The debate between Einstein and Bohr continued for decades regarding what “reality” meant in the context of quantum mechanics. Einstein and his followers insisted that an objective reality exists whether it is observed or not. Their most powerful argument was explained in the EPR paper, in which Einstein and his colleagues proposed that “elements of reality” (hidden variables) must be added to quantum mechanics to explain the behavior of entangled particles, negating the concept of quantum entanglement, which seems to require action at a distance. In EPR’s view, the entangled particles must be born with certain preprogrammed behaviors (those hidden variables) that make them act appropriately when measured without having to communicate with their twins.
These hidden variables would be microscopic properties of particles we are unable to observe directly by means of experimental measurement. Einstein anticipated that we might have the technology to measure them in the future, but for the time being they are “hidden.” EPR assumed that if we knew more about these hidden variables, we could finally explain the otherwise mysterious behavior of particles demonstrated by quantum mechanics.
On the other hand, Bohr and his followers insisted that the Uncertainty Principle would remain valid for an entangled system, and that the entangled particles would somehow have to interact at the moment one of a pair is measured, even if they are separated by a vast amount of space. True to the Copenhagen Interpretation, Bohr insisted that these variables are not just unobservable, but rather that they simply don’t exist outside of the context of an observation.
This discussion was believed to be purely philosophical and irresolvable, since an experiment would require measurements to be made, yielding the same exact results when an observer is introduced into the picture. In Einstein’s words:
I think that a particle must have a separate reality independent of the measurements. That is: an electron has spin, location and so forth even when it is not being measured. I like to think that the Moon is there even if I am not looking at it.
So, is the Moon there when no one is looking at it? Obviously, this is a purely philosophical question that seems to be impossible to answer.
In 1964, nine years after Einstein’s death, Irish physicist John Bell figured out that each of the alternative answers to the EPR paradox would actually show subtle differences in some particular experiments. Specifically, Bell found that if local hidden variables exist,‡ then a minimum level of coincident detections for the entangled particles would be obtained when certain specific measurements were made. If, on the other hand, quantum entanglement is correct, the number of coincident counts would be below this same level.
The concept is difficult to explain without heavy math. However, Cornell University’s N. David Mermin has written a number of ingenious papers in which he explains Bell’s Theorem in an easy-to-understand way. We will make use of Mermin’s simplified explanations, but we would like to strongly encourage you to read Mermin’s original papers to understand the various subtleties of Bell’s argument.42–44 Before we start, remember that you can only measure the polarization of a photon once. This is because, the photon will acquire the polarizer’s polarization as soon it is measured (Figure 121b). The “measurement” of a photon’s polarization only gives us a “yes” or “no” type of answer based on the probability of the photon passing through or being absorbed by the polarizer according to its angle of polarization. The actual angle of polarization can only be measured to a certain value for a large number of photons.
Okay. Following Bell’s argument,§ let’s pose the following questions about a single photon:
1. Does the photon have a definite polarization at 0°?
2. Does the photon have a definite polarization at 120°?
3. Does the photon have a definite polarization at 240°?
According to EPR, the photon’s polarizations at these three angles are linked to some element of reality (a hidden variable), if they can be predicted without disturbing the photon in any way. We could ascertain the photon’s polarization at any one angle, so we could determine its element of reality for that one angle. However, the million-dollar question is whether the photon has elements of reality simultaneously at 0°, 120°, and 240°. As we know, we can only measure the photon’s polarization once, so we can’t know what the photon would have done when encountering polarizers at the other two angles.
According to EPR, elements of reality exist to define the behavior of the photon when encountering a polarizer at 0°, 120°, and 240°, even though we can’t actually measure more than one of them at a time. So, at an angle 0°, there may be a hidden variable A; at 120°, a hidden variable B; and at 240°, a hidden variable C. Einstein’s followers imagined that one day we may be able to measure these hidden variables and be able to predict with exactitude what a photon would do when it meets a polarizer at 0°, 120°, and 240°.
Figure 137 shows the thought experiment proposed by John Bell as a way of capturing EPR’s view on reality. Let’s suppose that a photon has three hidden variables A, B, and C. Variable A defines how the photon will behave when it encounters a polarizer at 0°, same for B for a polarizer at 120°, and C for a polarizer at 240°. That is, hidden variable A tells the photon whether or not to cross through a polarizer set at 0°, and so on for B and C. Somehow, these variables would be preprogrammed into the photon at the time it is created, and do not depend at the time of their “programming” on the condition of the polarizer the photon will encounter in the future.
