Is wavefunction collapse “global”?

Question

I have the feeling that the premises of this question may not be coherent (so to speak), but here goes:

Suppose we have a system $X$ in a quantum superposition between states $0$ and $1$, say, with equal amplitudes. Suppose we have two observers, Alice and Bob, who have never interacted with each other in any way. Alice measures $X$ to be $0$. Then, later, Bob measures $X$.

Will Bob “always” measure $X = 0$ if he never interacts with Alice, or will the wavefunction not have collapsed for him yet? Or does their both having measured the same system mean that they've interacted in a way that ensures they'll observe the same thing?

Answer 1

Yes, wave function collapse is global in the sense that you mean, though I would use the word "objective" instead of "global". The measurements that have been performed and their outcomes are an objective element of reality.

alanf's answer is correct but makes things more complicated than they need to be. NUU misunderstood the question (it isn't about spacelike separated measurements), and their answer says

If B knows the outcome of A, he must use the 'collapsed' wave function. If he does not, he must use what he knows, i.e. the original state X.

which is very wrong. Alice's measurement has an objective effect, and Bob must incorporate that into his world model, or else he will make wrong predictions. If he knows the outcome of Alice's measurement, he must use the post-measurement collapsed state. If he knows what measurement she made but doesn't know the outcome, he must use the post-measurement mixed state (or model her as part of the system, as in alanf's answer, which amounts to the same thing). If he doesn't know what Alice did, he doesn't know the state and can't reliably make predictions.

There are various nits you could pick with this, so I want to clarify that measurements are objective in the same way that anything in a classical theory is objective. If your model of the present state of the universe is wrong about the Andromeda galaxy, your predictions about Earth will be right for the next 2.5 million years, and even after that they will probably be pretty close. Nevertheless, your starting state was wrong—just not wrong in a way that affected the predictions you made. In the same way, if you don't incorporate Alice's measurement into your model of the world, your model is objectively wrong, though still fine for many purposes.

Answer 2

Wave function collapse is not global, it is fictional. Let's suppose that the state is $\alpha|X=0\rangle+\beta|X=1\rangle$, where $|\alpha|^2+|\beta|^2=1.$

When Alice measures the state, an operation is applied that correlates both Alice and the environment with the value of $X$, like so $|X=j\rangle|0\rangle_A|0\rangle_E|0\rangle_B\to|X=j\rangle|j\rangle_A|j\rangle_E|0\rangle_B$. The environment is just everything around the system other than Alice. The $A,E,B$ subscripts stand for Alice, the environment and Bob respectively.

Bob might get the measurement result directly from the system, or from Alice or the environment. In any case, the end result will be $$\alpha|X=0\rangle|0\rangle_A|0\rangle_E|0\rangle_B+\beta|X=1\rangle|1\rangle_A|1\rangle_E|1\rangle_B.$$ After the measurement there are two versions of Bob: one version sees 0, the other sees 1. There is no version of Bob that sees both outcomes or some weird mix of 0 and 1, and there is a large literature that explains why this is the case, for an example see

http://arxiv.org/abs/quant-ph/0703160.

The short version is that only information contained in the eigenvalues of some observable, or a subset of such information, can be copied from one system to another. Bob won't see any other information because none of the rest of the information will be copied to him. This follows from quantum mechanics with no collapse postulate. So the collapse postulate is unnecessary for explaining that result.

Answer 3

Here's how I understand your question:

A and B are space-like separated and make a measurement on a single particle that has equal (or just non-vanishing) probabilities of being in A's or B's region.

You now ponder how the measurement process works on a deeper level. Could the collapse be a dynamical (i.e. time dependent) process? I think it can not. If it were, A and B would both try to 'pull' the wave-function to their side. You also get in trouble with relativity and the question who starts the process first.Thinking deeper, this will probably lead to the conclusion that the wave-function is not to be considered as 'real' (or 'ontic'). But I have not done this or seen this done.

It is thus better to regard the wave-function just as information. As soon as A or B have measured the particle, they have gained information and can update (=collapse) the wave-function.

Back to the specific question: For the physical outcome, it does not matter WHEN you collapse the wave-function. QM always assures that only one of A,B can measure the particle. Asking 'when' the collapse 'occurs' is not a sensible question, because the collapse is not dynamic.

Added: If B knows the outcome of A, he must use the 'collapsed' wave function. If he does not, he must use what he knows, i.e. the original state X.