Can someone explain why this QM FTL communication setup is wrong?

Question

So I thought I understood the double-slit experiment and EPR paradox until this setup occured to me. It combines EPR entanglement with "which-way" double slit setups. I know it must be wrong, but I can't see how:

Suppose you had a device that sends out entangled e-p pairs in streams towards point A and point B. By "stream" I mean that it sends e-p pairs in regular time intervals headed in the same direction every time.

At point A, there is a device for measuring the particles spin. At point B there is a double slit setup. The catch is that this is a hypothetical double slit experiment which has "spin slits" instead of spatial slits. Ignoring the details of how this would work, let's just say that this means that there is an interference pattern created by incoming particles possibly having spin up or spin down. Like "which-way" double slit experiments, this pattern will go away if you measure the spin before the particles pass through the "spin slits".

If observer A does nothing, observer B sees an interference pattern aggregated from the stream of electrons. But if observer A decides to measure the spin of the particles, then shouldn't this make the interference pattern at B disappear? Since the act of observing the electrons in the stream collapses their spin wave-functions, they can no longer exibit the interference pattern at B which results from passing through both "spin" slits at the same time. It's essentially a "which-way" experiment where the "which-way" detection device is separated in space.

What am I missing?

Note: the difference between seeing an "interference pattern" and not seeing one is not a definite thing. Given a pattern created by say, 100 electrons, we could only say this "looks" like an interference pattern. But this is still information.

Answer 1

Lots of people have similar ideas and in fact I answered this very question in another post ("FTL" Communication with Quantum Entanglement?), which isn't actually a duplicate question (just the answer is duplicated).

Basically the answer in a nutshell is that entanglement always destroys quantum coherence. What Bob (observer B) observes by himself, no matter what, is no-interference pattern, in the form of a pattern $\frac 12 |f_0(x)|^2 + \frac 12 |f_1(x)|^2.$ Alice's initial measurement will, if you use it to filter out Bob's observations, trigger a splitting of this setup into its constituent parts $\frac 12 |f_0(x)|^2$ and $\frac 12 |f_1(x)|^2$.

However, it is possible for Alice to measure something else: she can measure the x-projection or y-projection of her spin, effectively erasing her ability to measure the usual z-projection of spin. (The x-projection in particular corresponds to measuring a qubit in the Hadamard basis $|+\rangle, |-\rangle$, as discussed in the above link.) This has a very interesting effect because it instead splits Bob's measurement into different constituent parts, $\frac 14 |f_0(x) \pm f_1(x)|^2,$ which are wavy. However, if you sum them together, you indeed see that what Bob sees is fundamentally the same: it's only how Alice's observation "picks out" Bob's observations that looks wavy or not.