Bell’s Inequality

Irish physicist, John Stewart Bell, (1928-1990) was nominated for a Nobel Prize the year he died from a sudden brain hemorrhage. Nobel rules prevent the awarding of prizes to people who have died. Bell never learned of his nomination.
Someone nominated Irish physicist, John Stewart Bell, (1928-1990) for a Nobel Prize during the year he died from a sudden brain hemorrhage. Nobel rules prevent the awarding of prizes to people who have died. Bell never learned of his nomination.

UPDATE: June 17, 2016   Ali Sundermier just published the best description of quantum entanglement for non-scientists I’ve seen. Here is the link. An excellent beginner’s overview of quantum mechanics by Cathal O’Connell is in this link.

UPDATE: February 4, 2016  Here is a link to the August 2015 article in Nature, which makes the claim that the last testable loophole in Bell’s Theorem has been closed by experiments conducted by Dutch scientists. Quantum entanglement is real.

UPDATE: Nov. 14, 2014    David Kaiser has proposed an experiment to determine Is Quantum Entanglement Real?  Click the link to redirect to the Sunday Review, New York Times article. It’s a good non-technical explanation of some of the science related to Bell’s Theorem. 

John Stewart Bell‘s Theorem of 1964 followed naturally from the proof of an inequality he fashioned (now named after him), which clearly demonstrated that quantum particle behavior in experiments violated certain rules of logic. It is the most profound discovery in all science, ever, according to Henry Stapp — retired member of the Lawrence Berkeley National Laboratory, and a former associate of both Wolfgang Pauli and Werner Heisenberg. Other physicists, like the late Richard Feynman, have said Bell simply stated the obvious.

Beta Barium Borate crystals can be used to “down-convert” photons into entangled pairs.

Here is an analogy to help give an idea of just what it is in these quantum experiments that violates Bell’s Inequality: imagine two yellow tennis balls; in quantum experiments, they might be atomic particles like electrons or photons; even molecules as large as buckyballs have been tested. 

These tennis balls are created in such a way that they become entangled — that is to say, they share their properties and destinies. For the purposes of this analogy, imagine they share identical color and shape, for example.  [In fact, many entangled particles like fermions display opposite properties, as required by the Pauli exclusion principle.  But it’s easier, for now, to ignore technicalities.]

Imagine that whatever one tennis ball does, so does the other; whatever happens to one tennis ball happens to the other, instantly it turns out. The two tennis balls (the quantum particles) are entangled.

[Don’t worry, right now, about how particles become entangled in nature, or how scientists create these paired particles.  Believe simply that entanglement is a pervasive natural phenomenon that can be replicated in science labs. Entangled atomic particles play important roles, for example, in many previously misunderstood biological processes like photosynthesis, cell enzyme metabolism, animal migration, metamorphosis and olfactory sensing. Video referenceVideo reference 2.]

This cat is entangled. Whatever it does, the yarn does. Whatever happens to the yarn, happens to the cat.
This cat is entangled. Whatever it does, the yarn does. Whatever happens to the yarn, happens to the cat.

Imagine that these yellow tennis balls are shot from cannons in opposite directions. Any measurement (or disturbance) made on the ball going to the left will have the same effect on the entangled ball traveling in the opposite direction.  

So, if a test on the left-side ball allows it to pass through a color-detector on the left, then the entangled ball on the right can be thought of as having passed through an identical color-detector on the right with the exact same result. If the ball on the left gets through the color-detector, then so did the ball on the right, whether or not the color test was performed on it. If the ball on the left doesn’t get through, then neither did the ball on the right. Remember, these tennis balls are entangled.

Now imagine that cannons shoot thousands of pairs of entangled tennis balls in opposite directions, both to the left and the right. The yellow-detector on the left is calibrated to allow half of the yellow balls through. When looking for tennis balls coming through the left-hand detector, observers always see yellow balls, but only half get through.

Spin is one of the characteristics of a quantum object, much like yellow is a characteristic of a tennis ball.
Spin  describes a particle property of quantum objects like electrons—in the same way color or roundness describe tennis balls. The property is confusing, because no one believes electrons (or any other quantum objects) actually spin. The math of spin is underpinned by the complex-mathematics of spinors, which transform spin arrows into multi-dimensional objects not easy to visualize or illustrate. Look for an explanation of how spin is observed in the laboratory later in the essay. Click links for more insight.

