High School How Does Quantum Darwinism Relate to the Born Rule and Observer Agreement?

[cube137]

No. All objects are quantum objects, in the sense that they are composed of quantum building blocks. But the number of building blocks in the object makes a difference. An object with only 1 building block, like an electron, is very different from an object with $10^{25}$ building blocks, like a piece of wood. Just because everything is a quantum object doesn't mean everything has to behave exactly the same.
First of all, this isn't strictly true. If the system is already in an eigenstate of the observable being measured, then the measurement doesn't change its state. But that's not really a practical issue, because if we already know the system is in an eigenstate, we don't need to measure it anyway because we already know its state.

However, once again, the size of the disturbance relative to the size of the object matters. If you are measuring an object that has only one quantum building block, like an electron, any measurement you make is going to disturb it significantly--heuristically, because the measurement itself has a minimum size which is basically one quantum building block. (For example, if we try to measure the electron by bouncing photons off of it, the minimum measurement we can make is to use one photon.) But if you are measuring an object with $10^{25}$ building blocks, like a piece of wood, there are lots of ways to measure it without significantly affecting its state, simply because of the huge number of building blocks. In fact, measuring an object of that size is really no different from what its environment is continually doing to it anyway--which is part of Zurek's point. The reason macroscopic objects like pieces of wood or Buckingham Palace look the same to everybody is that none of us have to do anything special to measure them; they're already being measured, all the time, just by being embedded in their environment. All Zurek is doing is giving more details about how that works and why it privileges particular states, the ones we think of as "classical" states and are intuitively familiar with.
No. See above.

Hi Peterdonis, you said above "The reason macroscopic objects like pieces of wood or Buckingham Palace look the same to everybody is that none of us have to do anything special to measure them; they're already being measured, all the time, just by being embedded in their environment".

For an isolated quantum system like electron.. you can measure them in different basis and the results would be different. But in macroscopic object which is already in an eigenstate.. you can't change it by measuring it again. I read Zurek paper again. His whole point is about being afraid to perturb the system so he has to rely on fragments being measured by observers without perturbing the system. But for isolated quantum system, re-preparing them can change the measurements.. so Zurek was not referring to small quantum objects since observers can change its properties by measuring them. So Zurek must be talking about macroscopic object. But is it not macroscopic object are already in eigenstate. Why did you say "The reason macroscopic objects like pieces of wood or Buckingham Palace look the same to everybody is that none of us have to do anything special to measure them". What would happen to the block of wood if we indeed do special to measure them directly without dealing with the fragments? Can you give an actual example how one can still perturb the wood by measuring it directly when it is already in an eigenstate. Unless Zurek means the wood is not in eigenstate. If so.. what kind of measurement can affect the wood when directly perturbed by the observers without intercepting any fragments (just for sake of discussions)?

 

[secur]

As I read it, the state $|2>_E$ is one of the basis states of the environment, so it can't possibly be a 2-dimensional subspace. The other two dimensions in the environment (for the case of a 3D environment) are in the $|0>_E + |1>_E$ subspace; and if the environment has more than 3 dimensions, they are also in the latter subspace (note that he says "the state $|+>_E$ exists in (at least) 2D subspace").

Yes the notation does indicate that. The environmental bases are called |0>, |1>, |2> (and |+> is the sum of the first two). But the 2-d System bases are called |0> and |2>. So the notation by itself doesn't imply dimensionality.

The key thing is that the two probabilities are 2/3 and 1/3. That's why he has to expand the first into 2-d subspace. Having done that he uses symmetry to assign equal probabilities (as well-explained in preceding pages) to each and thus "derives" the 2 : 1 ratio, the correct Born probabilities, ostensibly "from scratch".

It seems entirely artificial. For instance if the |2> S had the probability 2/3, instead, he would have assigned the 2 dimensions of E to that one. Suppose instead of 2/3 and 1/3 they were 3/4 and 1/4. Then he would assign 3 bases of E to the first and 1 to the second. Or if they were 3/10 and 7/10, he'd need 10 bases, 3 for the first and 7 for the second. But there's no justification for this except the need to match the Born rule.

What's missing is an argument that when one coefficient is sqrt 2/3 and the other 1/3 then the states of E they're entangled with must have 2 and 1 dimensions respectively. But as far as I know the number of E basis elements is not particularly related to those coefficients. It's as though he's supposing an equipartition law, treating the bases as degrees of freedom. But in fact those probabilities are affected by many different things. Given the ratio 2 : 1 of E dimensions, AFAIK the first coefficient might be sqrt .1 and the second .99. Isn't that so?

This paper gives only a brief sketch but I did find more extensive treatments on the net. Just google quantum darwinism. You could add "ancilla" because that's what the discussion seems to center on. Nothing I found contradicts what is said above. I'm probably missing something, but it seems fishy to me.

