# Quantum Eraser under Lorentz Boost

Suppose I am conducting the Quantum Eraser experiment. The results of this experiment are easy to understand with the traditional quantum mechanical interpretation of a pair of entangled photons.

Specifically, suppose that I am performing the "eraser" part of the experiment in which one photon is diagonally polarized so that when the entangled partner passes through the circular polarizers of the double-slit apparatus it cannot be determined through which slit it passed (thus reproducing an interference pattern).

Now suppose (It's like supposition Inception! Supposings inside supposings inside supposings.) that I Lorentz boost into a reference frame in which the particle passing through the double-slit apparatus is received at the detector (and consequently has its polarization measured) before its entangled partner gets diagonally polarized.

It appears that there are two possible outcomes: either (a) I disagree with the results recorded by an observer in the commoving frame of the experiment (i.e. I see no interference pattern.) or (b) I must conclude that the future polarization of the entangled partner somehow reached back in time and changed the outcome of my measurement.

So which is it? Do I get different results or do I need to embrace a further layer of quantum non-locality?

This is a tough one for me to understand because of its complementary experiment: Imagine putting the polarizer much farther away than the double slit apparatus so that there is no interference pattern created (because the entangled partner does not get diagonally polarized until after the measurement has already occurred). In that case, I could boost into a frame where an interference pattern should occur because the partner gets polarized first. (Or am I missing something? Will there be - or not be - an interference pattern regardless?) If the answer is that I get the same results as the commoving measurement, then why is the commoving frame preferred? Why is it its results that are maintained in all reference frames and not the (expected) results from my boosted frame (which should be just as valid)?

Also, I think that this version of the experiment is fundamentally different from the Delayed-Choice Quantum Eraser experiment and Wheeler's thought experiment since both of those experiments are easily explained by representing the particle in a different eigenbasis (namely, as being in a superposition of the interfering and the non-interfering states - instead of being in a "collapsed" state of another observable). However, I am also interested to know whether I am wrong in this estimation. Are these experiments basically the same? Can conclusions about or explanations of one of them be generalized to the others?

Edit: Because of the question in the comments, I've added a little additional explanation for why I think the experiments are basically different.

Primarily, the difference comes from the fact that both Wheeler's thought experiment and the Delayed-Choice Quantum Eraser can be explained in such a way that apparent "cause" always precedes "effect." Causality is still violated, but at least its not giving us information about the future. For instance, if we didn't know whether or not the entangled partner beam would get polarized a year from now (in the quantum eraser experiment), we could predict whether it does or not today by determining whether or not there was an interference pattern at the double-slits. If there exists any boosted frame in which the beam gets polarized first, then that result must carry over to the frames where it gets polarized after.

So "future-telling" can happen in the relativistic quantum eraser, but not in the other two experiments. For example, in the Wheeler experiment, we polarize the photon and then after the fact decide at random whether to measure its polarization or not. If we measure the polarization, we get no interference; if we don't measure the polarization, we do get interference. Often this is ascribed to a sort of "predictive" non-locality, but just as easily, we could say that the polarized state of the photon is actually a superposition of states (namely, interfering and non-interfering) and when we measure its polarization we collapse it into the non-interfering state (and if we don't, we collapse it into interference). Likewise for the delayed-choice quantum eraser: instead of thinking that the beam-splitter in the future decided the interference in the past, you could just as easily conclude that the interference (or not) in the past decided the outcome of the beam-split in the future.

• All delayed choice experiments share the feature that something not settled until time $t_0+\Delta t$ affects something determined at time $t$. You seem to have built another way to say the same thing, but why would you expect a different outcome (i.e. other than the final results are consistent). – dmckee Dec 22 '13 at 22:58
• I don't think that I would expect different results, but since that conclusion is not logically excluded (and since I don't really understand what's going on in my example), I didn't want to disregard it out of hand. It seems to me that my example and the other delayed-choice experiments that I mention are different because, in the other experiments, it is always possible to construct an explanation of the events which allows only past events to influence future events. Those explanations still violate local causality, but they are temporally well-ordered in the commoving reference frame. – Geoffrey Dec 22 '13 at 23:18
• Perhaps I am misunderstanding, but this is how I am reasoning. A boost can change the time-ordering of space-like separated events, but it can not change the time-ordering of time-like separated events. In all reference frames the entanglement must come before any polarization or measurement, so your experiment is homomorphic with the Delayed Choice Quantum Eraser: one path is measured before a choice is made on the other path that affects the outcome of the measurement on the first. – dmckee Dec 23 '13 at 3:13
• I think we're on the same page, but my point is that the events at the ends of the two beams are space-like separated and can, therefore, have their order reversed. However, the structure of DCQE allows room for us to explain the consequences of the experiment with simple quantum theory regardless of the order of the events (it is symmetric under ordering flip). Conversely, QE does not: events at the end of one beam can change the results at the end of the other but not vice versa (regardless of the interference outcome at the DS the other photon will always be polarized the same way). – Geoffrey Dec 23 '13 at 4:45