Now, imagine doing a test for roundness on the balls shot to the right. The test is performed after the yellow-test on the left, but before any signal or light has time to travel to the balls on the right. The balls going right don’t (and can’t) learn what the detector on the left observed. The roundness-detector is set to allow three-fourths of all round tennis balls through.

When the round balls on the right are counted, three-eighths of them are passing through the roundness-detector, not three-fourths. We might speculate that the roundness-detector is acting on only the half of the balls that passed through the color-detector on the left. And we would be right. 

These balls share the same destinies, we recall. Apparently, the balls on the right learned instantly which of their entangled twins the color-detector on the left allowed to pass through, despite all efforts to prevent it.

So now do the math. One-half (the fraction of the yellow balls which passed through the left-side color-detector) multiplied by three-fourths (the fraction calibrated to pass through the right-side roundness-detector) equals three-eighths. That’s what is seen on the right—three-eighths of the round yellow tennis balls pass through the right-side roundness-detector during this fictionalized and simplified experiment.

Polarization is another characteristic of a quantum particle, much like roundness is for a tennis ball.
Polarization is a term used to describe a wave property of quantum objects like photons.  Polarizing filters are rotated in experiments to determine some of the properties of atomic particles, like spin.

According to Bell’s Inequality, twice as many balls should pass through the right side detector (three-fourths instead of three-eighths). Under the rules of classical physics (which includes relativity) communication between particles does not exceed the speed of light. 

There is no way the balls on the right can know if their entangled twins made it through the color detector on the left. The experiment is set up so that the right-side balls do not have time to receive a signal from the left-side balls. The same limitation applies to the detectors.

The question scientists have asked is: how can these balls (quantum particles)—separated by large distances—know and react instantaneously to what is happening to their entangled twins? What about the speed limit of light? Instantaneous exchange of information is not possible, according to the theory of relativity, described by Einstein.  

The French quantum physicist, Alain Aspect, suggested his way of thinking about it in the science journal, Nature (March 19, 1999).

Alain Aspect
Alain Aspect, French physicist, is best known for his work on quantum entanglement.

He wrote: The experimental violation of Bell’s inequalities confirms that a pair of entangled photons separated by hundreds of meters must be considered a single non-separable object—it is impossible to assign local physical reality to each photon.

Of course, the single non-separable object can’t have a length of hundreds of meters, either. It must have zero length for instantaneous communication between its endpoints. But it’s been well established by the distant separation of detectors in experiments done in labs around the world that the length of this non-separable quantum object can be arbitrarily long; it can span the universe.  

When calculating experimental results, it’s as if a dimension (in this case, distance or length) has gone missing. It’s eerily similar to the holographic effect of a black hole where the three-dimensional information that lives inside the event-horizon is carried on its two-dimensional surface. (See my technical comment at the end of the article.)

Schematic of physicist Alan Aspect's experimental apparatus which verified that the act of measurement influenced distant entangled calcium electrons instantaneously.
Here is a drawing of an apparatus the French physicist, Alain Aspect, designed to quickly change the angle of polarity-measurements for emitted photons. In experiments, he used the logic of Bell’s Inequalities and the speed of his switches to show that it was not possible for photons to carry specific (or unique) polarity-angles until after they were measured by the polarization detectors.  Once measured, Alain showed that the new, narrowly-defined polarity states of his photons always propagated to their distant entangled twins, instantly.  Video link here.

Another way physicists have wrestled with the violations of Bell’s Inequality is by postulating the concept of superposition. Superposition is a concept that flows naturally from the linear algebra used to do the calculations, which suggests that quantum particles exist in all their possible states and locations at the same time, until they are measured. 

Measurement forces wave-particles to “collapse” into one particular state, like a definite position. But some physicists, like Roger Penrose, have asked: how do all the super-positioned particles and states that weren’t measured know instantaneously to disappear?

Superposition, a fundamental principle of quantum mechanics, has become yet another topic physicists puzzle over. They agree on the math of superposition and the wave-particle collapse during measurement but don’t agree on what a measurement is or the nature of the underlying reality. Many, like Richard Feynman, believe the underlying reality is probably unknowable.

