Previously in series: Entangled Photons

(Note: So that this post can be read by people who haven't followed the whole series, I shall temporarily adopt some more standard and less accurate terms; for example, talking about "many worlds" instead of "decoherent blobs of amplitude".)

The legendary Bayesian, E. T. Jaynes, began his life as a physicist. In some of his writings, you can find Jaynes railing against the idea that, because we have not yet found any way to predict quantum outcomes, they must be "truly random" or "inherently random".

Sure, today you don't know how to predict quantum measurements. But how do you know, asks Jaynes, that you won't find a way to predict the process tomorrow? How can any mere experiments tell us that we'll never be able to predict something—that it is "inherently unknowable" or "truly random"?

As far I can tell, Jaynes never heard about decoherence aka Many-Worlds, which is a great pity. If you belonged to a species with a brain like a flat sheet of paper that sometimes split down its thickness, you could reasonably conclude that you'd never be able to "predict" whether you'd "end up" in the left half or the right half. Yet is this really ignorance? It is a deterministic fact that different versions of you will experience different outcomes.

But even if you don't know about Many-Worlds, there's still an excellent reply for "Why do you think you'll never be able to predict what you'll see when you measure a quantum event?" This reply is known as Bell's Theorem.

In 1935, Einstein, Podolsky, and Rosen once argued roughly as follows:

Suppose we have a pair of entangled particles, light-years or at least light-minutes apart, so that no signal can possibly travel between them over the timespan of the experiment. We can suppose these are polarized photons with opposite polarizations.

Polarized filters block some photons, and absorb others; this lets us measure a photon's polarization in a given orientation. Entangled photons (with the right kind of entanglement) are always found to be polarized in opposite directions, when you measure them in the same orientation; if a filter at a certain angle passes photon A (transmits it) then we know that a filter at the same angle will block photon B (absorb it).

Now we measure one of the photons, labeled A, and find that it is transmitted by a 0° polarized filter. Without measuring B, we can now predict with certainty that B will be absorbed by a 0° polarized filter, because A and B always have opposite polarizations when measured in the same basis.

Said EPR:

"If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity."

EPR then assumed (correctly!) that nothing which happened at A could disturb B or exert any influence on B, due to the spacelike separations of A and B. We'll take up the relativistic viewpoint again tomorrow; for now, let's just note that this assumption is correct.

If by measuring A at 0°, we can predict with certainty whether B will be absorbed or transmitted at 0°, then according to EPR this fact must be an "element of physical reality" about B. Since measuring A cannot influence B in any way, this element of reality must always have been true of B. Likewise with every other possible polarization we could measure—10°, 20°, 50°, anything. If we measured A first in the same basis, even light-years away, we could perfectly predict the result for B. So on the EPR assumptions, there must exist some "element of reality" corresponding to whether B will be transmitted or absorbed, in any orientation.

But if no one has measured A, quantum theory does not predict with certainty whether B will be transmitted or absorbed. (At least that was how it seemed in 1935.) Therefore, EPR said, there are elements of reality that exist but are not mentioned in quantum theory:

"We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete."

This is another excellent example of how seemingly impeccable philosophy can fail in the face of experimental evidence, thanks to a wrong assumption so deep you didn't even realize it was an assumption.

EPR correctly assumed Special Relativity, and then incorrectly assumed that there was only one version of you who saw A do only one thing. They assumed that the certain prediction about what you would hear from B, described the only outcome that happened at B.

In real life, if you measure A and your friend measures B, different versions of you and your friend obtain both possible outcomes. When you compare notes, the two of you always find the polarizations are opposite. This does not violate Special Relativity even in spirit, but the reason why not is the topic of tomorrow's post, not today's.

Today's post is about how, in 1964, ~~Belldandy~~ John S. Bell irrevocably shot down EPR's original argument. Not by pointing out the flaw in the EPR assumptions—Many-Worlds was not then widely known—but by describing an experiment that disproved them!

It is experimentally impossible for there to be a physical description of the entangled photons, which specifies a single fixed outcome of any polarization measurement individually performed on A or B.

This is Bell's Theorem, which rules out all "local hidden variable" interpretations of quantum mechanics. It's actually not all that complicated, as quantum physics goes!

We begin with a pair of entangled photons, which we'll name A and B. When measured in the same basis, you find that the photons always have opposite polarization—one is transmitted, one is absorbed. As for the first photon you measure, the probability of transmission or absorption seems to be 50-50.

What if you measure with polarized filters set at different angles?

