What would be the outcome of an experiment wherein the spin of a qubit is measured in two or more orthogonal directions simultaneously?

Question

Out of curiosity, I am seeing the Leonard Susskind’s Theoretical Minimum lecture series on YouTube and have just started watching the second lecture.

As I understand so far, the qubit can have a spin value of +-1 in any orientation and if the spin is not already prepared by previous measurements, it will have an equal probability of having either or the spin values.

However, once the spin is prepared by previous measurements, the probability is dependent on the angle between the previous and current measurements.

This make me curious to contemplate what would happen if one measures the spin in more than one direction simultaneously? I believe if the qubit would have classical properties, and the spin value in one direction is +-1 in other orthogonal directions, it should be zero which is not possible in quantum realm but also since the measurement is simultaneous, it should not be either +1 or -1.

The question is moot if it is not possible to measure the spin in two different directions simultaneously.

Answer 1

A measurement is a physical process that creates a record of some property of a physical system. Since a measurement is a physical process it is constrained by the laws of physics, which include the equations of motion of quantum theory.

A record is a piece of information that can be copied indefinitely often and that constraint plus the equations of motion of quantum theory implies that a measurement produces a result that is an eigenvalue of an observable represented by a Hermitian operator:

https://arxiv.org/abs/0903.5082

So any particular measurement result represents the outcome of one measurement at a time.

Continuous measurements, repeeated measurements, measurements of continuous observables and imperfect measurements make this picture a bit more complicated:

https://arxiv.org/abs/1604.05973

The reason is that in practice you don't get a record that gives you an exact single value for what you're trying to measure. For example, in practice with measurements of continuous observables you have to make do with different outcomes having a very low probability of overlap. But this just means that real measurements are still measurements of a particular observable but they give you less information about it than an unphysical perfect measurement would.

Answer 2

You can't "measure a qubit in two bases at the same time". A measurement necessarily involves a collapse of the wavefunction, so "measuring in the Z basis" means forcing the state to be either $|0\rangle$ or $|1\rangle$, while measuring in the X basis means forcing the state to be either $|+\rangle$ or $|-\rangle$. You can see from this framing that it doesn't really make sense to "measure in both bases at the same time".

You can of course measure the qubit $\rho$ in one basis and then in the other. But that won't give you what you might think. Even assuming that after the first measurement in the Z basis you are left with a corresponding state $|i\rangle$, that state won't have any information left about the original state (at least not for this choice of measurements), so the second measurement isn't telling you anything else about $\rho$.

It might be interesting to note however that you can perform other kinds of measurements that give you full information about the original state. Any informationally-complete POVM will do that. For example, if you evolve $\rho$ together with an ancillary qubit through the unitary evolution given by the Fourier transform (many other choices work just as well) then measuring the resulting two qubits in the computational basis gives you tomographically complete information about $\rho$. You might consider this as a form of "measuring multiple bases at the same time", though using such terminology would be a bit misleading in my opinion.