oknow said:
In the quantum eraser experiment, the eraser is in the apparatus for a reason -- to erase the information derived from a measurement. After all, if no measurement had been made, there would be nothing for the eraser to erase.
The quantum eraser underscores the importance of observation in QM and, in this case, the critical distinction between measurement and observation.
Every interaction (measurement) between two objects, A and B, results in a change of state for the involved objects. If third party C cannot observe the resulting measurement information, from C's perspective, it is as if that measurement, which did indeed occur, had not taken place. In such case, C remains in superposition relative to the change of state of A and B.
In the quantum eraser experiment, which-way measurement occurs within the instrumentation, but that which-way information is subsequently erased such that it is not accessible to the experimenter. Consequently, from that experimenter's point of view, things remain as if the measurement had not occurred.
To get back to this point: It is not that easy. There are two basic "kinds" of quantum eraser experiments.
1) The typical version attributed to Walborn puts wave plates at the double slit positions. These serve as which way markers. For example, they shift horizontal polarization to right-circular polarization, vertical polarization to left-circular polarization,right-circular polarization to vertical polarization and left-circular polarization to horizontal polarization. This is a reversible interaction and it is essentially loss-free. All of the light passes the waveplates. This approach also works for delayed choice experiments.
One could also perform an irreversible interaction by putting polarizers at the slit positions. These are filters that just let a part of the light field pass. This obviously constitutes a measurement for the photons that do not pass the filter. However, these do not form an interference pattern anyway. The photons that pass the filter will have a polarization that matches the filter orientation. However, there is no way to know whether a photon is indeed present unless we actually detect it. Accordingly, if we place perpendicular polarizers at the two slits, the photons that make ith through will not show an interference pattern if detected directly. One can now insert another filter oriented at 45 degrees to the two initial filters. Again, some photons will not make it through the filter. Those that make it through will show a polarization aligned with the filter and will show interference when detected by a detector placed behind the double slit and all of the filters. For the photons that actually make it to this detector, the absorption at the detector is the one measurement (strong measurement, to be more precise) we have here. Beforehand, information about the path would have been available to us if we had performed a measurement. However, we did not. Thus, it would be more accurate to say that we erased the possibility to gain which-way information instead of erasing which-way information. In order to get real which-way information along the way, we would need some kind of signal or trace that actually tells us that a photon has just passed a certain flter. This is, however, not how they work. We can just take note of the photons that get absorbed at these filters. This whole kind of setup does not work well in delayed choice scenarios, which is why they are not used in sophisticated scenarios.2) The Scully version instead uses a very clever design that relies on nonlocal filtering. However, as you mentioned only the quantum eraser and not the delayed choice version, discussing this might be out of the scope of the discussion.
Also, even more importantly it is not true that "Every interaction (measurement) between two objects, A and B, results in a change of state for the involved objects". However, it is a very important and fundamental point to understand why that is the case which really helps in understanding quantum mechanics. Simply consider a photon bouncing off a mirror. If this "interaction" changed the state of both the photon and the mirror, this would already constitute which-way information. We could just check the state of the mirror and it would tell us which way the photon went. Mach-Zehnder interferometers would not work if that was indeed happening. What indeed happens is that the mirror is in some initial state that is characterized by some position and momentum and some uncertainty of both (at the very least due to Heisenberg uncertainty). This is the state of the mirror. When reflecting off the mirror, the photon will transfer momentum to the mirror, but it will be so little momentum that the amount is small compared to the initial momentum uncertainty of the mirror. The initial and the final state of the mirror have a large overlap - usually much more than 99.99% - so that one cannot say whether an interaction happened judging from a single photon reflection process. Taken the other way around, this also defines what a "good" detector or measurement process is. If the initial and the final state of your "detector" before and after the interaction are orthogonal (have 0 overlap), then one consider the action of this detector as a measurement in the most common sense in QM. For example, if you make the mirror much much lighter and mount it on a spring, the momentum transferred by a single photon might be sufficient to put the mirror into a different state. Some precision measurements work this way.
There are of course other ways to treat the other cases (POVMs and so on), but the example above illustrates the typical strong measurement that people usually have in mind when they talk about measurements in QM.
