trentmkelly/LessWrong-43k huggingface dataset:
Decoherence is Falsifiable and Testable
by Eliezer Yudkowsky — 2008-05-07T07:54:34.000Z

The words “falsifiable” and “testable” are sometimes used interchangeably, which imprecision is the price of speaking in English. There are two different probability-theoretic qualities I wish to discuss here, and I will refer to one as “falsifiable” and the other as “testable” because it seems like the best fit.

As for the math, it begins, as so many things do, with:

P(Ai|B)=P(B|Ai)P(Ai)ΣjP(B|Aj)P(Aj).

This is Bayes’s Theorem. I own at least two distinct items of clothing printed with this theorem, so it must be important.

To review quickly, B here refers to an item of evidence, Ai is some hypothesis under consideration, and the Aj are competing, mutually exclusive hypotheses. The expression P(B|Ai) means “the probability of seeing B, if hypothesis Ai is true” and P(Ai|B) means “the probability hypothesis Ai is true, if we see B.”

The mathematical phenomenon that I will call “falsifiability” is the scientifically desirable property of a hypothesis that it should concentrate its probability mass into preferred outcomes, which implies that it must also assign low probability to some un-preferred outcomes; probabilities must sum to 1 and there is only so much probability to go around. Ideally there should be possible observations which would drive down the hypothesis’s probability to nearly zero: There should be things the hypothesis cannot explain, conceivable experimental results with which the theory is not compatible. A theory that can explain everything prohibits nothing, and so gives us no advice about what to expect.

P(Ai|B)=P(B|Ai)P(Ai)ΣjP(B|Aj)P(Aj).

In terms of Bayes’s Theorem, if there is at least some observation B that the hypothesis Ai can’t explain, i.e., P(B|Ai) is tiny, then the numerator P(B|Ai)P(Ai) will also be tiny, and likewise the posterior probability P(Ai|B). Updating on having seen the impossible result B has driven the probability of Ai down to nearly zero. A theory that refuses to make itself vulnerable in this way will need to spread its probability widely, so that it has no holes; it will not be able to strongly concentrate probability into a few preferred outcomes; it will not be able to offer precise advice.

Thus is the rule of science derived in probability theory.

As depicted here, “falsifiability” is something you evaluate by looking at a singlehypothesis, asking, “How narrowly does it concentrate its probability distribution over pos... ,,,,,,,,,,,,,,,,,,,,,,
Decoherence is Simple
by Eliezer Yudkowsky — 2008-05-06T07:44:04.000Z

An epistle to the physicists:

When I was but a little lad, my father, a PhD physicist, warned me sternly against meddling in the affairs of physicists; he said that it was hopeless to try to comprehend physics without the formal math. Period. No escape clauses. But I had read in Feynman’s popular books that if you really understood physics, you ought to be able to explain it to a nonphysicist. I believed Feynman instead of my father, because Feynman had won the Nobel Prize and my father had not.

It was not until later—when I was reading the Feynman Lectures, in fact— that I realized that my father had given me the simple and honest truth. No math = no physics.

By vocation I am a Bayesian, not a physicist. Yet although I was raised not to meddle in the affairs of physicists, my hand has been forced by the occasional gross misuse of three terms: simple, falsifiable, and testable.

The foregoing introduction is so that you don’t laugh, and say, “Of course I know what those words mean!” There is math here. What follows will be a restatement of the points in Belief in the Implied Invisible, as they apply to quantum physics.

Let’s begin with the remark that started me down this whole avenue, of which I have seen several versions; paraphrased, it runs:

The many-worlds interpretation of quantum mechanics postulates that there are vast numbers of other worlds, existing alongside our own. Occam’s Razor says we should not multiply entities unnec­essarily.

Now it must be said, in all fairness, that those who say this will usually also confess:

But this is not a universally accepted application of Occam’s Razor; some say that Occam’s Razor should apply to the laws governing the model, not the number of objects inside the model.

