Everett's interpretations and Macroscopic quantum states

[deneve]

Max Tegmark in his paper “Many worlds in context” http://arxiv.org/abs/0905.2182

Argues that …. .“Everett’s MWI is simply standard QM with the collapse postulate removed, so that the Schrödinger equation holds without exception”. He also argues that from this we can deduce that not only microscopic-superpositions are inevitable, but also that macroscopic-superpositions (say of a cat being dead and of being alive) are perfectly legitimate quantum states.

However, the assumption that macroscopic superpositions are valid seems puzzling to me. The well known Schrödinger’s cat scenario gives:|psi> a|up>|cat alive> + b|down>|cat dead>.But is this really viable? cats are very complicated Can a cat really be in a macroscopic pure quantum state like |cat Alive> just because we can write it on the page as a bra vector? Is it the intention that the state vector of the cat can be explained as the tensor product of states of all the particle states that make up the cat?

|cat alive> = |molecule 1> ⊗ |molecule 2> ⊗ |molecule 3> ⊗...

If instead we suppose the pure cat state to be an energy eigenstate, then would it not be in a stationary state. Its only dependence on time being through some phase factor which makes no physically distinguishable difference - surely a cat in this state would be in a frozen state (i.e. |cat dead> )?

If the cat was said to be in a mixed state then, for this interpretation is it the implication that it would need to be an "improper mixture" with some off diagonal terms persisting in the off diagonal terms of the density matrix in some basis?

 

[bhobba]

This is one of those misconceptions about MW.

There is no collapse in MW - everything evolves as per Schroedinger's equation etc etc.

But the interpretation, at least in modern times, relies on decoherence.

What's going in Schroedinger's Cat according to MW is, first the observation occurs at the particle detector - that's where the weirdness is, everything else is common-sense classical from that point on. So you have two states - |detected> and |not detected>. But this is an improper mixed state ∑p1 |detected><detected| + p2 |not detected><not detected|. The cat etc is best viewed as part of the rest of the world with these states ie |detected> is really |rest of world>|detected>.

Here it depends on a crucial idea that often is not emphasised - namely the difference between proper and improper mixed states:
http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf

Personally I find a lot of Tegmark's stuff like Quantum Suicide, to be blunt, rubbish.

Thanks
Bill

 

[deneve]

Thanks bhoppa for that, I had Hensen's paper on my reading list.

However,

From your answer I see that you are considering |rest of the world> = |row> to be a (pure) quantum state. which is clearly a macroscopic one, which can be part of a superposition like |ψ>=|∑a_i|system>|detector ready>|row>. Does this mean you think such states can form macroscopic superpositions?

I thought this would have to be so because |ψ> obeys S's equation at all times (in Everett).
So quantum weirdness can still happen. The full density matrix would be something like (with cat included in |row> or acting as the detector!)
ρ = |ψ><ψ|
=∑p1 |system up>||up detected>|row u><row u|<up detected|<system up|+p2|system down>||down detected>|row d><row d|<down detected|

Tracing over the "row" would give

≅∑p1 |system up>||up detected><up detected|<system up|+p2|system down>||down detected><down detected |system down|

The approximately equal bit being due to the fact that off diagonal terms albeit very small are still hanging around since
<row d|row u> remains finite but ever more getting smaller as the quantum effect of the measurement leaks into the environment.

This was my understanding of why the expression above was called an improper mixture - essentially because quantum weirdness is still there I thought this was what Hensen was getting at. Indeed for an Everett interpretation I thought that all mixtures must necessarily be improper !
If you have a copy of J Barrett's "The Quantum Mechanics of Minds and Worlds" footnote p96 seems to concur with my thinking here. See also http://arxiv.org/abs/quant-ph/0402094v2
Am I understanding this correctly?

 

[bhobba]

From your answer I see that you are considering |rest of the world> = |row> to be a (pure) quantum state. which is clearly a macroscopic one, which can be part of a superposition like |ψ>=|∑a_i|system>|detector ready>|row>.

Whew. You lost me. Yes |row> is a pure state, so is, for example, |detected>, and we have |row>|detected> as its combined pure state. Now |detected> is a state of the detector so that would preclude a superposition like you mention.

I am making an assumption that |row> evolves in a purely deterministic classical way depending on if the particle was detected or not, because of the setup.

This was my understanding of why the expression above was called an improper mixture - essentially because quantum weirdness is still there

It is an interpretive assumption of MW that the 'parts' of an improper mixed state are separate worlds. By assumption quantum weirdness is gone in each world. By itself the concept of improper mixture explains 'apparent collapse' - actual collapse requires some further assumption - which in MW is that they are separate worlds.

