In chapter 1 of Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang, I came across this paragraph on quantum teleportation,

Intuitively, things look pretty bad for Alice. She doesn’t know the state $\lvert\psi\rangle$ of the qubit she has to send to Bob, and the laws of quantum mechanics prevent her from determining the state when she only has a single copy of $\lvert\psi\rangle$ in her possession. What’s worse, even if she did know the state $\lvert\psi\rangle$, describing it precisely takes an infinite amount of classical information since $\lvert\psi\rangle$ takes values in a continuous space. So even if she did know $\lvert\psi\rangle$, it would take forever for Alice to describe the state to Bob.

So Alice and Bob share a qubit each from an EPR pair created long ago and now Alice wishes to teleport the state $\lvert\psi\rangle$ to Bob by only sending classical information.

I do not understand why describing $\lvert\psi\rangle$ takes an infinite amount of classical information, since to my knowledge, only the amplitudes of the basis vectors need to be known($\lvert\psi\rangle=\alpha \lvert 0\rangle+\beta \lvert 1\rangle$). Maybe I did not understand properly what it means for Alice to know a state $\lvert\psi\rangle$. Any guidance would be helpful. Thank you.

PS: I'm not from a Quantum mechanics background.

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    $\begingroup$ related on physics: physics.stackexchange.com/q/382655/58382 $\endgroup$ – glS Oct 25 at 3:56
  • $\begingroup$ Will check the link out thank you. $\endgroup$ – Jamāl Oct 25 at 6:44
  • $\begingroup$ @glS Yes, it cleared up my confusion thank you! $\endgroup$ – Jamāl Oct 27 at 9:52

Amplitudes $\alpha$ and $\beta$ in the description of Alice's state are complex numbers; to describe them precisely, you're going to need infinite number of bits of information. If you're only using a finite number of bits, you get an approximation of the state, which can be a very good one but still not an exact representation.

(Why? There are uncountably many real numbers with absolute value less than 1, but only countably many finite strings of 1's and 0's that you could use to try and represent them.)

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  • $\begingroup$ I'm not sure if i understand. Why does alice need to send binary information? For example, to send $(\lvert 0\rangle+\lvert 1\rangle)/\sqrt{2})$ cant she just send the coefficients by writing on a piece of paper? And I'm not too sure if infinite bits will be required in all cases. $\endgroup$ – Jamāl Oct 24 at 21:22
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    $\begingroup$ That's the setup of the problem - she only has a channel of classical information that can send 0 and 1. Of course, if she could just walk to Bob, she could just give him the qubit, but that's not allowed. And what if the coefficients are not that nicely expressed? There are numbers that you can't write out as concisely as your example. $\endgroup$ – Mariia Mykhailova Oct 25 at 1:07
  • $\begingroup$ Oh i see, she can only send classical bits and the scheme should work for any coefficient. Thank you. $\endgroup$ – Jamāl Oct 25 at 6:40

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