Figure 137 A simplified form of John Bell’s view of EPR’s argument assumes that a photon has three hidden variables A, B, and C. Variable A defines how the photon will behave when it encounters a polarizer at 0°, same for B for a polarizer at 120°, and C for a polarizer at 240°. (a) Somehow, these variables would be preprogrammed into the photon at the time it is created, and do not depend at the time of their “programming” on the condition of the polarizer the photon will encounter in the future. (b) We can only measure the photon’s polarization once, so we can’t know what the photon would have done when encountering polarizers at the other two angles. However, EPR would have asserted that the photon has elements of reality simultaneously at 0°, 120°, and 240°.
Bell has then defined reality by simply telling us that the photon’s polarizations at 0°, 120°, and 240° exist whether or not we try to measure them. Table 7 shows the eight possible permutations of A, B, and C. A photon would thus have to be “programmed” with one of these.
TABLE 7 The Eight Possible “Programs” that a Photon may have as Hidden Variables in Bell’s Definition of EPR’s Realitya
Let’s now replace the single-photon generator of Figure 137 by an entangled-photon generator, as shown in Figure 138. Let’s also assume the machine produces pairs of photons at a regular interval and detectors placed behind the polarizers report on detections when photons would be expected. This way, not detecting a photon at its expected time of arrival actually means it was absorbed by the polarizer, and not that the photon didn’t exist.
Figure 138 Using an entangled-photon generator instead of the single-photon generator of Figure 137 allows us to test for two of the three hidden variables simultaneously, since, according to EPR, both photons would carry the same “program.”
According to Bell’s interpretation of the EPR argument, both photons would be programmed with the same hidden variables, so this setup allows us to conduct an experiment equivalent to making two measurements on the same photon. In this way, we can at least perform a direct measurement on two out of the three hidden variables.
Obviously, the behavior for cases A, B, and C (Table 7) are the same for both polarizers. However, let’s rework our table to show coincident measurements for the various possible cases of hidden variables and polarizer settings. Let’s use binary notation for the outcomes (YES = 1, NO = 0), and please note that two zeros (neither of the photons went through its polarizer or, in other words, both photons were absorbed by their respective polarizers) also constitutes a coincidence (see Table 8).
TABLE 8 Probability of Coincident Measurements on a Pair of Entangled Photons by Two Independent Analyzing Polarizersa
Now, please pay attention, because this is the really important part: Table 8 shows us that regardless how the photons are programmed, the average probability of detecting a coincidence is at least 0.333. That is, regardless of how we set up our experimental apparatus of Figure 138, if we perform many measurements, we should end up detecting coincidences between detectors placed behind the polarizers at least one-third of the time. The coincidence detection rate should never be less than one-third as long as you check cases [AB], [BC], and [AC] evenly. The rate should actually be a bit higher than one-third because of the existence of cases 1 and 8.
Now, let’s suppose Bohr was right, and hidden variables don’t exist. As we will see, Bell chose polarizer angles 120° apart because quantum mechanics (without hidden variables) would predict¶ that the coincidence detection rate for either [AB], [BC], or [AC] is equal to the cosine square of the angle between the two polarizers. Since cos2(120°) = cos2(240°) = 0.25, quantum mechanics would predict that the average probability of coincident detections is around 0.25.
Bell’s Theorem, then, tells us that the probability of coincident detections for any local hidden-variables theory (probability > 0.333) is incompatible with the predictions of quantum mechanics (probability ≈ 0.25). Bell had found a way of telling whether the moon is there when nobody is looking! Well… maybe that’s too grandiose. However, Bell’s Theorem provides a way of knowing whether Einstein was right or wrong about the existence of local hidden variables.
Bell’s test for the existence of hidden variables is thus simply checking that a large number of measurements in an experiment conducted as we have described above will yield a probability of coincident detections above one-third. This is known as Bell’s Inequality. On the other hand, experimental results violating Bell’s Inequality with a sufficiently large statistical confidence would indicate hidden variables don’t exist, giving weight to the Copenhagen Interpretation.
In 1969, American physicists John Clauser, Michael Horne, Abner Shimony, and Richard Holt proposed a different version of Bell’s Inequality (now known as the CHSH Test from the initials of the authors’ last names) that is more suited for performing actual experiments to distinguish between the entanglement hypothesis of quantum mechanics and local hidden-variable theories. Since then, other Bell-like inequalities have been proposed to close experimental shortcomings and loopholes. However, as for the original test proposed by Bell, these refined tests are all based on the statistics of counting coincidences between detections. In some of these inequalities, the coincidence limit is a lower bound, like in Bell’s original inequality, but others define it as an upper bound, indicating that quantum entanglement would give particles more information than they could actually carry as local hidden variables. Also, like Bell’s Inequality, these inequalities must be obeyed under local realism but are violated by particles behaving according to the Copenhagen Interpretation of quantum mechanics.
With this background, let’s discuss how to produce and detect entangled photons in the lab to experimentally determine who was right in the protracted debate between Einstein and Bohr on the existence of an objective reality independent of observers.
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