In last month’s Insights column, we explored a puzzle that is a simple analogue of one of the most astonishing results of quantum mechanics — Bell’s theorem. Bell showed that if quantum mechanical predictions are correct, then we have to give up one of three reasonable assumptions about the world. In a recent Quanta article Natalie Wolchover explains how:
… when two particles interact, they can become “entangled,” shedding their individual probabilities and becoming components of a more complicated probability function that describes both particles together. This function might specify that two entangled photons are polarized in perpendicular directions, with some probability that photon A is vertically polarized and photon B is horizontally polarized, and some chance of the opposite. The two photons can travel light-years apart, but they remain linked: Measure photon A to be vertically polarized, and photon B instantaneously becomes horizontally polarized, even though B’s state was unspecified a moment earlier and no signal has had time to travel between them. This is the “spooky action” that Einstein was famously skeptical about in his arguments against the completeness of quantum mechanics in the 1930s and ’40s.
In 1964, the Northern Irish physicist John Bell found a way to put this paradoxical notion to the test. He showed that if particles have definite states even when no one is looking (a concept known as “realism”) and if indeed no signal travels faster than light (“locality”), then there is an upper limit to the amount of correlation that can be observed between the measured states of two particles. But experiments have shown time and again that entangled particles are more correlated than Bell’s upper limit, favoring the radical quantum worldview over local realism.
As Wolchover further describes in the article, there is a third possible assumption found in Bell’s analysis — “freedom of choice” — the assumption that the experimenters are free to place the polarizers at any angle that they want.
Our puzzle modeled the experiment described above for a pair of anti-correlated particles, and is based on an intuitive formulation of Bell’s theorem described by the physicist David Mermin. If you vary polarizer orientations, quantum mechanics correctly predicts that the correlation between the photons is given by the formula 1 – cos2(θ/2), where θ is the angle between the two polarizers. Our puzzle explores what would be necessary to create this amount of correlation in an analogous situation from everyday life.
Two students, A and B, who are polar opposites of each other, are gearing up to do a course on quantum mechanics. Thirty-seven days before the course (Day –37) they take a computer test consisting of 100 true/false questions. Every question that A answers as true, B answers as false, and vice versa — their answers are perfectly anti-correlated. At the start of the course (Day 0), the two take the same test again. Some of their answers are now different from what they were the first time, but they are still perfectly anti-correlated. Thirty-seven days later (Day +37), they take the same test for the third time. Again, some of their answers are different, but they are still perfectly anti-correlated.
You and a friend sit at separate computer terminals and compare the tests. You can bring up just one of A’s tests on your computer screen at any given time, while your friend can bring up just one of B’s. First, the two of you pull up the tests the students took on the same day, comparing A’s Day –37 test with B’s Day –37 test, and so on. Sure enough, they are all perfectly anti-correlated, with no matching answers at all. Next, you compare A’s Day 0 test with B’s Day –37 test. In this case, there are exactly 10 answers that match. Similarly, B’s Day 0 test has 10 answers that match those in A’s Day +37 test. Finally, you compare B’s Day –37 test with A’s Day +37 test. And here comes the surprise …
As I explained, these experiments map directly to our puzzle. A and B’s same-day tests are the anti-correlated photons, and you and your friend are the experimenters. The days of the tests represent the angles, in degrees, of your respective polarizers. If the polarizers are at the same angle (same-day tests), the photons are 100 percent anti-correlated, just as the students are. Since the situations are isomorphic, we should be able to replicate the photon correlation results with the test correlation results — the situations should give identical numerical answers for all angles (days) under the same assumptions as Bell’s theorem. These common-sense assumptions are: Completed tests with definite answers exist (realism), they cannot influence one another while the grading is being done (locality), and the examiners are free to compare any of A’s tests with any of B’s (freedom of choice). (In order to simulate the probabilistic nature of quantum mechanics precisely, you would have to imagine that each test had a very large number of questions, and that you and your friend can compare only small fractions of the two tests that are nevertheless large enough to yield consistently reliable probabilistic results. This condition does not change the numerical answers.)
Question 1: What are the minimum and maximum numbers of matching answers you would expect for these two tests (B’s Day –37 test and A’s Day +37 test)?
The Day 0 tests of A and B have exactly opposite answers. If B’s –37 day test has 10 answers in common with A’s Day 0 test, then B must have chosen the opposite answer for 10 questions compared with his own Day 0 test. Now if A chose the same 10 questions to answer differently in her Day 37 test, then our two target tests would again be perfectly anti-correlated. On the other hand, if A chose 10 different questions from the ones B had chosen to answer in the opposite way, then they would agree on 20 of their answers, but no more.
Question 2: If you found that there were 36 answers that matched, how would you explain it?
Answer 2: Again, as Michael rightly pointed out, the only explanation can be that your computers have been hacked! The only way for 36 answers to match for the comparison between (A Day +37, B Day –37) after you’ve confirmed that the correlations between (A Day 0, B Day –37) and (B Day 0, A Day –37) are both 10, is that the answers of the tests are being changed in real time depending on what tests you and your friend call up on your computers for comparison. Analogically, the quantum results can be explained (assuming realism and freedom of choice) only by assuming superluminal connections between widely separated particles that enforce the correlations at the instant of measurement itself.
Question 3: Where do all the numbers in the above scenario (–37, 0, +37, 10 and 36) come from?
