Talk:Quantum decoherence

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[edit] Relevant Link?

IANAP, but can someone who is confirm or deny the relevance of the link to The Decoherence of Measurement recently added? In particular the following:

There seems to be a guiding principle (that of the statistical increase of order in the Universe). This guiding principle cannot be communicated to quantum systems with each and every measurement because such communication would have to be superluminal. The only logical conclusion is that all the information relevant to the decrease of entropy and to the increase of order in the Universe is stored in each and every part of the Universe, no matter how minuscule and how fundamental.
It is safe to assume that, very much like in living organisms, all the relevant information regarding the preferred (order-favoring) quantum states is stored in a kind of Physical DNA (PDNA). The unfolding of this PDNA takes place in the physical world, during interactions between physical systems (one of which is the measurement apparatus).

A Google_Test for "PDNA" doesn't show any relevant results but the article itself. CCooke 11:34, 3 Jan 2005 (UTC)

The whole paragraph is simply too vague. YMH 19:53, 9 May 2006 (UTC)

  • Introduuced section on "Decoherence in computation" at suggestion of discussion page of Uncertainty principle.--CarlHewitt 2005 July 6 23:03 (UTC)

I disagree that it is patent nonsense. It looks OK to me up thrugh the section on "Mathematics of decoherence". Maybe just delete after that?--CarlHewitt 2005 July 6 23:32 (UTC)

Perhaps it is not 'a patent nonsense' but link should be deleted. It is 'original research' and has no place in an encyklopedia. It is not an hypothesis either, rather a bit poetic parable. —Preceding unsigned comment added by FrishP (talk • contribs) 12:04, 6 January 2008 (UTC)

[edit] Speedy tag

This is definitely not a speediable article. It looks legit to me, though my physics background is not good enough to work as a hoax detector. I'm pulling the speedy tag - this ought to go through VfD if it's going to go through anything. Denni☯ 2005 July 7 00:00 (UTC)

[edit] Decoherence in computation

Decoherence also takes place in digital computation when a special device called an arbiter leaves a metastable state…

Does this section belong in an article on quantum decoherence? Its use of the term "decoherence" sounds more like macroscopic electronics than quantum physics, unrelated to quantum wave function "collapse". ~ Jeff Q (talk) 02:10, 14 July 2005 (UTC)

This is a good question. Leaving a metastable state would seem to have all the right properties of decoherence from a mixed state. Of couse decoherence is suppose to replace the old "collapse" interpretation.--Carl Hewitt 02:24, 14 July 2005 (UTC)
It still sounds like we're confounding two meanings of the term "decoherence". The article on metastability makes no mention of any meaning in quantum physics. "Mixed state" is a very overloaded term; it could be talking about mixed quantum states, mixed electronic states, or mixed state election results. This section sounds quite reasonable — for an article on electronics, not one on quantum physics. ~ Jeff Q (talk) 03:49, 14 July 2005 (UTC)
I have added some material to clarify this article and quantum indeterminacy.--Carl Hewitt 14:30, 14 July 2005 (UTC)
Well, the updated information is beyond my ability to discern appropriateness. (Based on your résumé, CarlHewitt, it certainly seems within yours.) I guess my biggest problem is that I've never heard of an "arbiter" in electronics, and my attempts to understand the text in terms of macro-scale digital electronics seem to be out of place. When you get a chance, could you come up with an article on Arbiter (electronics) that provides the context for this aspect of digital computation? Thanks for your help. ~ Jeff Q (talk) 05:16, 21 July 2005 (UTC)
To start with, I have put some external links in Metastability in electronics.--Carl Hewitt 05:22, 21 July 2005 (UTC)

[edit] Lost in Hilbert space ?

Hi all, I rewritten and expanded some areas. In particular I've expanded the intuitive picture of decoherence by trying to make an analogy between Hilbert spaces and ordinary space to explain quantum interference, related decoherence more the various interpretations of QM and linked in density matrices more. See what you think. I haven't touched the maths section -- that probably needs expanding and clarifying as well. --Michael C. Price talk 18:25, 24 June 2006 (UTC)

Two new sections. An explanation of decoherence using Dirac's bra-ket notation. And a section on how decoherence destroys quantum interference. Based heavily on Zurek's articles as cited. --Michael C. Price talk 00:40, 1 July 2006 (UTC)