Suppose that I measure A with a filter set at 0°, and find that A was transmitted. In general, if you then measure B at an angle θ to my basis, quantum theory says the probability (of my hearing that) you also saw B transmitted, equals sin² θ. E.g. if your filter was at an angle of 30° to my filter, and I saw my photon transmitted, then there's a 25% probability that you see your photon transmitted.

(Why? See "Decoherence as Projection". Some quick sanity checks: sin(0°) = 0, so if we measure at the same angles, the calculated probability is 0—we never measure at the same angle and see both photons transmitted. Similarly, sin(90°) = 1; if I see A transmitted, and you measure at an orthogonal angle, I will always hear that you saw B transmitted. sin(45°) = √(1/2), so if you measure in a diagonal basis, the probability is 50/50 for the photon to be transmitted or absorbed.)

Oh, and the initial probability of my seeing A transmitted is always 1/2. So the joint probability of seeing both photons transmitted is 1/2 * sin² θ. 1/2 probability of my seeing A transmitted, times sin² θ probability that you then see B transmitted.

And now you and I perform three statistical experiments, with large sample sizes:

(1) First, I measure A at 0° and you measure B at 20°. The photon is transmitted through both filters on 1/2 sin² (20°) = 5.8% of the occasions.

(2) Next, I measure A at 20° and you measure B at 40°. When we compare notes, we again discover that we both saw our photons pass through our filters, on 1/2 sin² (40° - 20°) = 5.8% of the occasions.

(3) Finally, I measure A at 0° and you measure B at 40°. Now the photon passes both filters on 1/2 sin² (40°) = 20.7% of occasions.

Or to say it a bit more compactly:

A transmitted 0°, B transmitted 20°: 5.8%
A transmitted 20°, B transmitted 40°: 5.8%
A transmitted 0°, B transmitted 40°: 20.7%

What's wrong with this picture?

Nothing, in real life. But on EPR assumptions, it's impossible.

On EPR assumptions, there's a fixed local tendency for any individual photon to be transmitted or absorbed by a polarizer of any given orientation, independent of any measurements performed light-years away, as the single unique outcome.

Consider experiment (2). We measure A at 20° and B at 40°, compare notes, and find we both saw our photons transmitted. Now, A was transmitted at 20°, so if you had measured B at 20°, B would certainly have been absorbed—if you measure in the same basis you must find opposite polarizations.

That is: If A had the fixed tendency to be transmitted at 20°, then B must have had a fixed tendency to be absorbed at 20°. If this rule were violated, you could have measured both photons in the 20° basis, and found that both photons had the same polarization. Given the way that entangled photons are actually produced, this would violate conservation of angular momentum.

So (under EPR assumptions) what we learn from experiment (2) can be equivalently phrased as: "B was a kind of photon that was transmitted by a 40° filter and would have been absorbed by the 20° filter." Under EPR assumptions this is logically equivalent to the actual result of experiment (2).

Now let's look again at the percentages:

B is a kind of photon that was transmitted at 20°, and would not have been transmitted at 0°: 5.8%
B is a kind of photon that was transmitted at 40°, and would not have been transmitted at 20°: 5.8%
B is a kind of photon that was transmitted at 40°, and would not have been transmitted at 0°: 20.7%

If you want to try and see the problem on your own, you can stare at the three experimental results for a while...

(Spoilers ahead.)

Consider a photon pair that gives us a positive result in experiment (3). On EPR assumptions, we now know that the B photon was inherently a type that would have been absorbed at 0°, and was in fact transmitted at 40°. (And conversely, if the B photon is of this type, experiment (3) will always give us a positive result.)

Now take a B photon from a positive experiment (3), and ask: "If instead we had measured B at 20°, would it have been transmitted, or absorbed?" Again by EPR's assumptions, there must be a definite answer to this question. We could have measured A in the 20° basis, and then had certainty of what would happen at B, without disturbing B. So there must be an "element of reality" for B's polarization at 20°.

But if B is a kind of photon that would be transmitted at 20°, then it is a kind of photon that implies a positive result in experiment (1). And if B is a kind of photon that would be absorbed at 20°, it is a kind of photon that would imply a positive result in experiment (2).

If B is a kind of photon that is transmitted at 40° and absorbed at 0°, and it is either a kind that is absorbed at 20° or a kind that is transmitted at 20°; then B must be either a kind that is absorbed at 20° and transmitted at 40°, or a kind that is transmitted at 20° and absorbed at 0°.

So, on EPR's assumptions, it's really hard to see how the same source can manufacture photon pairs that produce 5.8% positive results in experiment (1), 5.8% positive results in experiment (2), and 20.7% positive results in experiment (3). Every photon pair that produces a positive result in experiment (3) should also produce a positive result in either (1) or (2).