So it is good that we are all acknowledging the contrary arguments, and telling both sides of the story—

But suppose you had to calculate the simplicity of a theory.

The original formulation of William of Ockham stated:

Lex parsimoniae: Entia non sunt multiplicanda praeter necessitatem.

“The law of parsimony: Entities should not be multiplied beyond necessity.”

But this is qualitative advice. It is not enough to say whether one theory seems more simple, or seems more complex, than another—you have to assign a number; and the number has to be meaningful, you can’t just make it up. Crossing this gap is like the difference between being able to eyeball which things... ,,,,,,,,
Spooky Action at a Distance: The No-Communication Theorem
by Eliezer Yudkowsky — 2008-05-05T02:43:09.000Z

Previously in series: Bell's Theorem: No EPR "Reality"

When you have a pair of entangled particles, such as oppositely polarized photons, one particle seems to somehow "know" the result of distant measurements on the other particle. If you measure photon A to be polarized at 0°, photon B somehow immediately knows that it should have the opposite polarization of 90°.

Einstein famously called this "spukhafte Fernwirkung" or "spooky action at a distance". Einstein didn't know about decoherence, so it seemed spooky to him.

Though, to be fair, Einstein knew perfectly well that the universe couldn't really be "spooky". It was a then-popular interpretation of QM that Einstein was calling "spooky", not the universe itself.

Let us first consider how entangled particles look, if you don't know about decoherence—the reason why Einstein called it "spooky":

Suppose we've got oppositely polarized photons A and B, and you're about to measure B in the 20° basis. Your probability of seeing B transmitted by the filter (or absorbed) is 50%.

But wait! Before you measure B, I suddenly measure A in the 0° basis, and the A photon is transmitted! Now, apparently, the probability that you'll see B transmitted is 11.6%. Something has changed! And even if the photons are light-years away, spacelike separated, the change still occurs.

You might try to reply:

"No, nothing has changed—measuring the A photon has told you something about the B photon, you have gained knowledge, you have carried out an inference about a distant object, but no physical influence travels faster-than-light.

"Suppose I put two index cards into an envelope, one marked '+' and one marked '-'. Now I give one envelope to you, and one envelope to a friend of yours, and you get in a spaceship and travel a few light-years away from each other, and then you open your envelope and see '+'. At once you know that your friend is holding the envelope marked '-', but this doesn't mean the envelope's content has changed faster than the speed of light.

"You are committing a Mind Projection Fallacy; the envelope's content is constant, only your local beliefs about distant referents change."

Bell's Theorem, covered yesterday, shows that this reply fails. It is not possible that each photon has an unknown but fixed individual tendency to ...
Bell's Theorem: No EPR "Reality"
by Eliezer Yudkowsky — 2008-05-04T04:44:54.000Z

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 tr...
Entangled Photons
by Eliezer Yudkowsky — 2008-05-03T07:20:50.000Z

Previously in series: Decoherence as Projection

Today we shall analyze the phenomenon of "entangled particles". We're going to make heavy use of polarized photons here, so you'd better have read yesterday's post.

If a particle at rest decays into two other particles, their net momentum must add up to 0. The two new particles may have amplitudes to head off in all directions, but in each joint configuration, the directions will be opposite.

By a similar method you can produce two entangled photons which head off in opposite directions, and are guaranteed to be polarized oppositely (at right angles to each other), but with a 50% prior probability of going through any given polarized filter.

You might think that this would involve amplitudes over a continuous spectrum of opposite configurations—an amplitude for photon A to be polarized at 0° and for photon B to be polarized at 90°, an amplitude for A to be 1° polarized and for B to be 91° polarized, etc. But in fact it's possible to describe the quantum state "unknown but opposite polarizations" much more compactly.

First, note that the two photons are heading off in opposite directions. This justifies calling one photon A and one photon B; they aren't likely to get their identities mixed up.