The part about the number of degrees of freedom being 3x the number of particles is out of left field and I'm not even sure what the author of the comment meant. If they were implying that spatial coordiantes are the only degrees of freedom of a particle system, that's crazy. I've removed it. Birge 22:31, 12 August 2006 (UTC)

You obviously didn't read what the comment said before you deleted it. No relationship between number of degrees of freedom and number of particles is implied in any way. The other implication you mention is simply fantasy. --Michael C. Price talk 22:36, 12 August 2006 (UTC)
When you say the dimension of hilbert space is three times the number of particles, that's pretty much what you're saying, isn't it? Anyway, I ended up not deleting it and waiting for comment. So, why don't you enlighten me and every other reader of the article as to why the nonrelativistic hilbert space dimension is always three times the number of free particles, or provide a reference or link? Anyway, the whole section is rife with unhelpful parenthetical comments (sometimes a couple of levels deep, which is a terrible grammar) and is in high need of editing. Birge 20:27, 15 August 2006 (UTC)
Change "particles" to "free particles" in your first sentence you'll be correct. This is not an easy subject to understand so don't expect any easy explanations:
"The dimension d of the Hilbert space is taken to be equal to the number of degrees of freedom of the system"[1]
"degrees of freedom in the volume (or equivalently on the dimension of Hilbert space)"[2]
"“dimension” in the sense “the number of. degrees of freedom”" [3]
--Michael C. Price talk 22:33, 15 August 2006 (UTC)
The problem is I think you're wrong, unless I'm misunderstanding what you're saying. The complete state of even a single electron requires an infinite dimensional Hilbert space. Think of the wavefunction representation of a single free particle: that's an infinite dimensional vector space. Out of curiosity: did you write the entire "Lost in Hilbert Space" section? Birge 23:36, 15 August 2006 (UTC)
Re single electron: Only if you treat each point in space as a separate dimension/degree of freedom, as is normal in QFT, which is a different sort of dimensionality. The 3-N aspect only refers to non-relativistic QM, as the article says. I pretty much wrote every thing under "mechanisms". --Michael C. Price talk 00:01, 16 August 2006 (UTC)
There's really only one type of dimensionality when you're talking about a vector space. And each point in space IS a separate degree of freedom in the wavefunction of a particle. Forget about QED, we're not even talking about that. In the "references" you cited (a problem set is not normally considered a reference) they are talking about finite dimension Hilbert spaces that arise from considering ONLY spin of spin 1/2 particles. This assumes the other degrees of freedom of the particle are constrained or irrelevent. I didn't see a single mention of your 3 dimensions per particle assertion, which is, frankly, ridiculous. I have to ask this, though I'm afraid there's no polite way to do so: exactly what training or qualifications do you have to be making substantial edits and revisions of other's work on the topic of quantum mechanics? You don't appear to really have a solid grasp of any of this.
I could ask the same questions of you, since you seem not to have heard of the tensor product. You also seem ignorant of some other rather basic concepts, such as the analogy between classical phase space and a hilbert space. In non-relativistic QM the wavefunction of an N-particle system is a complex function of the co-ordinates of the N-particles, i.e. over 3N dimensions:
ψ(x1,x2,...,xN)
In classical mechanics a probability distribution in phase space would be a function over 6-N dimensions, since we would include the momenta as well:
ψ(x1,p1,x2,p2,...,xN,pN)
Which part of that do you find so difficult/unbelievable? --Michael C. Price talk 07:13, 16 August 2006 (UTC)
(1) In general, there are more degrees of freedom to system of particles than their position wavefunctions (e.g. spin) and (2) You were talking about the dimensionality of the hilbert space. Even if the only degrees of freedom are spatial, that is an infinite dimensional hilbert space. I'm not sure why you're dropping the notion of classical phase space here. It's completely irrelevent to our little argument. Birge 16:18, 16 August 2006 (UTC)
(1) Spin is just a discrete-valued label on the wavefunction whereas x or p are continuous variables. Personally I can't see how spin can be conscrued as a contributing a dimension, but it might come down to definitions. (2) Do you accept that a classical N-particle system with prob(x1,p1,x2,p2,...,xN,pN) inhabits a 6N-dimensional phase space? If so, how many dimensions do you think ψ(x1,x2,...,xN) occupies? I think you are identifying the number of eigenvalues = number of basis vectors spanning the hilbert space = dimensions of the Hilbert space, whereas I am using the word dimension in the classical spatial sense. --Michael C. Price talk 16:52, 16 August 2006 (UTC)
Yes, that's exactly the disconnect. The number of basis vectors spanning the space IS the dimension of the hilbert space. So isn't it incorrect to say the dimension of the Hilbert space of a single particle is three? That's exactly what you said in the parenthetical comment that started all of this. Birge 17:11, 16 August 2006 (UTC)
Perhaps the term is used to mean different things in different contexts. I refer you back to the links I found that support H'dimension = number of degrees of freedom. To avoid confusion I'll change the terminology. --Michael C. Price talk 17:42, 16 August 2006 (UTC)
Michael, as I pointed out already, those links referred to systems considering ONLY particle spin, which have discrete and finite eigenvalues. Thus, the degrees of freedom were finite. I'm using the terms "degrees of freedom" and "hilbert space dimension" EXACTLY as those references did. I believe you misread them. Birge 17:51, 16 August 2006 (UTC)
Believe what you like, but the facts are this: The first link only mentioned spin after it made the general identification of H-space dimensionality with the degrees of freedom, the second is talking about the holographic principle/Bekenstein bound/black holes and the third is a general paper on geometry and algebra. --Michael C. Price talk 18:10, 16 August 2006 (UTC)
Nothwithstanding my previous response, it is obvious that my use of terminology was ill-advised. I believe a helpful change would be to speak only in terms of "phase space" and not "Hilbert space"? --Michael C. Price talk 18:40, 16 August 2006 (UTC). --Michael C. Price talk 18:10, 16 August 2006 (UTC)
Done, and retitled "phase space picture". --Michael C. Price talk 19:34, 16 August 2006 (UTC)
Agreed. Birge 19:44, 16 August 2006 (UTC)