Quantum behavior is non-intuitive and mysterious. It violates the traditional ideas of what makes sense. As soon as certainty is established for one measurement, other measurements, made earlier, become uncertain.  

It’s like a game of whack-a-mole. The location of the mole whacked with a mallet becomes certain as soon as it is struck, but the other moles scurry away only to pop up and down in random holes so fast that no one is sure where or when they really are.

Physicists have yet to explain the many quantum phenomena encountered in their labs except to throw-up their hands to say—paraphrasing Feynman—it is the way it is, and the way it is, well, the experiments make it obvious.

Richard Feynman (1918-1988) downplayed Bell’s Inequality because, he said, it simply pointed out what was already obvious from experiments.

But it’s not obvious, at least not to me and, apparently, many others more knowledgeable than myself. Violations of Bell’s Inequality confound our understanding of quantum mechanics and the world in which it lives. A consequence has been that at least a few scientists seem ready to believe that one, perhaps two, or maybe all four, of the following statements are false:

1) logic is reliable and enables clear thinking about all physical phenomenon;

2) the Universe exists independently of any conscious observer;

3) information does not travel faster than light.

4) a model can be imagined to explain quantum phenomenon. 

I feel wonder when the idea sinks into my mind that one of these three seemingly self-evident and presumably true statements might be  false—possibly all four—because repeated quantum experiments suggest they must be. Why don’t we hear more about it on TV and radio?

Quantum mechanics (1)
Some scientists think non-physicists cannot grasp quantum mechanics. This little girl disagrees.

The reason could be that the terrain of quantum physics is unfamiliar territory for a lot of folks. Unless one is a graduate student in physics—well, many scientists don’t think non-physicists can even grasp the concepts. They might be right.

So, a lot is being said, all right, but it’s being said behind the closed doors of physics labs around the world. The subtleties of quantum theory don’t seem to suit the aesthetics of contemporary public media, so little information gets shared with ordinary people. Despite the efforts of enthusiastic scientists—like Brian CoxSean M. CarrollNeil deGrasse Tyson and Brian Greene—to serve up tasty, digestible, bite-size chunks of quantum mechanics to the public, viewer ratings sometimes fall flat.

When physicists say something strange is happening in quantum experiments that can’t be explained by traditional methods, doesn’t it deserve our attention? Doesn’t everyone want to try to understand what is going on and strive for insights?  I’m not a physicist and never will be, but I want to know. 

Even me—a mere science-hobbyist who designed machinery back in the day—wants to know. I want to understand. What is it that will make sense of the universe and the quantum realm in which it rests?  It seems, sometimes, that a satisfying answer is always just outside our grasp.

Here is a concise statement of Bell’s Theorem from the article in Wikipedia—modified to make it easier to understand: No physical theory about the nature of quantum particles which ignores instantaneous action-at-a-distance can ever reproduce all the predictions about quantum behavior discovered in experiments.

Familiarity with concepts like wave polarization and particle-spin can help demystify some aspects of quantum mechanics. One aspect that can’t be demystified: in experiments quantum objects display the properties of both waves and particles.

To understand the experiments that led to the unsettling knowledge that quantum mechanics—as useful and predictive as it is—does indeed violate Bell’s proven Inequality, it is helpful not only to have a solid background in mathematics but also to understand ideas involving the polarization of light and—when applied to quantum objects like electrons and other sub-atomic particles—the idea of spin.  Taken together, these concepts are somewhat analogous to the properties of color and roundness in the imaginary experiment described above.  

This article is probably not the place to explain wave polarization and particle spin, because they would take up too much space, and I don’t understand the concepts all that well, anyway.  (No one does.) 

But, basically, it’s like this: if a beam of electrons, for example, is split into two and then recombined on a display screen, an interference pattern presents itself. If one of the beams was originally passed through a polarizer, and if that polarizer is later rotated a full turn (that is, 360°), the interference pattern on the screen will reverse itself. If the polarizer-filter is rotated another full turn, the interference pattern will reverse again to what it was at the start of the experiment. 

So it takes two spins of the polarizer-filter to get back the original interference pattern on the display screen, which means the electrons themselves must have an intrinsic “one-half” spin. All so-called matter particles like electrons, protons, and neutrons (called fermions by physicists) have one-half spin.

