Relativity and Entanglement

Question

Say we have two particles which are entangled so that they have opposite spins. If one is up, the other is down. They are sent off to two spatially separated observers A and B. Both observers can claim the following: The probability of measuring their particle in the spin up state is 1/2 = probability of measuring spin down state. Now, suppose observer A measures his particle in the spin up state. Now, the probability of observer B measuring spin down = 1, and probability of measuring spin up = 0. I think it is fair to say that the Observation made by A influenced the outcome measured by B. He could have gotten spin up if he had measured his particle before A. However, there is no transfer of information due to the random outcomes. B doesn't know whether he measured spin down due to A measuring up, or if he was the first to measure his spin. Observations made by A are random too, and so if we repeat this experiment many times, B will measure spin up 1/2 the time and spin down 1/2 the time. However, the ordering of which measurements report spin up and spin down will be different than if B always measured before A. The probability would still be spin up 1/2 the time, and spin down 1/2 the time.

Are we left to say that relativity is about information and not this subtle influence? Einstein certainly was not of this opinion, "The following idea characterises the relative independence of objects far apart in space (A and B): external influence on A has no direct influence on B; this is known as the Principle of Local Action, which is used consistently only in field theory. If this axiom were to be completely abolished, the idea of the existence of quasienclosed systems, and thereby the postulation of laws which can be checked empirically in the accepted sense, would become impossible." -- A. Einstein

Answer 1

The majority opinion is that Einstein was wrong. However, I see some problems with the standard quantum mechanics (SQM) approach that you outlined. SQM contains two major parts: unitary evolution (described, e.g., by the Dirac equation) and the measurement theory (e.g., collapse, or the projection postulate, which, loosely speaking, states that, after measurement of some observable, the system stays in an eigenstate of that observable with the relevant eigenvalue). Both of these parts are used in your reasoning: on the one hand, unitary evolution ensures that the particles have zero total spin all the time, and the projection postulate ensures that, if particle A was measured to have spin up, it is in the relevant eigenstate. So far so good. The problem is that the two parts of SQM are mutually contradictory (the notorious measurement problem in quantum mechanics - http://plato.stanford.edu/entries/qt-measurement/ ). For example, unitary evolution cannot introduce irreversibility or turn a superposition of two states into their mixture, whereas the projection postulate does just that. So if you include the instrument (and the observer, if you wish) into the system containing particle A and then run unitary evolution to describe the measurement process, you will get a superposition of states with different spin projections and, strictly speaking, will not get a definite outcome. Actually, your reasoning is close to that used in the proof of the Bell theorem to prove that the Bell inequalities can be violated in quantum mechanics. However, such proof contains mutually contradictory assumptions. On the other hand, there have been no loophole-free experiments demonstrating violations of the Bell inequalities, so there are both theoretical and experimental difficulties with your reasoning, although it is widely used. Therefore, I tend to think that your question has not been resolved yet, as its resolution may demand either resolution of the measurement problem in quantum mechanics or loophole-free Bell experiments.