An Entangled Parable of Acausal Connections

One of the most surprising aspects of quantum entanglement is access to shared information between widely separated particles, enabling them to coordinate their state parameters (for example spin or polarization states) over great distances. Bell’s theorem, proposed by (Northern) Irish physicist John Stewart Bell in 1964, offers a litmus test to see if a greater deterministic framework with local realism or the possibility of “hidden variables” (advocated by Albert Einstein and David Bohm, respectively) might have conveyed information between the particles in an entangled state, or if such a back-channel might be ruled out.

To understand the difference between the presence or absence of local realism (as gauged by the two possible outcomes for Bell’s theorem), let’s tell a story two different ways. In the first version of the tale, akin to a detective story, we’ll be as realistic as possible — leading to a conclusion Einstein and Bohm would support. In the second telling, we’ll make it a fantasy story and add an element of magic. Quantum entanglement is not magic, of course, but if we didn’t posit the notion of instant, non-local anticorrelation due to an entangled quantum state, it would sure seem that way.

First the realistic version. Imagine a family with two identical twin boys, Fred and George. They look so much alike, that they sometimes fool their friends. Therefore they are raised with a strict rule. No matter which way you dress, pick some characteristic that is different. In their town, which is somewhat isolated, clothing options are limited. Shirts come in red or blue, but not other colors. They are either plaid or striped, long-sleeved or short-sleeved. Thus, if, on a certain day, Fred chooses plaid as his distinguishing characteristic, George must wear striped to distinguish the two, no matter what the color. But if, on the other hand, George decides one day that he *absolutely* must wear a different colored shirt that day than Fred’s, if he dons a blue shirt, George is slated for red, no matter what the pattern.

We have used color, pattern, and length as metaphors for the three axes, x, y, and z, in which the magnetic fields might be directed to measure spin. The binary options in each case — red or blue, plaid or striped, long-sleeved or short, represent “spin up” or “spin down” as the measured results in those directions. Fred and George’s daily preferences (color, pattern, or length) are akin to the directional choices (x, y, or z) made by experimenters for the paired electrons they are measuring.

With Fred and George at home, their parents can readily coordinate one of their clothing characteristics so they are wearing opposites each day. But what happen when they go to separate schools that each have soccer teams, requiring changing each day into a freshly-cleaned shirt (which the schools provide) before practice? Each afternoon, at exactly 1:00 PM, Fred and George each simultaneously pick either color, pattern, or length as their distinguishing option, and the school presents each with a soccer shirt. Strangely enough, if Fred and George each happen to pick “color,” one gets red and the other gets blue, with about a fifty-fifty chance, over time, for each to get a certain hue. They absolutely never would get the same color, if that is their preference. In contrast, if Fred and George each happen to pick “length” as their key difference that day, they’d each have a 50 percent chance of getting a short-sleeved shirt, with their brothers simultaneously being given — at the other school — long-sleeves. Focused on their sleeve lengths, they wouldn’t even bother to look at color. Finally, if Fred picks “pattern” and George, at the same time, picks “length,” they’d each be given one of the binary options for each, without view to any of the other characteristics. Therefore, in that case they would be no noted correlation or anti-correlation, as the case may be, between their choices, if those were noted over time — say for a full year.

As a matter of fact, with their parents’ full consent, both kids are involved in a twin study for a psychology project about sibling choices. As part of it, psychologists at each school record the boys’ soccer clothing preferences each day (once again, color, pattern, or length), and the outcomes. After a full year of observation, the researchers meet, compare notes and discuss.

During their discussions, the psychologists note the anti-correlation effect when both boys happen to choose the same characteristic. They take each boy aside and ask him if he coordinates in advance with his brother before coming to school (there are no mobile devices allowed at the school, so they can’t phone each other or send each other messages). Each boy swears — in the way the psychologists believe — that they absolutely don’t make their choices in advance. Rather, they decide in the spur of the moment, right before each soccer practice. The school officials also attest that the don’t know in advance what selections the kids will make.

As scientific observers, the psychologists infer that there must have been some kind of hidden information sent to the school, and preparation for all of the possible outcomes. They surmise that each school must have kept all of the clothing options in stock. Thus, for blue shirts, each must keep plaid and striped varieties available, just in case the boy pupil happens to choose “pattern” that day instead of color. Similarly, for red, they likely have both plaids and striped, long-sleeves and short-sleeves in stock, perhaps in equal measure, just to account for all of the possibilities. Realism demands that just because something isn’t measured, observed, or chosen, it must still be available, just in case, to allow for the full range of possible selections.

To establish the validity of their realism hypothesis, they develop a theoretical set of inequalities between various options that would hold if all the items are already in stock, but not hold if they somehow magically appeared. One inequality they come up with is the following: the number of times Fred chooses color and gets red, while George chooses length and gets long-sleeves, plus the number of times Fred chooses pattern and gets striped, while George chooses length and gets short-sleeves, must be greater than or equal to the number of times Fred chooses color and gets red, while George chooses pattern and gets plaid. The researchers also compose other equivalent variations of that inequality. Otherwise, their conjecture about a supply of various shirt types with all options available doesn’t hold up mathematically.

Analyzing their collected data, sure enough realism holds up. Presented with the evidence, Fred and George’s parents confess to the psychologists about a secret scheme they have developed along with the school. Every morning they send a private note with each kid, tucked into a hidden compartment of his schoolbag, detailing a list of possible responses to each clothing option. For instance, Fred might have a note expressing the parents’ wish that if he selects “color” he should get blue that day, if he selects “pattern” he should get striped, if he selects “length” he should get “short-sleeved.” Fred would get a note with the opposite requests that day: red, plaid, and long-sleeved. The choices each day would be completely random, with 50 percent likelihood for each, just anti-correlated for each category. That way, the scheme would not arouse suspicion amongst the boys, who like to think they’re independent from their parents. Each school would check the notes each day, serving as “hidden variables,” while keeping all possible combinations in stock. The psychologists find the whole plan complicated, but at least it offers a logical explanation for an otherwise baffling set of events.

Now let’s cast the same characters in a fantasy story. Fred and George still go to different schools, still play soccer, still need to select one aspect of the soccer shirts they wear each day, and still find their shirts anti-correlated if that happen to chose the same characteristic, such as color, pattern, or sleeve-length. They are also being watching by psychologists, who record their choices. That much is the same. There are only two things different. First of all, the school only stocks one kind of shirt — plain and of a single length. Secondly, the boys are from a family of wizards. From birth, they have been magically entangled. If one chooses a clothing characteristic and the other chooses the same property, they instantly and automatically anti-coordinate. For example, if each puts on a soccer shirt and says “color,” one automatically turns blue or red (with equal likelihood) and the other instantly takes on the opposite hue. The school knows that the boys are wizards and allows them such a daily ritual just to practice their powers in a benign way. The psychologists, who don’t know about the magics, compose their “shirt options inequality,” test it against their collected data, but it just doesn’t add up. They conclude that there simply must be some kind of mysterious entanglement between the boys that evades all realistic desciptions. End of story.

Similarly, contemporary quantum experimentalists, with Bell’s inequality in hand, are well able to distinguish between the realistic (hidden variables) and entangled (standard quantum) options. However, in designing their experiments, they must make sure that there are no “loopholes” allowing for alternative explanations. Rather than be trusting, like the psychologists in our stories who believe that the boys don’t share notes or send messages to each other, circumstances would need to be such that preparation and (light-speed or less) communication would be impossible.