SEE SPOT RUN: A SIMPLE PROOF OF BELL'S THEOREM
Nick Herbert firstname.lastname@example.org
Abstract: Nick Herbert's simple proof of Bell's Non-locality Theorem is revisited. The proof is reduced to its bare bones, wherein Nick shows that quantum reality is non-local because (in a particular optical experiment) 1 + 1 = 3.
By the end of the 19th century, physicists had achieved an almost complete picture of the world comprised of two basic entities, particles and fields. The particles, such as the electron, behaved like tiny bits of matter; and the fields, such as electricity, gravity and magnetism, were spread out over vast regions and when disturbed could vibrate and travel as waves.
The particles were the source of fields and were also moved by fields. The charged electron, for instance, produces an electric field which will influence the motion of electrons and other charged particles.
The quantum revolution began in 1900 with Max Planck's discovery of the basic quantum of action and was completed by Heisenberg, Schrödinger and Dirac who formulated the basic quantum mathematics. In the new quantum picture waves acquired particle properties and particles behaved like waves. In the quantum world everything has both a particle and a wave aspect which can manifest in different measurement contexts. The wave/particle duality at first seemed paradoxical, but once physicists managed to assimilate this subtle marriage of opposites into their mathematics, quantum theory took off and soon became our deepest and most successful theory of the natural world.
As early as 1935, Erwin Schrödinger noted a peculiar property of this new theory which he called "entanglement", namely that when two quantum systems are brought together and then separated, they remain still connected, at least in the theory, by an instantaneous new kind of wholeness. Quantum entanglement, said Schrödinger, was not one but THE main difference between the new quantum theory and the old classical ways of describing nature.
The most curious feature of this new quantum connection was that it appeared only in the theory--no possible measurements on the entangled particles themselves could ever reveal directly that they were part of an instantaneous wholeness. The very theory that described this entanglement firmly predicted that this entanglement could never be used to send instantaneous signals from one particle to its entangled partner. Thus this most conspicuous feature of the quantum world seemed ironically destined to remain forever hidden from direct view and to be verified only indirectly by the stunningly accurate predictions of quantum theory as a whole.
One way of formulating the dilemma of entanglement is to invoke the notion of "locality" which holds that one particle influences another only by direct contact or via some intermediary field; and furthermore that this influence can travel no faster than light. "Non-locality", on the other hand, would mean that one particle could influence another distant particle without anything passing between them, in an instantaneous manner, faster-than-light. Non-local interactions, if they existed, would be a kind of "physics voodoo" where one particle influences another, not via a conventional force field, but simply because they touched one another sometime in the distant past.
Einstein scorned such hypothetical non-local influences as "spooky actions at a distance" and most physicists have followed his lead. No unmediated, faster-than-light influences have ever been observed in the world after all. And quantum mechanics, our best theory of the world, tells us that, despite its own inner non-locality, no non-local effects will ever be observed.
Quantum theory is manifestly non-local--on paper its parts connect with one another faster-than-light. But the quantum facts--whatever we can observe--are always local. Theory = non-local. Facts = local. What about quantum reality? Is quantum reality local or not?
We have a good theory. We have good measurements. How can we even dare to ask questions about reality? And, more to the point, how can we ever expect to get answers to such questions?
The genius of Irish physicist John Stewart Bell was to show how to ask questions concerning the nature of reality and how to obtain answers to these questions--an astonishing achievement which philosopher Abner Shimony has christened "experimental metaphysics."
What is reality? Quantum theory provides a method of calculating the results of any experiment we can imagine but it gives us no picture of what is going on to produce these results. Quantum reality would be some kind of model or picture that explains the quantum results, some story we can tell ourselves about "what is really happening" behind the scenes.
Quantum theory is non-local; quantum facts are always local. What about quantum reality?