... what kind of measurement can affect the wood when directly perturbed by the observers without intercepting any fragments (just for sake of discussions)?

Fragments are the only way observers (or, test apparatus) can receive information. So you can't do a measurement without intercepting fragments. AFAIK.

 

[PeterDonis]

in macroscopic object which is already in an eigenstate.. you can't change it by measuring it again.

An eigenstate of what observable? If you measure a different observable than the one it's in an eigenstate of, you certainly can change it by measuring it, whether it's microscopic or macroscopic.

Furthermore, Zurek does not appear to me to be claiming that macroscopic objects like a block of wood are in eigenstates of any particular observable. He says they're in mixed states (improper mixtures obtained by tracing over the environment, per my previous post on that). Mixed states aren't eigenstates.

What would happen to the block of wood if we indeed do special to measure them directly without dealing with the fragments?

What "special" thing would you do? If you look at it, you're looking at photons bouncing off of it, which the environment is already doing, all the time. Ditto for touching it (air molecules are hitting the wood all the time), listening to the sound it makes (air vibrations are being propagated from the wood all the time), smelling it (the molecules you smell are interacting with the surroundings all the time), etc.

In other words, when you think of yourself as "measuring the wood directly", you aren't really; you are really interacting with the same "fragments" that are part of the environment and are storing copies of information about the wood. To interact with the wood directly, without involving the environment, you would need to do something like cut into it--which would certainly change its state.

 

[PeterDonis]

the notation by itself doesn't imply dimensionality.

I wasn't basing my statement on the notation. I was basing it on the description of how the states are defined.

What's missing is an argument that when one coefficient is sqrt 2/3 and the other 1/3 then the states of E they're entangled with must have 2 and 1 dimensions respectively.

I'm actually not sure that's the general strategy he's trying to describe. For one thing, it doesn't work with coefficients that don't square to rational numbers; for example, $\sqrt{\pi}$ and $\sqrt{1 - \pi}$. But I need to look into this aspect more; I agree there's certainly something missing from the paper we've been looking at.

 

[Simon Phoenix]

So proper mixed state and improper mixed state are not standard usage

As far as I know they are standard terms.

Let's suppose I send you a bunch of spin-1/2 particles and I tell you that I've prepared each one in either the state $|0 \rangle$ or the state $|1 \rangle$ where I've made the choice entirely at random with probabilities $p$ and $q=1-p$, respectively.

The density matrix describing the particles is$$\rho = p |0 \rangle \langle 0| + q |1 \rangle \langle 1|$$ this can be interpreted as a statistical mixture of pure states. Each particle is actually 'in' a pure state (because it's been prepared that way) but there's no way for you to tell me which pure state should be attached to any given particle (without doing a measurement).

This is called a 'proper' mixture.

Now suppose I do the same kind of thing and send you a bunch of particles, but now what I'm sending you is the following; each particle I send you is one of the particles from an entangled pair given by the pure state $$| \psi \rangle = \sqrt {p} |0,0 \rangle + \sqrt {q} |1,1 \rangle$$ where the first label in the states on the right hand side describes my particles. The particles you have are now described by tracing out the global density operator over my states, and we end up with a density operator for your particles that is $$\rho = p |0 \rangle \langle 0| + q |1 \rangle \langle 1|$$ which is exactly what we get in the previous case.

In this second case we call this an 'improper' mixture - but the 2 mathematical descriptions are exactly the same. It means that there's no experiment you can do on your particles alone to distinguish whether you have a 'proper' or 'improper' mixture, even though conceptually we can see they derive from very distinct (global) physical situations.

So, if you're just working with the particles I've sent you - there's no way for you to tell whether I've prepared them according to the first prescription (as a statistical mixture of pure states) or whether I've sent you particles that have an entangled partner particle.

[EDIT : thanks Peter for pointing out where I wasn't wholly clear]

 

[PeterDonis]

As far as I know they are standard terms.

I assume you mean, with the particular definitions you are giving them (which are not the same as the ones cube137 was using earlier).

It means that there's no experiment we can do to distinguish between 'proper' and 'improper'

I assume you mean, no experiment on the particles you sent. Of course in the second case you can do experiments involving the other particles in each pair that will tell you about the global state and hence will distinguish it from the first case.

 

[Simon Phoenix]

I assume you mean, no experiment on the particles you sent. Of course in the second case you can do experiments involving the other particles in each pair that will tell you about the global state and hence will distinguish it from the first case

Yes - oops I should have been clearer. Original post now fixed (I hope)

 

[Simon Phoenix]

In other words, when you think of yourself as "measuring the wood directly", you aren't really; you are really interacting with the same "fragments" that are part of the environment and are storing copies of information about the wood.

Yes, and clearly that can contain only a very limited amount of information about the wood - so the environment isn't keeping a whole load of copies of $|wood \rangle$, but just copies of limited information (the 'observables'). Have I interpreted that correctly?