[edit] Time and Probabilities

Jerry Heath -

Small particles (at the quantum mechanical level) - if they are independent - are without time; that is, are outside the limits of time.

This does not mean that such a particle can go backwards in time. It does mean that such a particle can be "found" at a given location specified by the probabilities but unrelated to "how fast, how far." Time cannot be involved...the appearance is that the particle could "move" from one place to the other instantaneously (implying if not being infinite velocity).

Note that probabilistic behavior requires that there are no time limitations - otherwise the particle behavior would be required to follow the Newtonian, "how far, how fast" rules. The requirements of time would collapse the wave equation.

But time can be imposed on such small particles. The simple rule is that anything that reduces the possibilities imposes time. The imposition is to the extent that the possibilities are reduced. The limitations of possibilities forces the particle to operate more under "how fast, how far" rules rather than probabilistic rules. Such time based behavior appears to be related to the amount of the limitation of possibilities imposed.

Increasing the density in the region of the particle reduces the possibilities and imposes time. By density I mean the mass over the volume in the region.

Time is imposed on a particle by most measurement methodologies. If time is not imposed directly (the measurement depends on an imposed time limit) it is imposed by increasing the density in order to make the measurement.

Imposing time imposes "how far, how fast" on the particle.

Light in actuality has infinite speed (light has no time). But when we measure the "speed" of light we impose time on the light and determine a finite velocity. This suits our purpose.

Particles will exhibit probabilistic behavior only to the extent that that behavoir is not limited by imposing time on them...since probabilistic behavior cannot occur with "how far, how fast" (time) limits on the behavior.

Time is a contradiction to probabilities. Time collapses the wave equation. - Jerry Heath

ProfJerryHeath 19:29, 28 June 2006 (UTC)

Jerry Heath -

Both the small (quantum) particle and the distribution probabilities of such a particle cannot be in time. The distribution must form immediately and the particle cannot have time. The particle 'obeys' the distribution curve immediately.

But as we view such particle probabilities our view is limited by time because we are 'stuck in time.' Thus as we measure a timeless phenomena our time world imposes a time on the measurement. But the time impression we get is a phantom of timeless probabilities. The collapse of the wave equation due to density is also a result of this kind of process. Time and probability are mirror worlds.

Gravity is a special distribution that has time associated with it. As gravity becomes an important force, due to density, the probability distribution collapses into the time mirror of probabilities.