It would seem obvious that if the facts are always local, then reality must be local as well. For why would nature need to deploy a bizarre non-local reality to produce merely local effects? The non-locality present in quantum theory could then be dismissed as "the map is not the territory" similar to the way we dismiss the International Date Line as a method of time travel even though in theory there exists a discontinuity of one full day somewhere in the middle of the Pacific. Not everything in a theory, even so successful a theory as quantum mechanics, need necessarily correspond to an element of reality.
So common sense seems to suggest that the local quantum facts that we observe in every experiment ought to be able to be modeled by an equally local underlying local quantum reality.
However, what John Bell showed, in his profound and simple proof, is that NO CONCEIVABLE LOCAL REALITY CAN UNDERLIE THE LOCAL QUANTUM FACTS. Bell proved, in short, that REALITY IS NON-LOCAL.
In other words, even though all the quantum facts are local, these facts cannot be simulated by an underlying local reality. Any reality that fits the facts must be non-local.
Furthermore, Bell's Proof does not depend on the details of quantum theory but only on simple logic applied to a few experimental facts. Thus if someday quantum theory would fail or be supplanted by a more successful theory, Bell's Theorem would still survive quantum theory's demise. Reality would remain non-local no matter what kind of theory explained these facts.
I devote the rest of this paper to a short proof of Bell's remarkable theorem.
A DIGRESSION ON THE POLARIZATION OF LIGHT AND ITS MEASUREMENT BY A POLARISCOPE NAMED SPOT
In common with all physical phenomena, light possesses both wave and particle aspects. A light wave vibrates in a direction TRANSVERSE to its direction of motion. This means that, if the light wave is headed straight towards you and you could see its vibration, you would see an oscillatory motion that might be up-and-down (Vertical Polarization), or side-to-side (Horizontal Polarization) or any angle in between.
There are certain transparent crystals (notably calcite) that possess a special axis such that light polarized along that axis travels in one direction and light polarized at right angles to that axis travels in another direction through the crystal. Thus calcite will split a beam of light into two beams--each polarized either along the crystal axis or across the crystal axis. Placing a calcite crystal on a printed page one sees two images of the page, each image formed of light completely polarized in one direction.
A calcite crystal can be used to measure the polarization of a light beam by placing a light detector in each of the two beams. These detectors will measure the relative amount of light polarized in each of two directions.
For instance, light completely polarized along the calcite axis will excite only one of the detectors. Completely unpolarized light will excite both detectors to the same extent.
As a detector we could use the human eye or a photocell but these detectors are much too insensitive to clearly demonstrate quantum effects. What we want is a detector that is so sensitive that it directly senses the particle nature of light. When a light wave falls on such a detector it creates single pulses of energy attributed to photons-- individual particles of light. In reality all light detectors operate by detecting single photons but for most detectors (such as the eye) many photons must be present before an effect is actually observed. Certain solid-state detectors and electron photomultiplier tubes are sensitive enough to respond to single photons. For our ideal polarization detector we imagine that two ideal photon detectors are placed behind the calcite crystal so that we can actually count the number of photons polarized in each of two directions.
Let's mount the calcite crystal and the two detectors in a short tube, with a little arrow on the side to indicate the direction of the crystal axis. And voila, we have a quantum-sensitive polarization detector which I call SPOT, for Single Photon Orientation Tester.
We point SPOT at a source of light and SPOT puts out a sequence of pulses which we label "1" or "0" depending on which of the two detectors fires. We calibrate SPOT by aiming it at a source of Vertically polarized light, turning the tube till one detector fires all the time, calling that detector "1" and moving SPOT's little red arrow so that it points vertically.
In this calibration mode SPOT's output looks like this:
...1, 1, 1, 1, 1, 1, 1, 1, 1...
which we interpret to mean that SPOT is looking at a beam of light made up solely of Vertically-polarized photons.
If we turn SPOT by 90 degrees while viewing the same Vertically polarized light beam, its output will look like this:
...0, 0, 0, 0, 0, 0, 0...
which we interpret to mean that SPOT is looking at a beam of photons that are polarized exactly orthogonal to the direction SPOT's arrow is pointing.
