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In the Quantum Computation book by Nielsen and Chuang, the authors write in the context of quantum teleportation

"Even if she (Alice) did know the state |w> , describing it precisely takes an infinite amount of classical information since the state |w> takes values in continuum space "

I am not clear, what they mean to say by this statement. My professor told me that the coefficients for the basis state might be irrational, which would require infinite bits to describe classically, and that is what the authors want to convey.

But I am not quite sure that is the correct explanation because of two reasons:

  1. describing any irrational number in a classical system would take infinite bits too.

  2. Any irrational number can be approximated to a rational number with any arbitrary precision, which would then take finite bits to describe.

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  • $\begingroup$ Your point #1 is wrong. Any computable number could be represented by a finite number of bits. If you replace "irrational" with "real" then the statement is correct. Your second point is correct. However, when he says "precisely" in the original quote, I think he means that there can't be any error, so you really would need an infinite number of bits. $\endgroup$ – Brian Moths Dec 8 '15 at 18:25
  • $\begingroup$ I do not get you. Do you mean to say, describing any real number would take infinite bits? (Point 1) $\endgroup$ – Apiastos Dec 8 '15 at 18:30
  • $\begingroup$ I meant to say that describing any non-computable number requires and infinite number of bits. But computable numbers can be represented in a finite number of bits. $\endgroup$ – Brian Moths Dec 8 '15 at 18:33
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    $\begingroup$ out of context the statement does not seem to make sense, the argument can be applied to classical physics too $\endgroup$ – user83548 Dec 8 '15 at 19:02
  • $\begingroup$ @NowIGetToLearnWhatAHeadIs, I don't think this is correct. A computable number is not one which can be represented by a finite number of bits --- it's one which can be bounded to an arbitrary precision by a finite number of bits. No? $\endgroup$ – DilithiumMatrix Dec 8 '15 at 19:23
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Both of your 'reasons' are correct. But your professor's explanation is also how I would understand the textbook's quotation. I think the issue you are having is supposing that the statement:

"Even if she (Alice) did know the state |w> , describing it precisely takes an infinite amount of classical information since the state |w> takes values in continuum space",

for a quantum system is different from a classical (analog) state --- which is not the case. Describing something like an exact voltage, would also require an infinite amount of classical information (e.g. bits). Note also, that a quantum computer (or quantum information system) can be either discrete* or continuous (or a hybrid, apparently).

This answer, which discusses the difference between a bit and qubit, might be useful.

*Of course, a qubit, in general, will still be measured as 0 or 1.

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  • $\begingroup$ Ah, I get the point now. $\endgroup$ – Apiastos Dec 9 '15 at 19:01

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