*This post is part of the **Quantum Physics Sequence.
*

**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 splitting humans from another angle won't make their decoherence go away.

But this is an issue that you're bound to encounter if you pursue quantum mechanics, especially if you talk to anyone from the Old School, and so it may be worth expanding on this reply.

After all, if the short answer is as obvious as I've made it sound, then why, oh why, would anyone ever think you *could* eliminate an experimentally visible phenomenon like decoherence, by isomorphically reformulating the mathematical representation of quantum physics?

That's
a bit difficult to describe in one mere blog post. It has to do with
history. You know the warning I gave about dragging *history* into explanations of QM... so consider yourself warned: Quantum mechanics is simpler than the arguments we have about quantum mechanics. But here, then, is the history:

Once upon a time,

Long ago and far away, back when the theory of quantum mechanics was first being developed,

*No one had ever thought of decoherence*. The question of why a human researcher only saw one thing at a time, was a Great Mystery with no obvious answer.

You had to *interpret* quantum mechanics to get an answer back out of it. Like reading meanings into an oracle. And there were different, competing interpretations. In one popular interpretation, when you "measured" a system, the Quantum Spaghetti Monster would eat all but one blob of amplitude, at some unspecified time that was exactly right to give you whatever experimental result you actually saw.

Needless to say, this "interpretation" wasn't *in* the quantum equations. You had to add in the *extra* postulate of a Quantum Spaghetti Monster *on top*, *additionally* to the differential equations you had fixed experimentally for describing how an amplitude distribution evolved.

Along came Hugh Everett and said, "Hey, maybe the formalism just describes the way the universe *is,* without any need to 'interpret' it."

But people were so used
to adding *extra* postulates to interpret quantum mechanics, and so
*unused *to the idea of amplitude distributions as real, that they
couldn't see this new "interpretation" as anything *except* an additional Decoherence
Postulate which said:

"When clouds of amplitude become separated enough, the Quantum Spaghetti Monster steps in and *creates a new world* corresponding to each cloud of amplitude."

So then they asked:

"Exactly how separated do two clouds of amplitude have to be, quantitatively speaking, in order to invoke the instantaneous action of the Quantum Spaghetti Monster? And in which basis does the Quantum Spaghetti Monster measure separation?"

But, in the *modern* view of quantum mechanics - which is accepted by everyone except for a handful of old fogeys who may or may not still constitute a numerical majority - well, as David Wallace puts it:

"If I were to pick one theme as central to the tangled development of the Everett interpretation of quantum mechanics, it would probably be:

the formalism is to be left alone."

Decoherence is not an extra phenomenon. Decoherence is not
something that has to be proposed additionally. There is no
Decoherence Postulate *on top of* standard QM. It is implicit in
the standard rules. Decoherence is just what happens *by default,* given
the standard quantum equations, unless the Quantum Spaghetti Monster
intervenes.

Some still claim that the quantum equations are unreal - a mere model
that just happens to give amazingly good experimental predictions. But then decoherence
is what happens to the particles in the "unreal
model", if you apply the rules universally and uniformly. It is *denying* decoherence that requires you to postulate an
extra law of physics, or an act of the Quantum Spaghetti Monster.

(Needless to say, no one has ever observed a quantum system behaving coherently, when the untouched equations say it should be decoherent; nor observed a quantum system behaving decoherently, when the untouched equations say it should be coherent.)

*If you're talking about anything that isn't in the equations, you must not be talking about "decoherence".*
The standard equations of QM, uninterpreted, do not talk about a
Quantum Spaghetti Monster creating new worlds. So if you ask when the Quantum Spaghetti Monster creates a new world, and you can't answer the question just by looking at the equations, then you must not be talking about "decoherence". QED.

Which basis you use in your calculations makes no difference to standard QM. "Decoherence" is a phenomenon implicit in standard QM. Which basis you use makes no difference to "decoherence". QED.

Changing your view of the configuration space can change your view of the blobs of amplitude, but ultimately the same physical events happen for the same causal reasons. Momentum basis, position basis, position basis with a different relativistic space of simultaneity - it doesn't matter to QM, ergo it doesn't matter to decoherence.

If this were not so, you could do an experiment to find out which basis was the right one! Decoherence is an experimentally visible phenomenon - that's why we have to protect quantum computers from it.

Ah, but then where is the decoherence in

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

?

The decoherence is still *there.* We've just made it harder for a human to *see,* in the new *representation*.

The main interesting fact I would point to, about this amazing new representation, is that we can no longer calculate its evolution with *local causality.* For a technical definition of what I mean by "causality" or "local", see Judea Pearl's *Causality*. Roughly, to compute the evolution of an amplitude cloud in a *locally causal* basis, each point in configuration space only has to look at its infinitesimal neighborhood to determine its instantaneous change. As I understand quantum physics - I pray to some physicist to correct me if I'm wrong - the position basis is local in this sense.

(Note: It's okay to pray to physicists, because physicists actually exist and can answer prayers.)