As with yesterday, the polarization state (1 ; 0) is what passes a 90° filter. The polarization state (0 ; 1) is what passes a 0° filter. (1 ; 0) is polarized up-down, (0 ; 1) is polarized left-right.

If A is in the polarization state (1 ; 0), we'll write that as A=(1 ; 0).

If A=(1 ; 0) and B=(0 ; 1), we'll write that as

[ A=(1 ; 0) ∧ B=(0 ; 1) ]

The state for "unknown opposite polarization" can be written as:

√(1/2) * ( [ A=(1 ; 0) ∧ B=(0 ; 1) ] - [ A=(0 ; 1) ∧ B=(1; 0) ] )

Note that both terms are being multiplied by the square root of 1/2. This ensures that the squared modulus of both terms sums to 1. Also, don't overlook the minus sign in the center, we'll need it.

If you measure the A photon's polarization in the up-down/left-right basis, the result is pretty straightforward. Your measurement decoheres the entanglement, creating one evolution out of the A=(1 ; 0) ∧ B=(0 ; 1) configuration, and a second, noninteracting evolution out of the A=(0 ; 1) ∧ B=(1; 0) configuration.

If you find that the A photon is polarized...
Decoherence as Projection
by Eliezer Yudkowsky — 2008-05-02T06:32:17.000Z

Previously in series: The Born Probabilities

In "The So-Called Heisenberg Uncertainty Principle" we got a look at how decoherence can affect the apparent surface properties of objects: By measuring whether a particle is to the left or right of a dividing line, you can decohere the part of the amplitude distribution on the left with the part on the right. Separating the amplitude distribution into two parts affects its future evolution (within each component) because the two components can no longer interfere with each other.

Yet there are more subtle ways to take apart amplitude distributions than by splitting the position basis down the middle. And by exploring this, we rise further up the rabbit hole.

(Remember, the classical world is Wonderland, the quantum world is reality. So when you get deeper into quantum physics, you are going up the rabbit hole, not down the rabbit hole.)

Light has a certain quantum property called "polarization". Of course, all known physical properties are "quantum properties", but in this case I mean that polarization neatly exhibits fundamental quantum characteristics. I mention this, because polarization is often considered part of "classical" optics. Why? Because the quantum nature of polarization is so simple that it was accidentally worked out as part of classical mechanics, back when light was thought to be a wave.

(Nobody tell the marketers, though, or we'll be wearing "quantum sunglasses".)

I don't usually begin by discussing the astronomically high-level phenomena of macroscopic physics, but in this case, I think it will be helpful to begin with a human-world example...

I hand you two little sheets of semi-transparent material, looking perhaps like dark plastic, with small arrows drawn in marker along the sides. When you hold up one of the sheets in front of you, the scene through it is darker—it blocks some of the light.

Now you hold up the second sheet in front of the first sheet...

When the two arrows are aligned, pointing in the same direction, the scene is no darker than before—that is, the two sheets in series block the same amount of light as the first sheet alone.

But as you rotate the second sheet, so that the two arrows point in increasingly different directions, the world seen through both sheets grows darker. When the arrows are at 45° ...
The Born Probabilities
by Eliezer Yudkowsky — 2008-05-01T05:50:53.000Z

Previously in series: Decoherence is Pointless
Followup to: Where Experience Confuses Physicists

One serious mystery of decoherence is where the Born probabilities come from, or even what they are probabilities of. What does the integral over the squared modulus of the amplitude density have to do with anything?

This was discussed by analogy in "Where Experience Confuses Physicists", and I won't repeat arguments already covered there. I will, however, try to convey exactly what the puzzle is, in the real framework of quantum mechanics.

A professor teaching undergraduates might say: "The probability of finding a particle in a particular position is given by the squared modulus of the amplitude at that position."

This is oversimplified in several ways.

First, for continuous variables like position, amplitude is a density, not a point mass. You integrate over it. The integral over a single point is zero.

