Why do the alternatives separate from each other?

[Erland]

From Wikipedia on quantum decoherence:

"Let us choose an expansion where the resulting basis elements interact with the environment in an element-specific way. Such elements will—with overwhelming probability—be rapidly separated from each other by their natural unitary time evolution along their own independent paths. After a very short interaction, there is almost no chance of any further interference"

I never understood that part. Why do the elements separate rapidly from each other? Why don't they continue to interfere with each other and the environment in an ever more complex pattern? Is there any simple mathematical (or other kind of) reason for that?

 

[StevieTNZ]

Perhaps it is because we cannot fully write down the quantum state of the environment.

 

[PeterDonis]

From Wikipedia on quantum decoherence

I would not use Wikipedia as a source for this. (I personally wouldn't use it as a source for anything scientific, but even someone who would for some scientific topics would be well advised not to for this one.) This is a topic where pop science misconceptions abound. Unfortunately I don't know of a good online textbook-style reference that discusses decoherence; it is a new enough research area that the best source is going to be peer-reviewed papers.

Why do the elements separate rapidly from each other?

"Separate" here just means that they can't interfere with each other. It doesn't have to be physical separation. The key is that it becomes impossible (at least in a practical sense) to cause the elements to interact in such a way that their quantum phases would interfere with each other. The reason it becomes impossible is that the elements become entangled with the environment and information about their phases is lost. In principle, if we could keep track of every single quantum particle in the environment, this would not be the case; we would be able to keep track of exactly which quantum particles in the environment were entangled with each element, and how, and we would be able to make measurements whose results would depend on the specific phase information about each element, i.e., we would be able to make the elements interfere with each other. But in practice, we can't do that because there are too many quantum particles in the environment and we have no way of keeping track of them and how they interact with the elements.

 

[secur]

From Wikipedia on quantum decoherence:

Wikipedia says: "Let us choose an expansion where the resulting basis elements interact with the environment in an element-specific way."

This is supposed to mean the "resulting basis elements" are eigenstates of the observable. For instance they might be eigenstates of spin, energy, orbital numbers, or position. In other interpretations such as Copenhagen we just specify that the observable is the one being measured by the experimenter. But in MWI decoherence is supposed to "automatically" specify / determine the observable. It turns out they haven't found a satisfactory way to do this. AFAIK, in MWI there's always a vague phrase like "interact with the environment in an element-specific way" attempting to define the observable (or its eigenbasis).. Of course in a paper they'll unpack it more, but it still won't be convincing.

Wikipedia says: "Such elements will—with overwhelming probability—be rapidly separated from each other by their natural unitary time evolution along their own independent paths. After a very short interaction, there is almost no chance of any further interference"

Right. Once you have the "natural" basis, decoherence will operate to destroy coherence (or "entanglement") and send the off-diagonal elements of the density matrix to zero.

 

[Demystifier]

Why do the elements separate rapidly from each other?

Because wave functions for a large number of particles live in a highly-dimensional space.

For an intuition, think of two balls moving in some space. If they move in 1 dimension, they often collide. In 2 dimensions, they collide only occasionally. In 3 dimensions the collisions are rather rare... Now try to imagine how unlikely a collision is in $10^{23}$ (!) dimensions.

 

[DrClaude]

Unfortunately I don't know of a good online textbook-style reference that discusses decoherence; it is a new enough research area that the best source is going to be peer-reviewed papers.

I would say that Schlosshauer comes close: https://arxiv.org/abs/quant-ph/0312059v4

 

[stevendaryl]

From Wikipedia on quantum decoherence:

"Let us choose an expansion where the resulting basis elements interact with the environment in an element-specific way. Such elements will—with overwhelming probability—be rapidly separated from each other by their natural unitary time evolution along their own independent paths. After a very short interaction, there is almost no chance of any further interference"

I never understood that part. Why do the elements separate rapidly from each other? Why don't they continue to interfere with each other and the environment in an ever more complex pattern? Is there any simple mathematical (or other kind of) reason for that?

To be able to see interference between two possibilities $A$ and $B$, you need to have a possible final state $C$ that is reachable both via $A$ and $B$. In other words, in the final state $C$, the information about which path was taken--via $A$ or via $B$--must be erased. For example, in the double-slit experiment, a photon might come from one slit or the other, but winds up at the same spot on the photographic plate, with no record of which slit it came from. If there is such a record, the interference will be destroyed.

If states $A$ and $B$ include the environment, and the environment is different in the two states, then it will be impossible to find a state $C$ that is reachable from both $A$ and $B$. The environment would act like a record of which state held in the past, and such records destroy interference effects.

 

[jartsa]

To be able to see interference between two possibilities $A$ and $B$, you need to have a possible final state $C$ that is reachable both via $A$ and $B$. In other words, in the final state $C$, the information about which path was taken--via $A$ or via $B$--must be erased. For example, in the double-slit experiment, a photon might come from one slit or the other, but winds up at the same spot on the photographic plate, with no record of which slit it came from. If there is such a record, the interference will be destroyed.

If states $A$ and $B$ include the environment, and the environment is different in the two states, then it will be impossible to find a state $C$ that is reachable from both $A$ and $B$. The environment would act like a record of which state held in the past, and such records destroy interference effects.

Where in this can we add coherent waves that become non-coherent ?

One bit of information entering an object splits the object-wave into two almost coherent parts. Then the parts quickly become much less coherent, as the "0" bit spreads around in one part, and the "1" bit spreads around in the other part? And then those two parts do not interfere, because they are not coherent?