However, once you start breaking down the amplitude distribution into components like (Sensor-RIGHT - Sensor-LEFT), then the flow of amplitude, and the flow of causality, is no longer *local* within the new configuration space. You can still calculate it, but you have to use nonlocal calculations.

In essence, you've obscured the chessboard by subtracting the queen's position from the king's position. All the information is still there, but it's harder to *see*.

When it comes to talking about whether "decoherence" has occurred in the quantum state of a human brain, what should intuitively matter is questions like, "Does the event of a neuron firing in Human-LEFT have a noticeable influence on whether a corresponding neuron fires in Human-RIGHT?" You can choose a basis that will mix up the amplitude for Human-LEFT and Human-RIGHT, *in your calculations.* You cannot, however, *choose a basis* that makes a human neuron fire when it would not otherwise have fired; any more than you can *choose a basis* that will protect a quantum computer without the trouble of shielding, or *choose a basis* that will make apples fall upward instead of down, etcetera.

The formalism is to be left alone! If you're talking about anything that isn't in the equations, you're not talking about decoherence! Decoherence is part of the invariant essence that doesn't change no matter how you spin your basis - just like the physical reality of apples and quantum computers and brains.

There may be a kind of Mind Projection Fallacy at work here. A tendency to see the basis itself as real - something that a Quantum Spaghetti Monster might come in and act upon - because you spend so much time calculating with it.

In a strange way, I think, this sort of jump is actively encouraged by the Old School idea that the amplitude distributions *aren't real.* If you were told the amplitude distributions were physically real, you would (hopefully) get in the habit of looking past mere *representations*, to *see through* to some invariant essence inside - a reality that doesn't change no matter how you choose to represent it.

But people are told the amplitude distribution is not real. The calculation itself is *all there is*, and has no virtue save its *mysteriously excellent* experimental predictions. And so there is no point in trying to *see through* the calculations to something within.

Then why *not* interpret all this talk of "decoherence" in terms of an arbitrarily chosen basis? Isn't that all there is to *interpret* - the calculation that you did in some representation or another? Why not complain, if - having thus *interpreted* decoherence - the separatedness of amplitude blobs seems to change, when you change the basis? Why try to see through to the neurons, or the flows of causality, when you've been told that the calculations are all?

(This notion of *seeing through* - looking for an essence, and not being distracted by surfaces - is one that pops up again and again, and again and again and again, in the Way of Rationality.)

Another possible problem is that the calculations are crisp, but the essences inside them are not. Write out an integral, and the symbols are digitally distinct. But an entire apple, or an entire brain, is larger than anything you can handle formally.

Yet the form of that crisp integral will change when you change your basis; and that sloppy real essence will remain invariant. Reformulating your equations won't remove a dagger, or silence a firing neuron, or shield a quantum computer from decoherence.

The phenomenon of decoherence within brains and sensors, may not be any more crisply defined than the brains and sensors themselves. Brains, as high-level phenomena, don't always make a clear appearance in fundamental equations. Apples aren't crisp, you might say.

For
historical reasons, some Old School physicists are accustomed to QM
being "interpreted"
using extra postulates that involve crisp actions by the Quantum
Spaghetti Monster - eating blobs of amplitude at a particular instant,
or creating worlds as a particular instant. Since the equations aren't
supposed to be *real,* the sloppy borders of real things are not looked for, and the crisp calculations are primary. This makes it hard to see through to a
real
(but uncrisp) phenomenon among real (but uncrisp) brains and apples,
invariant under changes of crisp (but arbitrary) representation.

Likewise, any change of representation that makes apples harder to see, or brains harder to see, will make decoherence within brains harder to see. But it won't change the apple, the brain, or the decoherence.

As always, any philosophical problems that result from "brain" or "person" or "consciousness" not being crisply defined, are not the responsibility of physicists or of any fundamental physical theory. Nor are they limited to decoherent quantum physics particularly, appearing likewise in splitting brains constructed under classical physics, etcetera.

Coming tomorrow (hopefully): *The Born Probabilities*, aka, that mysterious thing we do with the squared modulus to get our experimental predictions.

I can't say that I've understood everything in the series on QM, but it has been immensely useful for me in beginning to understand it. And, in general, most of what I've read of OB I've found useful - especially the comments, because I find myself "catching up" with a lot of these ideas, and I have a tendency when I'm catching up to the ideas of someone more intelligent than me to not find fault where I would if I had a better grasp of the ideas. Though I'm still not very far along on the path to being a rationalist, I know that it's a path I've been trying to walk my whole life, despite the fact that much of it was spent stumbling and tripping through religion, popular politics, and arguments that were more about proving who was "right" rather than finding out what was right. I'm glad to have found yet another resource for walking the path, especially one as useful as this one. I haven't commented here before, but I just thought I'd toss in that I really appreciate the writing you do here (and yours as well, Robin) and I'm glad that I stumbled across this blog.

Posted by: Eric | April 30, 2008 at 11:03 AM

Hrm... I wonder if there are other basies with local behavior other than the usual positional one?

If yes, does what we see as decoherence automaticlaly "look decoherent" in that basis too?

