What happens that doesn't allow our measurement?

Question

Suppose pair production results in an electron and positron pair in an entangled system.

One person (A) measures the electron and another person (B) measures the positron. Another property of this system is the state of one particle will be the opposite of the other particle. Based on these properties person A measures the spin of the electron in the $x$ direction, using this information we can deduce the spin in the $x$ direction of the positron.

Now person B attempts to measure the spin in the $y$ direction, which will result in both of them knowing the spin in the $x$ and $y$ direction of both particles. But person B will be unable to measure the particle's $y$ spin. This is a well known scenario, and my question is how is it that person B is unable to measure the $y$-spin of the positron?

What prevents him, or how is he unable to do so? Do the states all change or does something else happen to restrict him from measuring it? Any help clearing this up would be helpful.

Answer 1

Another property of this system is the state of one particle with be the opposite of the other particle.

You have been sold a (rather common) misunderstanding of how entanglement works, and you're extrapolating wildly from there.

What you can make is an entangled state such that if you measure the same property on both systems, then the results (which are always random and uncontrollable) are completely anticorrelated. Anything beyond that is unwarranted extrapolation; in particular,

using this information we can deduct the spin in the x direction of the positron

does not follow. The positron does not "have" a spin projection that you then measure: you say what you're going to measure, and only then does quantum mechanics have something to say about values. If you don't perform a projective measurement on an observable, you cannot say anything about its value, period.

The way this actually looks is that you usually have your two spins in the maximally entangled state $$ |\psi⟩ = \frac{1}{\sqrt{2}}\big(|\!↑↓⟩+|\!↓↑⟩\big), $$ but if you only have access to one of the two spins (i.e. if you cannot perform correlated measurements) then you only have access to the reduced state of each of the spins, which is $$ \rho=\frac12\big( |\!↑⟩⟨↑\!| + |\!↓⟩⟨↓\!| \big). $$ This state is maximally mixed, it is completely incapable of supporting any (local) superpositions or interference, and it will always return a 50/50 split on the measurement of any spin direction. If, and when, you perform correlated measurements of the same spin direction, this can change (giving you perfectly random 50/50 splits on each, but which anticorrelate with each other), but only in the form of information on correlations.

And, certainly, no projective measurement tells you what the value of an observable "was" before the measurement.