Thanks
Bill

 

[deneve]

Whew. You lost me. Yes |row> is a pure state, so is, for example, |detected>, and we have |row>|detected> as its combined pure state. Now |detected> is a state of the detector so that would preclude a superposition like you mention.

I'm not sure I understand why |detector> precludes a state like
|ψ>=∑a_i|system>|detector ready>|row>.

→ a_1|system up>|detector up>|row up> + a_2|system down>|detector down>|row down>,

which is Everett's "universal wave function" and with say "detected" as meaning "detector read up". Maybe I misinterpret ?.
You say that

"It is an interpretive assumption of MW that the 'parts' of an improper mixed state are separate worlds. By assumption quantum weirdness is gone in each world. By itself the concept of improper mixture explains 'apparent collapse' - actual collapse requires some further assumption - which in MW is that they are separate worlds." (sorry can't figure how to do quotes properly yet!).

I agree with your first statement but I didn't think there ever was an actual collapse in Everett. So I thought that the quantum weirdness was always there but we'd never be able to perform an experiment which detected the interference information which leaks into the environment during decoherence. However I agree that FAPP it is generally accepted that the decohered worlds would effectively be separate worlds.

I suppose what my original post was addressing was where the cut between classical and quantum lies. Everettians would say there is no cut, but I don't yet understand how, if there is no cut, that such a state such as |cat alive>, or indeed |detector ready> can be prepared in a way that would make cats and detectors able to display interference when thrown at a pair of slits?

 

[Swamp Thing]

how, if there is no cut, that such a state such as |cat alive>, or indeed |detector ready> can be prepared in a way that would make cats and detectors able to display interference when thrown at a pair of slits?

deneve, some of your questions have worried me, too. But I did not know how to articulate them clearly as you have done. I have sometimes felt that there is a "preparation problem" that is perhaps complementary to the measurement problem. When we say |detector ready> is it not true that the detector could be "ready" in millions of ways (microstates) that we have (for our convenience) classified into the single macrostate called "ready"? And is it not true that |pointer up> and |pointer down> are also macrostates that could encompass all sorts of different micro configurations? In that case there could be all sorts of differenct amplitudes to go from one of the "ready" microstates to one of the "up" microstates. I don't know how to take this further towards any conclusion, but I would like to know if it makes sense up to this point.

 

[bhobba]

I'm not sure I understand why |detector> precludes a state like
|ψ>=∑a_i|system>|detector ready>|row>.\

I don't understand how it can. I don't even understand what you mean by |detetor ready> - we are considering when it becomes entangled with the radioactive source and decoherence has occurred.

I will again go through it.

You have this detector and it can be in two states |detected> and |undetected>. Due to some process in the detector, via decoherene, it will be in the improper mixed state Σ p1 *|detected><detected| + p2*|undetected><undetected|. Also, via decoherence, that is the correct basis because decoherence does solve the basis problem ie even though it is mathematically equivalent to different basis this one has been singled out. Now the physical interpretation of MW, and it is an assumption with actual consequences, is each element of that mixed state is a separate world. In one world it is in the state |detected>, in the other |undetected>. Now really of course the state is |detected>|row> and similarly for undetected. Let's take the world with |detected>|row>. Because of the set-up that evolves to the cat dying and being dead whether the box is opened or not. This is why Schroedinger's Cat is basically a crock - it isn't what people make it out to be with the observer causing collapse etc - the collapse happened at the detector. The detector is in the state |detected> - its not in a superposition with anything. Such is precluded by the MW assumption we are in a world where it detected the particle. Outside the universal wavefunction continues merrily along - things are still in superposition - but we are not considering outside - we are considering it inside where things are entangled.

If that doesn't explain it, I think I will have to leave it to someone else because I really can't explain it any better.

I will however mention Wallaces book explains this all in a much more detailed and sophisticated way using ideas from decoherent histories to bootstrap decoherence etc because there is an assumption of the Born Rule in tracing over the environment in decoherence. But you need to consult the book for that.

