It is common wisdom - and mathematically proven - that quantum entanglement cannot be used to bypass the relativistic speed limit and transfer information faster than light. So there must be something wrong with the following gedankenexperiment, but I can't figure out what:
Let there be a device which (nearly) simultaneously creates lots of entangled, coherent photon pairs and sends them to Alice and Bob, each of them receiving one photon of each pair. Both are far away from the light source and have placed screens into the - highly focused - light beam, so that they detect some small area of light on the screen.
Now Alice wants to transfer one bit of information to Bob. If it is a 0, she does nothing. If it is a 1, she puts an appropriate double-slit plate into the light path, immediately before the photon bunch arrives. This will create an interference pattern on her screen, and entanglement will instantly replicate the photons' paths and such the pattern onto Bob's screen. So if he sees an interference pattern instead of a dot, he knows that it's a 1, and vice versa.
Of course Bob's screen will not show an exact copy of what Alice sees: it will be olverlayed by noise, e.g. from photons which bounced off Alice's double-slit plate or decohered through environment interaction. Worst case a statistical accident may create a pattern out of a 0-bit which looks like the result of interference. But we may optimize the signal/noise ratio through the experimental setup, and besides of that any real-world communication channel is subject to partial information loss (which must be compensated by error correction).
So it seems like this experiment transfers information faster than light. What did I miss?