(Historical note: If "observing a particle's position" invoked a mysterious event that squeezed the amplitude distribution down to a delta point, or flattened it in one subspace, this would give us a different future amplitude distribution from what decoherence would predict. All interpretations of QM that involve quantum systems jumping into a point/flat state, which are both testable and have been tested, have been falsified. The universe does not have a "classical mode" to jump into; it's all amplitudes, all the time.)

Second, a single observed particle doesn't have an amplitude distribution. Rather the system containing yourself, plus the particle, plus the rest of the universe, may approximately factor into the multiplicative product of (1) a sub-distribution over the particle position and (2) a sub-distribution over the rest of the universe. Or rather, the particular blob of amplitude that you happen to be in, can factor that way.

So what could it mean, to associate a "subjective probability" with a component of one factor of a combined amplitude distribution that happens to factorize?

Recall the physics for:

(Human-BLANK * Sensor-BLANK) * (Atom-LEFT + Atom-RIGHT)
=>
(Human-LEFT * Sensor-LEFT * Atom-LEFT) + (Human-RIGHT * Sensor-RIGHT * Atom-RIGHT)

Think of the whole process as reflecting the good-old-fashioned distributive rule of algebra. The initial ...
Decoherent Essences
by Eliezer Yudkowsky — 2008-04-30T06:32:11.000Z

Followup to: Decoherence is Pointless

In "Decoherence is Pointless", we talked about quantum states such as

(Human-BLANK) * ((Sensor-LEFT * Atom-LEFT) + (Sensor-RIGHT * Atom-RIGHT))

which describes the evolution of a quantum system just after a sensor has measured an atom, and right before a human has looked at the sensor—or before the human has interacted gravitationally with the sensor, for that matter. (It doesn't take much interaction to decohere objects the size of a human.)

But this is only one way of looking at the amplitude distribution—a way that makes it easy to see objects like humans, sensors, and atoms. There are other ways of looking at this amplitude distribution—different choices of basis—that will make the decoherence less obvious.

Suppose that you have the "entangled" (non-independent) state:

(Sensor-LEFT * Atom-LEFT) + (Sensor-RIGHT * Atom-RIGHT)

considering now only the sensor and the atom.

This state looks nicely diagonalized—separated into two distinct blobs. But by linearity, we can take apart a quantum amplitude distribution any way we like, and get the same laws of physics back out. So in a different basis, we might end up writing (Sensor-LEFT * Atom-LEFT) as:

(0.5(Sensor-LEFT + Sensor-RIGHT) + 0.5(Sensor-LEFT - Sensor-RIGHT)) * (0.5(Atom-RIGHT + Atom-LEFT) - 0.5(Atom-RIGHT - Atom-LEFT))

(Don't laugh. There are legitimate reasons for physicists to reformulate their quantum representations in weird ways.)

The result works out the same, of course. But if you view the entangled state in a basis made up of linearly independent components like (Sensor-LEFT - Sensor-RIGHT) and (Atom-RIGHT - Atom-LEFT), you see a differently shaped amplitude distribution, and it may not look like the blobs are separated.

Oh noes! The decoherence has disappeared!

...or that's the source of a huge academic literature asking, "Doesn't the decoherence interpretation require us to choose a preferred basis?"

To which the short answer is: Choosing a basis is an isomorphism; it doesn't change any experimental predictions. Decoherence is an experimentally visible phenomenon or we would not have to protect quantum computers from it. You can't protect a quantum computer by "choosing the right basis" instead of using environmental shielding. Likewise, looking at splittin...
Decoherence is Pointless
by Eliezer Yudkowsky — 2008-04-29T06:38:54.000Z

Previously in series: On Being Decoherent

Yesterday's post argued that continuity of decoherence is no bar to accepting it as an explanation for our experienced universe, insofar as it is a physicist's responsibility to explain it. This is a good thing, because the equations say decoherence is continuous, and the equations get the final word.