Because we are 'stuck in time,' when we look at quantum processes we can only see them though the looking glass. Schrödinger’s cat is placed, philosophically, on the other side of that looking glass. - Jerry Heath

12.162.215.130 13:44, 11 July 2006 (UTC)

[edit] Huge reversion

I'm really sorry, but it appears that the user Michael C. Price has been making huge revisions to this article that are entirely and utterly incorrect. His edits are often plausible and borrow from correct notation, but are gibberish upon close review (such as nice looking dirac notation equations which multiply two ket vectors together, a meaningless operation in the context of quantum mechanics). Pleaser refer to my discussion with him below, as well as his user page for more examples of his run ins with other physics pages. If somebody knows the correct procedure for handling what amounts to a dedicated and repeat vandal, please help. I'm not sure what to do, but cranks like this are completely destroying the integrity of Wikipedia as they have far more time on their hands than the people who actually know science. I can't think of a good way to filter out the good edits from his bad ones (and many of the good edits were on technically incorrect entries made by him) so I've reverted to the last version several months back before he started taking over this article. I'm sure Michael will revert this revert, but we really need to do something about this. I don't have time to deal with this any more. Birge 03:13, 16 August 2006 (UTC)

I think rather the problem comes from technically ignorant authors who revert first, ask questions later. And then pronounce as "gibberish" operations which
multiply two ket vectors together, a meaningless operation in the context of quantum mechanics
Have you never heard of tensor products? --Michael C. Price talk 06:53, 16 August 2006 (UTC)
Yes, I know what a tensor product is, and you didn't write it as a tensor product. Birge 15:12, 16 August 2006 (UTC)
I didn't write it as "multiply" either. I was assuming a basic level of intelligence from the reader to realise that tensor product was implied by the use of, and link to, Dirac notation. Obviously I didn't dumb down enough and for that I apologise. --Michael C. Price talk 15:22, 16 August 2006 (UTC)
You're right. Looking back your notation is consistent. I was wrong and just saw two kets back to back and didn't look closely enough to see what you were doing. I take back that objection and apologize. Birge 15:33, 16 August 2006 (UTC)
I appreciate the retraction and amicable resolution -- not something you often see. Thanks! --Michael C. Price talk 15:49, 16 August 2006 (UTC)
Whoops. I just checked out the Dirac Notation page, and apparently ket ket is an accepted shorthand for the tensor product of two states. Again, I'm really sorry. I should make it clear here that I very much apologize for calling you a crank, not just for my criticism of the dirac section. Birge 19:47, 16 August 2006 (UTC)

[edit] Questions on Dirac Section

Ok, I'll try to deal with this more productively and in the spirit of wikipedia. In the Dirac section, there is the statement that unitarity of the evolution leads to orthogonality of the |E_i> states. Then, later, it's stated that decoherence makes them approximately orthogonal. This doesn't make any sense. The |E_i> are states that are defined. Either they are orthogonal or not, and they don't change upon evolving the system. (I.e. if a system starts in |E_1> it may evolve to be a superposition of other states, but |E_1> is a mathematical definition that doesn't change.) - Birge 16:11, 16 August 2006 (UTC)

Okay, I see the source of the confusion. | Ei > has different definitions in the two interaction categories defined. | Ei > in one section is | i,Ei > in the other. It is the joint total states (environment + object system) on which orthogonality is exact. The states on which the orthogonality is only approximate are subsets of the joint system (environment - object system). I'll revise to make this clearer. --Michael C. Price talk 17:14, 16 August 2006 (UTC)
For the moment I've left the two different uses in, but put them in different subsections. --Michael C. Price talk 09:48, 17 August 2006 (UTC)

Fair enough. I think another problem in the section is that the entire point of the formalism is this business of einselection which is somewhat glossed over. It's just stated as fact that we are able to pick a basis in the system that will end up evolving to yield these |E_i,i> states in the combined system, and I think some explanation of that is needed. I suspect that's really where the subtlety of the theory comes in.

If I may: isn't a key point in all this that the orthonormality of |E_i> can be *derived* from classical QM? If correct, then the article would be improved by simply stating this fact for the reader. (I am not familiar with the term einselection, but infer it is the phenomenon in which this orthonormality of the |E_i> states occurs.) --Randall B. Smith, January 2008 —Preceding unsigned comment added by 74.2.65.26 (talk) 08:22, 27 January 2008 (UTC)

