This just started to bother me after reading yet another entangled particle question, so I hate to ask one myself, but...
If we have two entangled particles and take a measurement of one, we know, with certainty, what the outcome of same measurement of the other particle will be when we measure it. But, as with all fundamental QM, we also say the second particle does not have a definite value for the measurement in advance of making it.
Of course, the particle pairs are always described by an appropriate wave function, and measurement of half the pair changes the probabilities associated with each term. But, and this is the part that bothers me, with knowledge of the first measurement, we know the outcome of the second measurement. There's nothing strange or forbidden or anything about this, no non-local affects, no FTL communication, etc.
But does it still make sense to talk about the second particle not being in a definite state prior to the measurement? Or can we consider a local measurement of one half of an entangled pair to be a measurement of the other half? The latter option feels distasteful because it implies communication between the particles when communication is possible, which is obviously wrong, but I find it hard to pretend the second particle in the pair, whose state I know by correlation, is somehow not in that state before I confirm it by measurement.
I should clarify that my uneasiness comes from a comparison with the usual "large separation" questions of entangled particles, where our ignorance of the first measurement leads us to write an "incorrect" wavefunction that includes a superposition of states. Although this doesn't affect the measurement outcome, it seems there is a definite state the second particle is in, regardless of our ignorance. But in this more commonly used scenario, it seems more fair to say the second particle really isn't in a definite state prior to measurement.
As a side note, I realize there's no contradiction in two different observers having different descriptions of the same thing from different reference frames, so I could accept that a local measurement of half the pair does count as a measurement of the other half, while a non-local measurement doesn't, but the mathematics of the situation clearly point to the existence of a definite state for the second particle, whereas we always imply it doesn't have a well-defined state (at least in the non-local case) before measurement.
At this point, my own thinking is that the second particle is in a definite state, even if that state could, in the non-local separation case, be unknown to the observer. But that answer feels too much like a hidden variables theory, which we know is wrong. Any thoughts?
Answer
According to quantum mechanics, the particles - whether it is the first particle or the second particle or any other particle in the Universe - refuse to have any well-defined state or property prior to the measurement.
The only "kind of" exception is the case - relevant for a maximally entangled scenario - in which we are interested in a property of the second particle - or any other particle - that is predicted to take a particular value with probability equal to 100 percent. Of course, if the probability is 100 percent, then you can be sure what the measured property will be, and you may assume that this value of the property exists even before the measurement.
However, for the very same particle that has some value of the quantity equal to something at 100 percent, there still inevitably exist other observables that are not known. (Just design a Hermitian observable with random off-diagonal matrix elements.)
In the basis of eigenstates of those other properties, the probability amplitudes are generic, and some of the options have probabilities that differ from 0 percent as well as 100 percent. For those quantities - and it is a majority of observables - the usual prescriptions of quantum mechanics hold: the value is not determined prior to the measurement. It is not just unknown to the physicists: it is unknown to Nature. A picture in which those observables are determined leads to wrong predictions and contradictions.
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