If we turn SPOT so its arrow points at 45 degrees--half-way between Vertical and Horizontal, his output will look like this:
...0, 1, 1, 0, 1, 0, 0, 0, 1...
a seemingly random 50/50 sequence of zeros and ones. This is the same output we would record if SPOT were looking at a beam of unpolarized light. The difference between an unpolarized beam and the Vertical beam is that the unpolarized beam will produce this output for every setting of SPOT's arrow while the Vertical beam only produces this 50/50 random result for a few special settings.
SPOT is a simple example of a polariscope--a device for measuring the polarization of light. Aim SPOT towards a light source and turn SPOT's arrow till the number of "1"s produced is maximum. This direction is the polarization direction of the light. If SPOT's output is 100% "1"s, as in the calibration case, then the light is "completely polarized". If SPOT's maximum output is less than 100%, then the light is said to be "partially polarized".
Most sources of light in nature are unpolarized (SPOT produces a 50/50 random sequence of "0"s and "1"s). However sunlight scattered in the atmosphere becomes partially polarized with a polarization angle and degree of polarization that varies across the sky. SPOT would be an good tool to measure the polarization pattern of scattered sunlight in the heavens but we will use SPOT instead to probe the nature of quantum reality here on Earth.
TWIN-LIGHT: A SOURCE OF QUANTUM-ENTANGLED PHOTONS
The quantum entanglement that so impressed Erwin Schrödinger is not so easy to come by in nature. Even in the laboratory, strongly entangled quantum systems are difficult to produce. Early experiments to verify Bell's Theorem by John Clauser and Alain Aspect used Calcium or Mercury vapor which when excited gave off pairs of entangled photons. But these sources were noisy (lots of unentangled photons) and inefficient (not many entangled pairs) and have now been replaced by laser-pumped non-linear crystals which, in effect, split a single blue photon into a pair of entangled red photons. Each red photon can be directed by mirrors to travel in opposite directions to distant corners of the lab where experiments can be carried out separately on each member of the pair to determine the nature of their connectedness.
Each pair of photons emerging from the crystal is produced in what I call a "twin state" --linked in a peculiar way only possible in quantum theory. Neither photon A nor photon B has a definite polarization (in the theory) but the two-photon system as a whole is identically polarized (A = B). It is not simply the case that the whole is greater than the sum of its parts; what seems to be happening here is this: in the theory, the two photons taken together have a completely definite description while each photon by itself is in a completely indefinite state.
Berkeley physicist Henry Stapp called this peculiar property of twin light "a correlated readiness to respond".
Each photon's polarization depends on the other's polarization but that polarization in turn depends on the first. This mutual dependence renders each photon polarization completely indefinite subject to the system-wide rule that should a situation ever arise (say in a measurement) that one photon acquires a definite polarization, then the other photon--no matter how far away--must instantly take on that same value for its polarization. Before they are measured, we describe the photons as not having any polarization direction at all. Each photon possesses merely, in Henry's felicitous phrase, a "readiness to respond" to whatever instrument it might encounter; likewise its partner photon possesses no definite polarization direction but has its own "readiness to respond" to any instrument it might encounter. And for photon pairs in the twin state these two readinesses are linked up in such a way that whatever happens to one photon influences the state of the other no matter how far apart they have separated.
All this in theory, of course. This instant, faster-than-light change in the description of one particle when its distant partner is measured is the very essence of quantum non-locality. So the theory that describes these twin-light photons is non-local. But, as we shall see, all measurements on these two photons give completely local results. And what about the quantum reality that produces these results? Is it local or not?
Using one source of Twin Light and two SPOT detectors I will now show how John Bell posed and answered the quantum reality question.
