subset $A'\subset A$ to the set $\Sigma^*$ of all finite sequences of symbols in $\Sigma$. If $A$ is separable, then we can choose $A'$ to be a dense countable subset of $A$ which allows us to approximate elements in $A$ using elements in $A'$ arbitrarily well. This is theoretically the case in the first example below.

Even more often, we impose a restriction where $E:A'\to\Sigma^{\le n}$ maps $A'\subset A$ to the set $\Sigma^{\le n}$ of sequences up to some maximum length $n$. This imposes a limit on the precision of the encoding since many interesting sets do not have a finite dense subset. This is the case in the second example below.

Example 1: LaTeX

Let $A$ be the set of pure quantum states and let $\Sigma$ be the set of characters in the ASCII character table. We can encode many elements of $A$ using LaTeX. For example, the state

$$ |\psi\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{i}{\sqrt2}|1\rangle $$

may be represented as

\frac{1}{\sqrt{2}}|0\rangle + \frac{i}{\sqrt2}|1\rangle

which is a string of classical information describing a quantum state. This encoding is very popular on QCSE!

Example 2: Single-precision floating point numbers

As before, let $A$ be the set of pure quantum states, but this time let $\Sigma=\{0, 1\}$. We can encode some elements of $A$ by encoding the real and imaginary parts of each amplitude as little-endian single-precision floating point numbers in IEEE 754 and concatenating the results. For example, the state $|\psi\rangle$ defined above is approximately encoded as

11110011 00000100 00110101 00111111
00000000 00000000 00000000 00000000
00000000 00000000 00000000 00000000
11110011 00000100 00110101 00111111

where the first eight bytes encode a complex number near $1/\sqrt2$ and the last eight bytes encode a complex number near $i/\sqrt2$.

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    $\begingroup$ This is a nice explanation! $\endgroup$
    – KAJ226
    Aug 4 at 21:29

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