# Why does describing a quantum state take an infinite amount of classical information? [duplicate]

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.

• related on physics: physics.stackexchange.com/q/382655/58382 – glS Oct 25 at 3:56
• Will check the link out thank you. – Jamāl Oct 25 at 6:44
• @glS Yes, it cleared up my confusion thank you! – 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.
• 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. – Jamāl Oct 24 at 21:22