A SHORT PROOF OF BELL'S THEOREM
We imagine two-widely separated SPOT detectors A and B looking at a source of twin-light photons located somewhere in between. Since the results of this experiment do not depend on the exact placement of the light source, let's put Miss Alice A's SPOT detector in Anaheim, CA, Mr Bob B's SPOT detector in Baltimore, MD and situate the twin-light source in Kansas City, MO. In operation the Kansas City light source emits pairs of phase-entangled photons, sending one photon of the pair to Anaheim and the other photon to Baltimore. In practice, the detectors in John Clauser's first entangled-photon experiments at Berkeley were separated by the length of a lab table. Later experiments using fiber optics have verified quantum-entanglement over many kilometers. In theory, two photons from a twin-light source should remain tightly entangled even if separated by distances of thousands of light years.
In Anaheim, Miss A uses her SPOT detector to measure the polarization of the stream of photons emanating from the twin-light source in Kansas City. The first thing she discovers is that no matter how she sets her SPOT detector, she always gets the same result, a 50/50 random sequence of "1"s and "0"s. Miss A's beam appears to be completely unpolarized.
Likewise Mr B in Baltimore measures a 50/50 random sequence of "1"s and "0"s in his SPOT detector no matter in what direction he sets its pointer.
In particular, no matter how Mr B decides to set his SPOT detector, Miss A's result does not change at all. Her results are always 50/50 random.
If Miss A's results did change when Mr B turned his SPOT detector, this would be an example of a non-local event in the actual world. If such non-local facts actually existed, they could be used to send signals faster-than-light as well as backwards in time. Non-local facts could be used to build a time machine. Fortunately, quantum theory predicts that Miss A's photons will appear unpolarized no matter how B sets his SPOT detector, so the quantum facts in this experiment are local. Nothing observable changes faster-than-light in this experiment and, in fact, nothing changes at all. No matter what Miss A or Mr B do with their SPOT detectors, the photons they measure in both Anaheim and Baltimore always appear as random sequences of "1"s and "0"s.
However, because of the special nature of the twin-light source, whenever the red arrows of Miss A and Mr B's SPOT detectors point in the same direction, their detected sequences of "1"s and "0" are EXACTLY THE SAME. If the two observers decide to measure the same polarization direction, they find that, although the sequence of polarizations appears separately random, an identical random sequence appears at each detector. When the mutual angle between the two detectors is 0 degrees, the Match between the two sequences is 100%
Furthermore, when the two SPOT detectors are turned at a right angle to one another, the two random sequences are completely mismatched--wherever Miss A measures a "1", Mr B will record a "0". When the mutual angle between the two detectors is 90 degrees, the Match between the two sequences is 0%.
Finally, when the two SPOT detectors are set at some mutual angle that lies between 0 and 90 degrees, the Match between the two sequences will lie somewhere between these two extremes. In fact the degree of Match between the two distance is expressed as the cosine squared of the angle between the two SPOT detectors.
Zero angle = 100% Match.
Right angle = 0% Match.
Angle between Zero and Right angle = Cosine Squared (Angle) Match.
Using these few facts, and the method pioneered by John Bell, it is now easy to show that REALITY MUST BE NON-LOCAL.
I first ran into Bell's Theorem in the early 70s when my physicist friend the late Heinz Pagels showed me a paper which Bell had published in 1965 in a very obscure physics journal. "Here's something strange that should interest you, Nick," said Heinz. My first response was that Bell was mistaken and that I would prove him wrong. I would do so by framing the problem in the simplest possible way. What happened was that I ended up producing one of the simplest proofs of Nature's necessary deep non-locality.
Here's how it works.
Imagine that each random sequence that comes out of the SPOT detectors is a coded message. When both SPOT detectors are aligned, these messages are exactly the same. When the detectors are misaligned, "errors" are generated and the sequences contain a certain number of mismatches. How these "errors" might be generated is the gist of this proof.
Step One: Start by aligning both SPOT detectors. No errors are observed.
Step Two: Tilt the A detector till errors reach 25%. This occurs at a mutual misalignment of 30 degrees.
