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:
describing any irrational number in a classical system would take infinite bits too.
Any irrational number can be approximated to a rational number with any arbitrary precision, which would then take finite bits to describe.