Yes, it’s weird. Anyway, people can read-up on the latest ideas by clicking this link. It’s fun stuff. For those familiar with QM (quantum mechanics), I’ve added a technical comment in the comments section below my essay.

Otherwise, my analogy is useful enough, probably. In actual experiments, physicists measure more than two properties, I’m told. Most common are angular momentum vectors, which are called spin orientations. We can think of these properties as color, shape or hardness, if we want to make explanations more familiar—as long as we remember that each quality is binary; color is white or black; shape is round or square; hardness is soft or hard.

Crystals can be used to “down-convert” photons into  entangled pairs.

Spin orientations are binary too—the vectors point in one of two possible directions. And it should be kept in mind that each entangled particle in a pair of fermions always has at least one property that measures opposite to that of its entangled partner.  

The earlier analogy might be improved if we imagine pairs of entangled tennis balls where one ball is black, the other white; one is round, the other square; we could add a third quality where one ball is hard, the other soft. Most important, the shape and color and hardness of the balls are imparted by the detectors themselves during measurement, not before.

Before measurement, concepts like color or shape (or spin or polarity) can have no meaning; the balls carry every possible color and shape (and hardness) but don’t display any of these qualities until a measurement is made. Concepts like these keep quantum physicists awake at night, some say.

Anyway, my earlier, simpler analogy gets the main ideas across, hopefully. And a couple of the nuances of entanglement can be found within it. I’ve added an easy to understand description of Bell’s Inequality and what it means to the end of this article. Here are two additional links with more depth: CHSH Inequality; Bell Test Experiments.

A carbord cut-out of a cat imaged by photons that never went through the cut-out itself. Credit: Gabriela Barreto Lemos
This cardboard cut-out of a cat was imaged by entangled photons. Lower energy photons interacted with the cut-out while their higher energy entangled twins interacted with the camera to create the picture.
Credit: Gabriela Barreto Lemos

In the meantime, scientists at the Austrian Academy of Sciences in Vienna recently demonstrated that entanglement can be used as a tool to photograph delicate objects that would otherwise be disturbed or damaged by high energy photons (light). They entangled photons of different energies (different colors).

They took photographs of objects using low energy photons, but sent their higher energy entangled twins to the camera where their higher energies enabled them to be recorded. New technologies like these involving the strange behavior of quantum particles are in development, it seems, and promise to transform our world in coming decades.

Perhaps entanglement will provide a path to faster-than-light communication, which is necessary to signal distant space-craft in real time. Most scientists say, no, it can’t be done, but there might be ways to engineer around the difficulties; technology may become available to create an illusion of instantaneous communication that is actually useful. Click on the link in this paragraph to learn more.

Non-scientists don’t have to know everything about the individual trees to know they are walking in a quantum forest. One reason for writing this essay is to encourage people to think and wonder about the forest and what it means to live in and experience it. The truth is, the trees (particles at atomic scales) in the quantum forest seem to violate some of the rules of the forest (classical physics). They have a spooky quality, as Einstein famously put it.

remu warrior night scene 3
The quantum forest is a spooky place, Einstein said. 

Trees that aren’t there when no one is looking suddenly appear when someone is looking. Trees growing in one place seem to be growing in other places no one expected. A tree blows one way in the wind, and someone notices a tree at the other end of the forest—where there is no wind—blowing in the opposite direction. As of right now, no one has offered an explaination that doesn’t seem to lead to paradoxes and contradictions when closely examined by specialists.

Henry Stapp,
Henry Stapp,

John Stewart Bell proved that the trees in the quantum forest violate the laws of nature and logic. It makes me wonder whether we will ever know anything at all we can fully trust about the fundamental underlying essence of reality.

Some scientists, like Henry Stapp (retired), have proposed that brains may enable processes like choice and experiences like consciousness through the mechanism of quantum interactions. Stuart Hameroff and Roger Penrose have proposed a quantum mechanism for consciousness they call Orch Or.

Others, like Wolfgang Pauli and C. G. Jung, have gone further—asking, when they were alive, if the non-causal coordination of some process resembling what we now understand to be entanglement might provide an explanation for the seeming synchronicity of some psychic processes—an arena of inquiry a few governments are rumored to have already incorporated (to great effect) into their intelligence gathering tool kits.