Step Three: Return A detector to its original position (100% match). Now tilt the B detector in the opposite direction till errors reach 25%. This occurs at a mutual misalignment of -30 degrees.
Step Four: Return B detector to its original position (100% match). Now tilt detector A by +30 degrees and detector B by -30 degrees so that the combined angle between them is 60 degrees.
What is now the expected mismatch between the two binary code sequences?
We assume, following John Bell's lead, that REALITY IS LOCAL.
Assuming a local reality means that, for each A photon, whatever hidden mechanism determines the output of Miss A's SPOT detector, the operation of that mechanism cannot depend on the setting of Mr B's distant detector. In other words, in a local world, any changes that occur in Miss A's coded message when she rotates her SPOT detector are caused by her actions alone.
And the same goes for Mr B. The locality assumption means that any changes that appear in the coded sequence B when Mr B rotates his SPOT detector are caused only by his actions and have nothing to do with how Miss A decided to rotate her SPOT detector.
So with this restriction in place (the assumption that reality is local), let's calculate the expected mismatch at 60 degrees.
Starting with two completely identical binary messages, if A's 30 degree turn introduces a 25% mismatch and B's 30 degree turn introduces a 25% mismatch, then the total mismatch (when both are turned) can be at most 50%. In fact the mismatch should be less than 50% because if the two errors happen to occur on the same photon, a mismatch is converted to a match.
Thus, simple arithmetic and the assumption that Reality is Local leads one to confidently predict that the code mismatch at 60 degrees must be less than 50%.
However both theory and experiment show that the mismatch at 60 degrees is 75%. The code mismatch is 25% greater than can be accounted for by independent error generation in each detector.
Therefore the locality assumption is false. Reality must be non-local.
Reduced to bare bones, the proof looks like this. A's move causes a mismatch of 1 quarter of the bits in the message. Likewise B's move causes a mismatch of 1 quarter of the total bits. Taking both moves together (and assuming a local reality) the most mismatch that can occur is 2 quarters.
1 + 1 = 2 (or less) is what a local reality predicts.
However quantum theory and quantum fact lead to an entirely different outcome. The combined moves of A and B result in a mismatch of 3 quarters of the total number of bits in the message.
1 + 1 = 3 is what nature actually produces.
No local reality can explain these facts. Therefore reality must be non-local. Furthermore this conclusion is not a supposition or speculation but a mathematical proof. John Bell found a way to ask a question about reality itself--not merely about theory or fact. And he obtained a clear and surprising answer: reality is non-local.
In this situation, a non-local reality means that what happens at Miss A's SPOT detector--whether this particular photon registers as "1" or "0"--cannot depend on causes in Anaheim alone but must somehow depend also on the setting of Mr B's distant detector in Baltimore. To explain results like these only a non-local reality will suffice.
So even though the facts are local--nothing we can measure at A changes when detector B is rotated--the nature of the strong correlations observed between A and B necessitate that the reality that underlies this experiment be non-local.
We see how Bell was able to ask questions about reality--by assuming that a local reality underlies this two-photon experiment. And we see how nature actually gave Bell an answer to his question. "No" she said, "reality is not local. I don't work like that."
But wait, there's more. Through the efforts of physicists over the last century we have become accustomed to hearing about the unspeakable strangeness of the quantum world. We know now, as Niels Bohr put it, that atoms are not things and we are not surprised to learn that the inhabitants of the microworld--its electrons, photons, quarks and neutrinos do not follow the rules of common sense. This proof of Bell, however, tells us more. For it shows that quantum weirdness is not confined to the microworld but extends up into the world of ordinary experience. The twin-light source is made from ordinary stuff (a laser, a crystal and some mirrors and lenses--stuff we can almost buy at Radio shack--and the SPOT detectors are constructed of equally ordinary crystals and silicon chips. It is true that physicists explain the operation of these objects by means of hypothetical entities called photons but this explanation plays no part whatsoever in Bell's proof. Any machine that can produce the "1+1 = 3" results will need a non-local reality to make it work. So Bell has proved that not only do we need a non-local reality to explain the behavior of tiny, invisible quantum particles, we need a non-local reality to explain the behavior of certain quite large machines that we can see and touch.