Now let us consider the continuity of decoherence in greater detail...

On Being Decoherent talked about the decoherence process,

(Human-BLANK) * (Sensor-BLANK) * (Atom-LEFT + Atom-RIGHT)
=>
(Human-BLANK) * ((Sensor-LEFT * Atom-LEFT) + (Sensor-RIGHT * Atom-RIGHT))
=>
(Human-LEFT * Sensor-LEFT * Atom-LEFT) + (Human-RIGHT * Sensor-RIGHT * Atom-RIGHT)

At the end of this process, it may be that your brain in LEFT and your brain in RIGHT are, in a technical sense, communicating—that they have intersecting, interfering amplitude flows.

But the amplitude involved in this process, is the amplitude for a brain (plus all entangled particles) to leap into the other brain's state. This influence may, in a quantitative sense, exist; but it's exponentially tinier than the gravitational influence upon your brain of a mouse sneezing on Pluto.

By the same token, decoherence always entangles you with a blob of amplitude density, not a point mass of amplitude. A point mass of amplitude would be a discontinuous amplitude distribution, hence unphysical. The distribution can be very narrow, very sharp—even exponentially narrow—but it can't actually be pointed (nondifferentiable), let alone a point mass.

Decoherence, you might say, is pointless.

If a measuring instrument is sensitive enough to distinguish 10 positions with 10 separate displays on a little LCD screen, it will decohere the amplitude into at least 10 parts, almost entirely noninteracting. In all probability, the instrument is physically quite a bit more sensitive (in terms of evolving into different configurations) than what it shows on screen. You would find experimentally that the particle was being decohered (with consequences for momentum, etc.) more than the instrument was designed to measure from a human standpoint.

But there is no such thing as infinite sensitivity in a continuous quantum physics: If you start with blobs of amplitude density, you ...
The Conscious Sorites Paradox
by Eliezer Yudkowsky — 2008-04-28T02:58:35.000Z

Followup to: On Being Decoherent

Decoherence is implicit in quantum physics, not an extra postulate on top of it, and quantum physics is continuous. Thus, "decoherence" is not an all-or-nothing phenomenon—there's no sharp cutoff point. Given two blobs, there's a quantitative amount of amplitude that can flow into identical configurations between them. This quantum interference diminishes down to an exponentially tiny infinitesimal as the two blobs separate in configuration space.

Asking exactly when decoherence takes place, in this continuous process, is like asking when, if you keep removing grains of sand from a pile, it stops being a "heap".

The sand-heap dilemma is known as the Sorites Paradox, after the Greek soros, for heap. It is attributed to Eubulides of Miletus, in the 4th century BCE. The moral I draw from this very ancient tale: If you try to draw sharp lines in a continuous process and you end up looking silly, it's your own darn fault.

(Incidentally, I once posed the Sorites Paradox to Marcello Herreshoff, who hadn't previously heard of it; and Marcello answered without the slightest hesitation, "If you remove all the sand, what's left is a 'heap of zero grains'." Now that's a computer scientist.)

Ah, but what about when people become decoherent? What of the Conscious Sorites Paradox?

What about the case where two blobs of amplitude containing people are interacting, but only somewhat - so that there is visibly a degree of causal influence, and visibly a degree of causal independence?

Okay, this interval may work out to less than the Planck time for objects the size of a human brain. But I see that as no excuse to evade the question. In principle we could build a brain that would make the interval longer.

Shouldn't there be some definite fact of the matter as to when one person becomes two people?

Some folks out there would just say "No". I suspect Daniel Dennett would just say "No". Personally, I wish I could just say "No", but I'm not that advanced yet. I haven't yet devised a way to express my appreciation of the orderliness of the universe, which doesn't involve counting people in orderly states as compared to disorderly states.

Yet if you insist on an objective population count, for whatever reason, you have Soritic problems whether or not you delve into quantu...