"Bell's inequality" is that any theory of hidden local variables implies (1) + (2) >= (3). The experimentally verified fact that (1) + (2) < (3) is a "violation of Bell's inequality". So there are no hidden local variables. QED.

And that's Bell's Theorem. See, that wasn't so horrible, was it?

But what's actually going on here?

When you measure at A, and your friend measures at B a few light-years away, different versions of you observe both possible outcomes—both possible polarizations for your photon. But the amplitude of the joint world where you both see your photons transmitted, goes as √(1/2) * sin θ where θ is the angle between your polarizers. So the squared modulus of the amplitude (which is how we get probabilities in quantum theory) goes as 1/2 sin² θ, and that's the probability for finding mutual transmission when you meet a few years later and compare notes. We'll talk tomorrow about why this doesn't violate Special Relativity.

Strengthenings of Bell's Theorem eliminate the need for statistical reasoning: You can show that local hidden variables are impossible, using only properties of individual experiments which are always true given various measurements. (Google "GHZ state" or "GHZM state".) Occasionally you also hear that someone has published a strengthened Bell's experiment in which the two particles were more distantly separated, or the particles were measured more reliably, but you get the core idea. Bell's Theorem is proven beyond a reasonable doubt. Now the physicists are tracking down unreasonable doubts, and Bell always wins.

I know I sometimes speak as if Many-Worlds is a settled issue, which it isn't academically. (If people are still arguing about it, it must not be "settled", right?) But Bell's Theorem itself is agreed-upon academically as an experimental truth. Yes, there are people discussing theoretically conceivable loopholes in the experiments done so far. But I don't think anyone out there really thinks they're going to find an experimental violation of Bell's Theorem as soon as they use a more sensitive photon detector.

What does Bell's Theorem plus its experimental verification tell us, exactly?

My favorite phrasing is one I encountered in D. M. Appleby: "Quantum mechanics is inconsistent with the classical assumption that a measurement tells us about a property previously possessed by the system."

Which is exactly right: Measurement decoheres your blob of amplitude (world), splitting it into several noninteracting blobs (worlds). This creates new indexical uncertainty—uncertainty about which of several versions of yourself you are. Learning which version you are, does not tell you a previously unknown property that was always possessed by the system. And which specific blobs (worlds) are created, depends on the physical measuring process.

It's sometimes said that Bell's Theorem rules out "local realism". Tread cautiously when you hear someone arguing against "realism". As for locality, it is, if anything, far better understood than this whole "reality" business: If life is but a dream, it is a dream that obeys Special Relativity.

It is just one particular sort of locality, and just one particular notion of which things are "real" in the sense of previously uniquely determined, which Bell's Theorem says cannot simultaneously be true.

In particular, decoherent quantum mechanics is local, and Bell's Theorem gives us no reason to believe it is not real. (It may or may not be the ultimate truth, but quantum mechanics is certainly more real than the classical hallucination of little billiard balls bopping around.)

Does Bell's Theorem prevent us from regarding the quantum description as a state of partial knowledge about something more deeply real?

At the very least, Bell's Theorem prevents us from interpreting quantum amplitudes as probability in the obvious way. You cannot point at a single configuration, with probability proportional to the squared modulus, and say, "This is what the universe looked like all along."

In fact, you cannot pick any locally specified description whatsoever of unique outcomes for quantum experiments, and say, "This is what we have partial information about."

So it certainly isn't easy to reinterpret the quantum wavefunction as an uncertain belief. You can't do it the obvious way. And I haven't heard of any non-obvious interpretation of the quantum description as partial information.

Furthermore, as I mentioned previously, it is really odd to find yourself differentiating a degree of uncertain anticipation to get physical results—the way we have to differentiate the quantum wavefunction to find out how it evolves. That's not what probabilities are for.

Thus I try to emphasize that quantum amplitudes are not possibilities, or probabilities, or degrees of uncertain belief, or expressions of ignorance, or any other species of epistemic creatures. Wavefunctions are not states of mind. It would be a very bad sign to have a fundamental physics that operated over states of mind; we know from looking at brains that minds are made of parts.

In conclusion, although Einstein, Podolsky, and Rosen presented a picture of the world that was disproven experimentally, I would still regard them as having won a moral victory: The then-common interpretation of quantum mechanics did indeed have a one person measuring at A, seeing a single outcome, and then making a certain prediction about a unique outcome at B; and this is indeed incompatible with relativity, and wrong. Though people are still arguing about that.

Part of The Quantum Physics Sequence

Next post: "Spooky Action at a Distance: The No-Communication Theorem"

Previous post: "Entangled Photons"