I have read time and time again that there is nothing mysterious going on in the quantum eraser experiment, but I can't get the math to add up. Please help me: Consider these two simplified illustrations from this page: The experimental setup, with emission points A and B, beam splitters BS_ and detectors D_
and the measurement results
The point I often hear made is that since the interference patterns only happen once you correlate a statistical amount of data from the two sides of the experiment, there is no retro-causality at play, just good old correlation. Which sounds good.
But let's say that a dark spot in the interference pattern, e.g. in the center of the $D0$ correlated with $D2$ data, is perfectly dark - zero photon detections happen at the center which are correlated with a $D2$ detection, no matter how many times we do the experiment. And say that the right photon is detected long before the left photon. Then let us say we run the experiment with a single photon in each direction, and the right photon is detected while $D0$ is in the center position. It would then seem like we can use this to predict that the left photon will not be detected in $D2$. But that would mean that the detection of the right photon influences the path of the left photon, which is of course not possible. What am I missing?
Let me try to restate the question mathematically: Let us describe the state of the left photon, $| \psi_L \rangle$ right before it is detected,
$$| \psi_L \rangle = \frac{1}{\sqrt{2}} (| A \rangle + | B \rangle) = \frac{1}{\sqrt{4}} (| D_1 \rangle + | D_2 \rangle + | D_3 \rangle + | D_4 \rangle ),$$
so it is equally likely to be detected by any detector. This would be true if it was propagating independently of the right photon. Detection of the right photon does not collapse it into $| A \rangle$ or $| B \rangle$, but it does mean that $\langle D_2 | \psi_L \rangle = 0$, which is not true according to the definition above.
How can this be described without saying that the detection of the right photon "tells" the left photon to avoid $D_2$?
