Question

The title says it all.

My understanding is that a qubit is a superposition of $|0\rangle$ and $|1\rangle$ , i.e. the answer to a binary question. So I imagine that specifying an event in spacetime would require an infinite sequence of qubits. I'm thinking of a binary search tree. Is there some other way to do it with a finite amount of qubits?

Answer 1

You'd never need a infinite sequence as of course there's a limit to the accuracy of the a position measurement. Given either by Planck's length (but no measurement gets near Planck length, because no imaging beam get near Plank's energy), in the non-Planckian regime, you can measure a position by scattering a beam of light energy E= hv, and get a position accurate to.

$\Delta x = \frac{\hbar}{2p} = \frac{\hbar c}{2E}$

If you're starting in a box of volume V you already know the particle is in, then you need,

$B = \log_2 \left ( \frac{V}{\Delta x^3}\right) = 3\log_2 \left ( \frac{ VE^3 }{\hbar^3c^3} \right)$

Bits of information to describe where you measured the particle to be.

Qubits are more complicated though, using qubit instead of digital bits, you can describe a distribution of the probabilities of the particle being at each location inside the volume. A measurement yielding qubits might does not need to collapse the wavefunction, so may describe the amplitude of each location