Posted by: Psy-Kosh | April 30, 2008 at 12:08 PM

Psy-Kosh: in the basis of eigenvalues of the Hamiltonian, not only is the equation local, but nothing even moves.

Posted by: Chris | April 30, 2008 at 12:20 PM

Chris, forgive me if this is a foolish question, but wouldn't the components corresponding to eigenvalues of the Hamiltonian change only by a constant complex factor, rather than not changing at all?

Posted by: Eliezer Yudkowsky | April 30, 2008 at 12:46 PM

Chris, Eliezer: Yeah, at least last what I recall studying, the time development for Hamiltonian eigenvectors basically has them spinning around the complex plane (with the rate of rotation being a function of the eigenvalue. In fact, I believe it is directly proportional)

Posted by: Psy-Kosh | April 30, 2008 at 01:42 PM

Actually, this discussion leads me to wonder something: What properties does a matrix have to have such that its eigenvectors form a complete basis?

Posted by: Psy-Kosh | April 30, 2008 at 03:45 PM

The eigenvectors of a matrix form a complete orthogonal basis if and only if the matrix commutes with its Hermitian conjugate (i.e. the complex conjugate of its transpose). Matrices with this property are called "normal". Any Hamiltonian is Hermitian: it is equal to its Hermitian conjugate. Any quantum time evolution operator is unitary: its Hermitian conjugate is its inverse. Any matrix commutes with itself and its inverse, so the eigenvectors of any Hamiltonian or time evolution operator will always form a complete orthogonal basis. (I don't remember what the answer is if you don't require the basis to be orthogonal.)

Posted by: Stephen | April 30, 2008 at 04:04 PM

It would be a pleasure and a treat to join the recent discussion on QM, especially the Ebborian interlude, but I cannot afford the dozens of hours of study and reflection it would take to get to the point where I could actually

contributeto the discussion.If I ever find myself with the luxury of being able to

studyQM, this blog or the book that comes from it is where I would go first for written study material. (I'd probably need a reliable mathematical treatment, too, but those are easy to find.)Physics

isthe study of ultimate reality!QM is in my humble opinion humankind's greatest achievement.

Posted by: Richard Hollerith | April 30, 2008 at 04:08 PM

Stephen: Thanks!

And yeah, I was wondering what the answer was if I don't necessarally demand them to be orthognal, just that I require them to span the space.

Anyways, am right now reading through Down with Determinants. Maybe that'll have the answer in there.

(Actually, the part which I get to, at least for finite dimensional spaces, is already effectively in there: The number of distinct eigenvalues has to equal the dimension of the space. Of course, the question of what has to be true about a linear operator for _that_ to hold is something I'm wondering. :))

Posted by: Psy-Kosh | April 30, 2008 at 04:33 PM

"The number of distinct eigenvalues has to equal the dimension of the space."

That may be a sufficient condition but it is definitely not a necessary one. The identity matrix has only one eigenvalue, but it has a set of eigenvectors that span the space.

Posted by: Stephen | April 30, 2008 at 04:52 PM

Stephen: whoops. Just realized that and came here to post that correction, and you already did. :)

Posted by: Psy-Kosh | April 30, 2008 at 04:59 PM

I don't really follow a lot of what you've written on this, so maybe this isn't fair, but I'll put it out there anyway:

I have a hard time seeing much difference between you (Eliezer Yudkowsky) and the people you keep describing as wrong. They don't look beyond the surface, you look beyond it and see something that looks just like the surface (or the surface that's easiest to look at). They layer mysterious things on top of the theory to explain it, you layer mysterious things on top of physics to explain it. Their explanations all have fatal flaws, yours has just one serious problem. Their explanations don't actually explain anything, yours renames things (e.g. probability becomes "subjective expectation") without clearing up the cause of their relationships -- at least, not yet.

Posted by: Brian Jaress | April 30, 2008 at 07:50 PM

Psy-Kosh, Stephen: A finite-dimensional complex matrix has a complete basis of eigenvectors (i.e. it is diagonalizable) if and only if every

generalizedeigenvector is also an eigenvector. Intuitively, this means roughly that there are n independent directions (where n is the size of the matrix) such that vectors along these directions are stretched or shrunk uniformly by the matrix.Try googling "jordan normal form", that may help clarify the situation.

I don't know the answer in the infinite-dimensional case.

Posted by: Dario Amodei | May 01, 2008 at 01:03 AM

Roland, make that two. Though this mooshed my head.

I used to really enjoy thinking about how weird QM was. Look! The little photon goes through

both holes at the same time!Not really any more though, it's starting to seem a little bit...ordinary.Which is a good thing, of course.

Quick question - since you can't integrate over a single point, does that preclude the existence of any 'motionless' particle? Anything that ceased to have an appreciable (Planck-length?) amplitude spread would, in effect, not be there? That would chime with the transform duality thingy between location and velocity.

Hope I get chatting to someone who thinks in terms of quantum/classical dualities at some point, purely so that I can use the line "you're very clever, old man, but it's all amplitudes, all the time."

Posted by: Ben Jones | May 01, 2008 at 10:12 AM