In this Quantum Computing article by Michael Nielsen he argues about some of the limitations imposed by quantum measurement. In particular how the amplitude $\alpha$ of a single qubit $\alpha |0> + \beta |1>$ cannot be measured.
One reason this is important is because it means you can’t store an infinite amount of classical information in a qubit. After all, $\alpha$ is a complex number, and you could imagine storing lots of classical bits in the binary expansion of the real component of $\alpha$. If there was some experimental way you could measure the value of $\alpha$ exactly, then you could extract that classical information. But without a way of measuring $\alpha$ that’s not possible.
I am somewhat confused regarding the point being made. I assume the idea is that a classical bit can only transmit 1 bit of information, and the measurement of a qbit (a single physical system) can only result in one bit of information after the measurement (measurement outcome of "0" or "1" in the computational basis).
However, I don't see anything quantum about it. Could you not send an infinite amount of information in a single classical physical system? For example: Send light with frequency
f such that
the binary expansion of f has an infinite amout of bits. So for example, classically, you might send light with f=3.14159265359... Hz having infinitely many digits. Or a small piece of metal measuring d=2.1828.... m cut with infinite precision.
Is the point that the quantum nature of matter limits the amount of information vs. the classical nature of matter which imposes no limits?