Einselection has its own article, so any specific points might be better raised on its virgin talk page. In general, though, you can always pick a basis in quantum theory -- indeed that's what, I suspect, some people don't like about QM, that it seems just too easy to pick a basis. --Michael C. Price talk 20:20, 16 August 2006 (UTC)
I agree that you can pick any basis you want, I'm just saying the notion that you can pick one that has such nice properties upon interaction with the combined environment system is not obvious. Perhaps you're right that it belongs on the einselection page. However, given that it's the kernel of the entire subject of decoherence theory, I'd say it warrants more explanation here. Birge 20:32, 16 August 2006 (UTC)
Oh, one more thing. I think there's a missing definition for what |k(i)> refers to. It seems like it's suppposed to be some subspace basis set in the complete system, but I think it needs to be defined. Birge 20:58, 16 August 2006 (UTC)
Yes, I just noticed that myself. Carry on listing the problems and I'll try to get back tomorrow. --Michael C. Price talk 21:08, 16 August 2006 (UTC)
Okay, problem solved by deletion -- it was irrelevant (and confused and probably wrong OR!). Also I've updated einselection to include what seems to be its defining characteristic, namely the orthonormality of the environment states, with a direct quote from Zurek. --Michael C. Price talk 09:48, 17 August 2006 (UTC)

[edit] environment effected in Dirac Notation

I am attempting to familiarize myself with what's going on here, and the following statement has me wondering if the wrong word is being used. Under the section, Dirac Notation, the following sentence appears:

There are two extremes in the way the system can interact with its environment: either (1) the system loses its distinct identity and merges with the environment (e.g. photons in a cold, dark cavity get converted into molecular excitations within the cavity walls), or (2) the system is not disturbed at all, even though the environment is effected (e.g. the idealised non-disturbing measurement).


I suggest that the phrase, "the environment is effected" be changed to "the environment is affected", as I don't believe the intent is to communicate that the environment is created by the system being discussed, but that the environment is influenced or altered by that system. -douglas

I copped out and changed it to "disturb". --Michael C. Price talk 06:14, 23 November 2006 (UTC)

[edit] One article too many?

Why do we have separate articles for quantum coherence and quantum decoherence? Borisblue 00:19, 30 November 2006 (UTC)

Because they are not the same thing, nor are they opposites or converses. --Michael C. Price talk 00:53, 30 November 2006 (UTC)

[edit] Measurement Problem

@Michael C. Price: Why did you remove the incompleteness claim? The paragraph mentioned explicitely, that in order to solve the measurement problem of the CI, decoherence must be supplied with some nontrivial interpretational considerations. This implies, that the incompleteness claim is not valid for all interpretations. The original statement was therefore completely correct.--Belsazar 15:45, 3 September 2007 (UTC)

The discussion is not limited to the CI. Also MWI claims that interpretation proceeds from the formalism: thus the incompleteness claim is not valid for all interpretations (and MWI in particular).--Michael C. Price talk 00:54, 4 September 2007 (UTC)
Also MWI claims that interpretation proceeds from the formalism -> This claim of MWI is a nontrivial interpretational consideration per se and rather supports the statement in the article.--Belsazar 09:05, 4 September 2007 (UTC)
I'm not sure what you mean (you seem to contradict yourself), but the claim that the MWI emerges from the formalism alone is one made explicitly by Bryce DeWitt. It is a non-trivial exercise to show this, but that is another matter. --Michael C. Price talk 09:15, 4 September 2007 (UTC)
I have mixed feelings about this, but I tend to side with MichaelCPrice, for somewhat different reasons. The measurement problem in plain quantum mechanics often gets discussed as a philosophical issue, but you can phrase it in purely operational terms: when measurements have a detectable effect, and there's no definition of what constitutes a measurement, you can't predict the behavior of any system in which a measurement might take place. The sorts of people who complain about the measurement problem generally have that fact in mind. Decoherence does solve that problem, by making the supposed difference in behavior so tiny that it's way beyond anything even in principle detectable. All that's left of the measurement problem is metaphysics. So the claim that decoherence doesn't solve the measurement problem seems like vague and somewhat dubious philosophy, whereas there seems to be a clear scientific sense in which it does solve it. -- BenRG 09:37, 4 September 2007 (UTC)
For me the issue is now solved by the clarifications in the article (even if the claims of DeWitt sound somewhat strange to me, but this is another issue). Thanks @Michael C. Price.--Belsazar 12:07, 4 September 2007 (UTC)

[edit] The lead

Is it possible for the first paragraph to be phrased in a way that is at all comprehensible to readers unfamiliar with the underlying topics?