Thanks
Bill

 

[Swamp Thing]

I don't even understand what you mean by |detetor ready> -

In any formulation of quantum mechanics, the apparatus starts in a “ready” state, which is a way of saying “it hasn’t yet looked at the thing it’s going to observe” (i.e., the particle). More specifically, the apparatus is not entangled with the particle; their two states are independent of each other. So the quantum state of the particle+apparatus system starts out like this:

(“spin is up” + “spin is down” ; apparatus says “ready”)

This is from http://www.preposterousuniverse.com...ion-of-quantum-mechanics-is-probably-correct/

In my post #6, I asked whether this "ready" state is really a quantum state or a macrostate that could consist of any of a huge set of possible microstates. For example, a grain of photo emulsion is not "prepared" in a given unique state, nor do we later check to find out about its internal degrees of freedom after it has interacted with a photon. The observer only cares that the emulsion grain scatters light in a different way if it has absorbed a photon. deneve, you may like to look at this: http://arxiv.org/abs/physics/0406014

 

[bhobba]

In my post #6, I asked whether this "ready" state is really a quantum state

In Schrodinger Cat this ready state is simply a variant of detected and not detected because the arbitrary time interval hasn't passed yet. It has exactly the same form except the p1 and p2 are different. It not a pure state - its a mixed state hence is not in superposition.

For example, a grain of photo emulsion is not "prepared" in a given unique state, nor do we later check to find out about its internal degrees of freedom after it has interacted with a photon.

In that situation the emulsion becomes entangled with the photon, decoherence happens, and you get an interaction at a particular position.

The ready state simply means it hasn't interacted with the photon yet.

Thanks
Bill

 

[deneve]

deneve, some of your questions have worried me, too. But I did not know how to articulate them clearly as you have done. I have sometimes felt that there is a "preparation problem" that is perhaps complementary to the measurement problem. When we say |detector ready> is it not true that the detector could be "ready" in millions of ways (microstates) that we have (for our convenience) classified into the single macrostate called "ready"? And is it not true that |pointer up> and |pointer down> are also macrostates that could encompass all sorts of different micro configurations? In that case there could be all sorts of differenct amplitudes to go from one of the "ready" microstates to one of the "up" microstates. I don't know how to take this further towards any conclusion, but I would like to know if it makes sense up to this point.

Yes it makes sense to me - but measuring instruments have got be "good" in the sense that the Hamiltonian has got to be such that any unitaristic evolution means the pointer will end up pointing in the right place indicating a correct outcome by deviation from any designed ready state.

 

[deneve]

I don't understand how it can. I don't even understand what you mean by |detetor ready> - we are considering when it becomes entangled with the radioactive source and decoherence has occurred.

I will again go through it.

You have this detector and it can be in two states |detected> and |undetected>. Due to some process in the detector, via decoherene, it will be in the improper mixed state Σ p1 *|detected><detected| + p2*|undetected><undetected|. Also, via decoherence, that is the correct basis because decoherence does solve the basis problem ie even though it is mathematically equivalent to different basis this one has been singled out. Now the physical interpretation of MW, and it is an assumption with actual consequences, is each element of that mixed state is a separate world. In one world it is in the state |detected>, in the other |undetected>. Now really of course the state is |detected>|row> and similarly for undetected. Let's take the world with |detected>|row>. Because of the set-up that evolves to the cat dying and being dead whether the box is opened or not. This is why Schroedinger's Cat is basically a crock - it isn't what people make it out to be with the observer causing collapse etc - the collapse happened at the detector. The detector is in the state |detected> - its not in a superposition with anything. Such is precluded by the MW assumption we are in a world where it detected the particle. Outside the universal wavefunction continues merrily along - things are still in superposition - but we are not considering outside - we are considering it inside where things are entangled.

If that doesn't explain it, I think I will have to leave it to someone else because I really can't explain it any better.

I will however mention Wallaces book explains this all in a much more detailed and sophisticated way using ideas from decoherent histories to bootstrap decoherence etc because there is an assumption of the Born Rule in tracing over the environment in decoherence. But you need to consult the book for that.

OK so I think you are saying that the atom is initially in a superposition but very quickly decoheres into a state which the the detector picks up, thereby then being in either the detected (or undetected) state. The worlds have effectively split and so now you have the mixed state you show. Thus there is no need to consider superpositions such as I indicated.
However, I've never actually thought that the observer opening the box caused the superposition to suddenly collapse. I think something more on the lines of Sean Carroll - see his summary on p3 of
http://arxiv.org/pdf/1405.7907.pdf although I'm not sure I completely agree with his separating an observer from the environment as he does. I think the cat is long either alive or dead before we peek in. When we see it, we can confirm which world decoherence has left us in.

Thank you for your helpful remarks. I have most of Wallace's papers and the volume by Barrett Saunders and Wallace.