Can post-selecting on the screen in the Delayed Choice Quantum Eraser experiment be used to predict the quantum-eraser measurement results?

Question

I'm curious about QM and spent the last 3 days thinking about Delayed Choice Quantum Eraser (DCQE) experiment, but I couldn't solve this issue:

1. Assume we do the DCQE experiment so that for the whole experiment (let's say, 1000 particles) finishes before the first particle reaches the detectors 1-4 (using the notation from this diagram from Wikipedia).

2. Assume also we're using a switch to control if the idler particle goes to which-way-detector or to the eraser.

3. This schematic shows what we should see in $D_0$, even though we don't know yet $R_{01-04}$

4. Now, knowing $D_0$, use the switch to send the 100 particles that reached the most-left side of D0 to the eraser detectors ($D_1$ and $D_2$). It seems much more likely that they reach $D_2$ than $D_1$, because it's on peak on $D_2$ (and on rest on $D_1$) even though it's going through a half-silvered mirror. This looks like a contradiction to me.

So, am I missing something or QM is just that weird?

Answer 1

You're over-interpreting these sketches - they are only sketches, and their specific details can't really be used to make any real predictions.

Here is a more accurate version of those sketches, with a proper underpinning on a solid model of the experiment's behaviour:

^{Mathematica source via Import["http://halirutan.github.io/Mathematica-SE-Tools/decode.m"]["https://i.sstatic.net/P6HYG.png"]}

As you can see "the leftmost part of $D_0$" is equally compatible with the patterns $R_{03}$ and $R_{04}$, as detected on the quantum-eraser detectors 1 and 2.

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Still, you're not entirely wrong, particularly in the sharper formulation you give in the comments:

Isn't it true that the patterns made by the particles that reach R01-04 follow distinct distributions on D0? If so, it seems reasonable to extrapolate that there are some regions that are inverse peaks for D1/D2

Yes, the patterns made on the $D_0$ screen when post-selecting on $D_1$ and $D_2$ detections are indeed different - and, in fact, they're complementary interference patterns, with the peaks on $R_{01}$ lining up with the troughs on $R_{02}$ and vice versa. (This is how they can add up to an interference-less $D_0$ pattern when there is no post-selection. It is crucial that you understand that both $R_{01}+R_{02}$ and $R_{03}+R_{04}$ add up to $D_0$, and what that means - the 1/2 and 3/4 pairs are just different ways of splitting up the $D_0$ counts, depending on information acquired later.)

This means that you can zero in on one of the peaks of the $R_{01}$ fringes, say, the green box below:

If use some fancy switching mechanism to ensure that you send all the particles that fell on that green box over to the $D_1$/$D_2$ quantum-eraser part of the idler-photon side of the experiment, then indeed, as you say,

it seems much more likely that they reach D1 than D2.

Is this a problem or a contradiction? No. The photons are not going through an arbitrary half-silvered mirror - they're going through a precisely calibrated beam splitter. The beam path that reaches $D_2$ includes a contribution from $M_b$ (red beam) and a contribution from $M_a$ (blue beam), and if those beams are coherent, they can interfere both destructively and constructively. Absent any information about what happened to the signal photon on $D_0$, the idler and the signal are entangled, and there is zero relative coherence between those two beams, and $D_2$ will click half the time. However, by post-selecting on $D_0$'s measurements on the green box, you're effectively fixing the phase between the two beams in such a way that they interfere destructively on the $D_2$ side (and constructively on the $D_1$ side), so no light goes through to $D_2$ (on those post-selected runs).

So, basically, what you've described is a fancy way to run the quantum-eraser apparatus in reverse, where by splitting the $D_0$ screen into sectors you're providing information that can be used in a post-selection scheme to recover the interference pattern that comes out of the BS$_\mathrm{c}$ beam splitter.

If that seems weird, then yes,

QM is just that weird.