--Ultra Megatron (talk) 01:47, 20 November 2007 (UTC)

It is very difficult for any information about quantum theory to be phrased such that anybody "unfamiliar with the underlying topics" will find it easily comprehensible. While I enjoy reading wiki articles about quantum physics and theory, I will never learn all about the subject from this source. No one should expect to. The subject matter is so complicated that simplistic introductions to even small(ish) parts of it seem to be problematic at best. Can it be clearer? I would hope so, but until an Alan Guth or Brian Greene comes along to make it all a little clearer, we will have to muddle through.

-- The lead in us completely unacceptable, vague, and poorly written. It is a cop out to claim that information about Quantum Mechanics is necessarily incomprehensible, poor writing makes it incomprehensible. I have a rudimentary grasp of decoherence from my professional life, but the intro paragraph left me scratching my head. A proper introduction should be as clear as renormalization [4] is, by giving context, general reason for inclusion, and a basic definition. While the entire article is poorly written (anytime an author appeals solely to the mathematics for a "description", it is inappropriate for wikipedia), at least the opening can be clarified in a timely manner. —Preceding unsigned comment added by 68.190.80.137 (talk) 14:52, 1 January 2008 (UTC)

-- I agree the opening paragraph could be improved. I suggest something like this: Quantum Decoherence is a phenomenon that occurs in classical quantum mechanics. It is exhibited by a quantum mechanical system interacting with its environment in such a way that the phenomenon of quantum interference becomes negligibly small. Calculations suggest that in most situations quantum decoherence will occur very rapidly, and that one must go to great lengths to carefully prepare a system and its environment if one wishes to avoid the phenomenon. Quantum decoherence is especially significant for two reasons: 1] it can explain why we normally observe systems that obey classical laws of probability, and 2] why wave functions appear to collapse. I don't have references at hand for the few places in there that could use them, but if anyone thinks this prose merits inclusion, I'll try to dig them up. Also would need to link-ify all the appropriate terms. (Randallbsmith (talk) 08:58, 27 January 2008 (UTC))

I suggest removing "Calculations suggest that ".--Michael C. Price talk 13:35, 27 January 2008 (UTC)

[edit] Entropy

Would it be possible to add a discussion on the effects, or otherwise, of decoherence on the entropy of the objects and the resulting combined system?

The article mentions that the process of decoherence is irreversible. Can I presume that this is "irreversible" in the statistical mechanics sense of "it is in principle reversible, but unimaginably vastly hugely improbable that it might reverse". -- 80.168.224.193 (talk) 10:16, 30 December 2007 (UTC) Irreversible in thermodynamics is a technical term. Reversible processes are quasi-stationary, and always close to equilibrium. While related, it not same as Zermelo's objection to Maxwell's theory, known as the Wiederkehreinwand. FrishP (talk) 11:49, 6 January 2008 (UTC)

[edit] Irreversibility

Yes, decoherence is "irreversible" in the (quantum) statistical sense. It is the mechanism by which the "quantumness" of a system is transferred to an environment, which is treated statistically. If we did not treat the environment statistically (i.e., we kept track of everything), we would not get decoherence to mixtures. Gamblorius (talk) 18:42, 9 March 2008 (UTC)

[edit] Reversibility and Irreversibility

John von Neumann's Mathematical Foundations of Quantum Mechanics presents an interesting discussion of this question in chapter V, "General Considerations, 1. Measurement and Revesibility".

According to von Neumann's analysis, there are two quantum processes relevant:

Process 1: measurement (collapse of the wavefunction) and

Process 2: reversible time evolution according to a time-dependent Schrödinger equation.

Wavefunctions of the system, measuring device, and environment evolve reversibly according to process 2. Process 1 entails irreversible collapse of the wavefunction into a statistical mixture. According to von Neumann page 357 "[process] 2 transforms states into states while [process] 1 transforms states into mixtures".

von Neumann goes on to say (page 398),"Although our entropy expression, as we saw, is completely analogous to classical entropy, it is still surprising that it is invariant in the normal evolution in time (process 2) and only increases with measurements (process 1)."

Decoherence is not the same as collapse of the wavefunction (process 1). If decoherence occurs by evolution of a time-depenedent Schrödinger equation (process 2), then, in the context of von Neumann's analysis, decoherence is not irreversible. von Neumann extends his idea of process 1 to macroscopic examples.

A discussion of how decoherence and measurement, subjective or objecitve, may lead from "one quantum world" to an ensemble of "many classical worlds" is given in Stapp H. P., Physical Review A, 46(11), 1992.

Joseph Alia (talk) 02:00, 3 May 2008 (UTC)