Answer 3: This question was redundant. The correlation between photons with polarizers separated by 37 degrees is given by 1 – cos2[(37-0)/2], which is approximately 10 percent, which reflects the correlation between A’s and B’s tests 37 days apart. Similarly the expression 1 – cos2[(37-(-37))/2] = 36 percent gives the correlation between photons with polarizer angles 74 degrees apart, which reflects the correlation between A’s and B’s tests 74 days apart.
Question 4: Using the above formula, what is the largest possible difference between the actual correlation for an angle 2θ and the maximum value calculated for 2θ from the given correlation for θ, under the three assumptions described above? At what angle between the polarizers does this largest possible difference take place?
Answer 4: The largest difference between the quantum and classical correlations in this case is 0.25 or 25 percent, at an angle of 60 degrees between the polarizers. You can calculate this by maximizing the difference between the classical and quantum expressions for correlations, as explained well by Michael. Alternatively, you can just set up a spreadsheet that calculates the difference with the angle going from 0 to 90 degrees, and plot the two curves to see the difference. The graph for classical correlations is a straight line, while that for the quantum correlations is S-shaped. The differences are maximum at angles of 60 and 120 degrees, and minimum (nil) at 0, 90 and 180 degrees.
This demonstration of Bell’s inequality dictates that we ditch one or more of our assumptions. Michael very nicely summarized the alternatives we are left with:
The balloon model obviously throws away locality. So does the Bohm model of quantum mechanics.
Superdeterminism obviously throws away freedom of choice. So does the Hall model mentioned in Natalie Wolchover’s article.
Standard quantum mechanics obviously throws away realism. The Copenhagen interpretation, for example, only allows that preparation and measuring devices are real (“classical”), with quantum mechanics being about the correlations between such devices. In this sense the correlations are real, when measured, but there is no underlying reality of, say, “photons” that cause these correlations.
This is a great summary of the received wisdom. But I think quantum mechanics already throws away locality. There are many convincing examples of this, but I’ll cite two that we have gotten so used to that we don’t even see them anymore.
The first is the matter of discrete quantum jumps themselves. In models of the atom, electrons jump from being smeared around the atom in one orbital to being smeared around in another with a completely different configuration, releasing a photon of a certain energy. There is no in-between state, no variation in frequency as would be expected if the electron was initially located in different points in these orbitals. This clearly requires non-locality. As the Columbia physicist I.I. Rabi, one of the original contributors to quantum mechanics, said: “The atom is in one state and moves to another, and you can’t picture what it is in between, so you call this a quantum jump. In quantum mechanics, you don’t ask what’s the intermediate state because there ain’t no intermediate state. It passes from one to the other in God’s mysterious way.”
The second huge affront to locality occurs in Feynman’s “path integral” version of quantum mechanics. This approach assumes that a particle goes from one point to another by simultaneously following all paths everywhere in the universe. Now here’s nonlocality with a vengeance. And yet it works beautifully.
Yes, nonlocality is literally everywhere in quantum mechanics, and we should willingly embrace it. What this means is that the internal composition of every quantum particle or entangled pair is nonlocal and inherently superluminal. As I said, perhaps every particle has its own wormhole, or something like that, to play in: ER=EPR, even for single states. This nonlocality never leaks out into the open, just as it doesn’t in case of an electron’s quantum jump, or in the EPR experiment. So relativity is not threatened at all.
Embracing nonlocality and superluminal internals in quantum objects is extremely freeing — it allows us to construct models that are physical and can be visualized, and are not just abstractly mathematical. For me, there are compelling arguments against the latter, as I’ve detailed in my responses to phayes and to Alex Livingston. One key point is that probabilities and interference require ensembles and cannot be generated by single particles (as they are in quantum mechanics), unless you assume that they have parts.
I think a compelling image is that every quantum particle is like a bubble that can split into infinitesimal superluminal bubblets in a wave distribution, which can reform the bubble at different locations. We can detect the particle only when we force the bubblets to make a choice between two opposite attributes or locations by making a measurement, which cause the bubblets to coalesce again into a full bubble. Imagine two taut and flat plastic sheets with a bubble trapped inside. If you apply a little pressure, the bubble fractionates into millions of tiny bubbles that occupy the whole area inside the sheets. Release the pressure, and the bubblets reform into the original bubble at a different location. We can postulate that the probability of coalescing at a particular point is proportional to the size of the bubblet at that point, which would correlate neatly with the Born probability interpretation rule.
Can such a fantasy really be true? I don’t see why it couldn’t be. As I explained, the bubblet model can explain the double slit experiment (which Feynman once said contains all the essential aspects of quantum mechanics), and it gives an intuitive sense of how measurements and environmental phenomena analogous to them create our reality. Of course, the bubblets could be at a level of physicality so fine that we may never get close to detecting them. But models based on them could help integrate quantum mechanics and relativity, and help elucidate the structure of space-time. Will they? Only time (and space!) will tell. In the meantime, such models can make quantum mechanics vivid again. And since I am not a physicist, I can get away with making them without losing my job! I am glad some of you found this interesting.
The Quanta T-shirt for this puzzle goes to Michael, for his contributions to this and several other previous columns. Ashish’s answers and comments were likewise excellent. Thank you to all who contributed. See you next week for new insights!