BELL'S THEOREM IN PRACTICE
It seems extremely odd that nature needs to deploy a superluminal deep reality to accomplish her everywhere local results. One might hope that if ordinary facts do indeed conceal a superluminal reality, then in certain rare circumstances, non-local reality might burst up through the surface and produce actual non-local effects, little time machines, perhaps. But if quantum theory is correct, none of these non-local effects (which are a feature of both quantum theory and quantum reality) will ever surface into quantum fact. If quantum theory is totally correct, its underlying non-localness will be forever hidden from our direct perception; the only way non-local reality will ever manifest is indirectly, via arguments such as those first devised by John Stewart Bell.
What a strange world we live in! It is as though you walked into a sleazy diner somewhere in the Midwest and peeled up the dirty Formica tabletop only to discover that the table was made of solid gold! What Bell's Theorem shows is that beneath our humdrum local world lies a hidden superluminal voodoo connected quantum reality that's necessary to make this ordinary world operate.
Because Bell's Theorem is about reality not fact, it strictly speaking belongs in the category of metaphysics not physics. So, as profound as it is, this remarkable theorem has no practical consequences whatsoever. Bell's Theorem, however, has indirectly spawned renewed enthusiasm for the phenomenon of quantum entanglement, leading to new methods for producing, manipulating and detecting entangled particles. The brand new fields of quantum cryptography and quantum computing have sprung from this Bell-spurred interest in entanglement. The strange phenomenon of quantum teleportation is also a direct result of quantum entanglement research.
The greatest impact of Bell's Theorem may lie not in new technology but in changing the way we think about our world. Einstein's relativity changed our notions of time and space. We know, for instance, that it is mistaken to believe that there is one Now, the same for all. The lessons that quantum theory has to teach are still obscure. After almost a century of contact with nature's peculiar quantum way of doing business--Heinz Pagels called it the Cosmic Code--we are still lacking a quantum world view that does justice to our new knowledge of the way the world really works. We are far from understanding what nature is trying to tell us about basic concepts such as "possibility", "actuality", "measurement", and (to cite Henry Stapp once more) "the nature of non-human-observed occurrences". When and if future thinkers succeed in formulating a better picture of quantum reality, they will owe a immense debt to John Stewart Bell, who saw further than most into the innermost mysteries of the quantum world.
BOULDER CREEK, CA
MARCH 28. 2007
NICK HERBERT is the author of "Quantum Reality", "Faster Than Light", "Elemental Mind" and a chapbook "Physics on All Fours". He invented the shortest proof of Bell's Theorem, had a hand in the quantum no-cloning theorem, recently discovered the quantum no-wedding theorem and "Nick's Theorem" which uses physics to derive limits on local psychic powers. Nick is currently obsessed with quantum tantra which he envisions as a brand new way of doing science and maintains a blog on these and other interests here.
1. Bell's theorem first surfaced in J. S. Bell, Physics 1, 195 (1965)
2. Earlier versions of this easy proof were published in N. Herbert, Am Jour Phys 43, 315 (1975) and in N. Herbert, New Scientist 111, 41 (1986) "How to be in two places at the same time". It has appeared in some textbooks as "Herbert's Proof" where I would have preferred "Herbert's Version of Bell's Proof".
3. Responding to a concern by physicist Bruce Rosenblum (the co-author of "The Quantum Enigma") that identifying only Alice's marital status in today's social climate might make the author appear sexist, I offer the following additional facts about the twin-light photon observation team: Alice A--single, two cats, living happily with a married man nearly twice her age, likes surfing and martial arts; Bob B--single, socially awkward, taking dance lessons to meet girls, living at home with his mother, can bench press 300 pounds.