Either in a future rewrite of this article or in a new one, I plan to speculate about how quantum mechanics might influence human thought and consciousness—and, possibly, vice-versa. I may add more content on this subject in the coming weeks, if new ideas worthy of publication present themselves. Check back from time to time.

Billy Lee

P.S.  A simplified version of Bell’s Inequality might say that for things described by traits A, B, and C, it is always true that A, not B; plus B, not C; is greater than or equal to: A, not C.  When applied to a room full of people, the inequality might read as follows: tall, not male; plus male, not blonde; is greater than or equal to: tall, not blonde. People have tried every collection of traits and quantities imaginable. The inequality is always true, never false; except for quantum objects.

wave equation schrodinger
Schrodinger’s Wave Equation describes how the quantum state of a physical system changes with time. It can be used to calculate quantized properties and probability distributions of quantum objects.

One way to think about it: all the “not” quantities are, in some sense, uncertain in quantum experiments, which wrecks the inequality. That is to say, as soon as we measure “A” (for example) ,”not B” becomes uncertain. If we measure “not B”, on the other hand, “A” becomes uncertain.This introduction of uncertainties into quantities that were—before measurement—seemingly certain and fixed doesn’t seem to occur in non-quantum collections where individual objects are big enough to make uncertainties unnoticeable. It’s called the uncertainty principle and is fundamental to physics.

But what’s really strange: when working with entangled particles, certainty about one particle implies certainty about its entangled twin; predicted experimental results are precise and never fail. Oddly, once entangled quantum particles are measured, the results, though certain, change from those expected by classical theory to those predicted by quantum mechanics. They violate Bell’s Inequality and our common sense about how things should work. 

Worse: Bell’s Theorem seems to imply that no one will ever be able to construct a physical model of quantum mechanics to explain the results of quantum experiments.  No “hidden variables” exist which, if only we knew them, would explain everything.  Another way to say it is this: the underlying reality of quantum mechanics is unknowable.  [A technical comment about the mystery of QM is included in the comments section.]

Billy Lee

One Reply to “Bell’s Inequality”

  1. This comment is for people who are familiar with the experiments involving polarized particle-waves, beam-splitters (and/or polarizing filters), and the Bell’s Inequality violations.

    The mystery of QM, it seems to me, lies in the difference between 1) measuring areas by the methods of classical physics (where areas are compared to determine the relative probabilities that objects — like tennis balls — will pass through or not), and 2) the method of QM, which creates probability amplitudes for sub-atomic particles based on a one-dimensional angle of opportunity in the measuring device.

    Instead of using the two-dimensional area as in classical physics, QM uses angles, but even then, not the angles themselves, but the sines and cosines of those angles to determine absolute values. QM squares these one-dimensional — essentially pure numbers — to calculate the probability amplitudes, which are known to violate Bell’s Inequality. It can’t get any more abstract.

    The method reminds me of the recent analysis by “black-hole” physicists who point out that the information content within any three-dimensional space can be projected onto the two-dimensional surface area of that space. It’s referred to as the “hologram” effect. In the case of quantum measurements involving entangled particles, it seems like, once again, a dimension drops out of the geometry necessary to calculate probabilities. And the distance between entangled particles behaves as if it is zero; the state of one particle is transferred to its far away entangled twin instantaneously.

    What possibility is there to construct a model we can understand of the underlying reality of QM, when the physically meaningful handles we need don’t seem to be reflected in the analysis of experimental results?

    It’s true that Richard Feynman developed a model whereby we can imagine (and calculate) every possible path and phase angle of a moving subatomic particle and add them together to compute the probabilities of the particle’s path. The model is in some ways not helpful, because it fails to simplify our understanding of what is going on. Imagining that a particle is everywhere all the time until we make a measurement is complicated and counterintuitive. We seem to lack not only the sense organs to detect the quantum world, but also the brain algorithms to enable any natural intuition about it. Perhaps we really do live on the holographic surface of a higher dimensional reality.

    So far, we have constructed measurement devices which appear to interact with quantum phenomenon, and we’ve devised mathematical schemes to help us make accurate predictions about their interactions with our machinery. We’ve made some first steps toward harnessing the fundamental powers we believe lie at the heart of QM in fields like electronics, computing, imaging and energy creation. Are more advances are on the way? An exciting quantum future seems certain.

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