I was intrigued by Zurek's allusion to error-correcting codes and Shannon's theorem. If we look at the 'noisy typewriter' channel then we have a channel that is like a broken typewriter so that if we hit the B key then we could get A,B or C being printed with equal probability, and so on for every key. If we only ever use the B,E,H,... keys then we can communicate error-free on this noisy channel. So in a very loose way I'm thinking that if B is the observable - then 'environment' states A,B or C contain the same information (about B).

 

[PeterDonis]

the environment isn't keeping a whole load of copies of $|wood \rangle$, but just copies of limited information (the 'observables').

Yes, I agree. Zurek seems to be suggesting that the environment is keeping many copies of information about the position of the block of wood as a whole--which basically means the position of its center of mass and its size. That is certainly a small fraction of all the possible information about the $10^{25}$ or so atoms in the wood. (This is still true even if we also imagine the environment to be storing many copies of information about the wood's surface color, texture, etc.)

 

[secur]

I'm actually not sure that's the general strategy he's trying to describe. For one thing, it doesn't work with coefficients that don't square to rational numbers; for example, $\sqrt{\pi}$ and $\sqrt{1 - \pi}$. But I need to look into this aspect more; I agree there's certainly something missing from the paper we've been looking at.

If probabilities (meaning, of course, squares of the coefficients - assuming Born rule) are irrational or transcendental he'll use converging sequence of rationals.

Extension of this proof to the case where probabilities are commensurate is conceptually straightforward but notationally cumbersome.

Probabilities being commensurate, meaning the ratios of them are rational. (Which is, actually, equivalent to the probabilities themselves being rational.)

The case of noncommensurate probabilities is settled with an appeal to continuity.

If they're incommensurate there are non-rationals involved, like $\sqrt{1/\pi}$ and $\sqrt{1 - 1/\pi}$. In that case he'll rely on the density of rationals in the reals.

See this paper https://arxiv.org/pdf/quant-ph/0405161.pdf:

D. Born’s rule: the case of unequal coefficients

We now introduce a counterweight / counter C. It can be thought of either as a subsystem extracted from the environment E, or as an ancilla that becomes correlated with E so that the combined state is: {formula} where {C k} are orthonormal. Moreover, we assume that {C k} are associated with subspaces of {The Hilbert Space} of sufficient dimensionality so that the ‘fine-graining’ represented by {formula} is possible.
...
This is Born’s rule. The extension to the case where {the probabilities} are incommensurate is straightforward by continuity as rational numbers are dense among reals.

(BTW I should use LaTex.) Here he goes into the strategy in more detail. It seems there's a vital piece missing, why should the subspace dimensions be related (at least, so directly) to probabilities / coefficients?

Note that reproducing Born rule constitutes one of the major difficulties in MWI. AFAIK it's never been solved. So I was particularly interested in how he'd handle it. The discussion preceding this, involving Schmidt decomposition and "swapping" to establish equiprobabilities, is fine and interesting. But is it a smokescreen, to cover this fine-graining, which is key to the whole approach? If it really works, it's an MWI breakthrough.

 

[cube137]

Yes, I agree. Zurek seems to be suggesting that the environment is keeping many copies of information about the position of the block of wood as a whole--which basically means the position of its center of mass and its size. That is certainly a small fraction of all the possible information about the $10^{25}$ or so atoms in the wood. (This is still true even if we also imagine the environment to be storing many copies of information about the wood's surface color, texture, etc.)

For those of us indoctrinated into Many worlds... so Zurek fragments only belong to one world or branch.. and the reasons we can't perturb the quantum object directly is akin to the reason we can't influence or affect all the worlds or branches at once?? This is especially apparent in Zurek own words at page 8 of https://arxiv.org/pdf/1412.5206v1.pdf "Quantum Darwinism, Decoherence, and the Randomness of Quantum Jumps"

"Repeatability leads to branch-like states, Eq. (13),
suggesting Everettian `relative states' [19]. There is no
need to attribute reality to all the branches. Quantum
states are part information. As we have seen, objective
reality is an emergent property. Unobserved branches
can be regarded as events potentially consistent with the
initially available information that did not happen. Information
we gather can be used to advantage|it can lead
to actions conditioned on measurement outcomes [5]."
<snip>
"Quantum Darwinism explains why we see only one
branch. One can dismiss other branches, e.g. with an appeal
to Everett [19]. So we can account for a perception
of collapse. Thus, while unitarity precludes fundamental
collapse, local observables that reveal branches do not
commute with the global observable whose eigenstates
are coherent superpositions of all the branches, Eq. (13).
Therefore, local observers have no way to probe (hence,
cannot perceive) the global state vector."

For all the superposition experiments that have been performed. Is the idea that superposition is just quantum information compatible or do both copies really exist at same time (such as spin up or spin down that is in superposition.. is there really a copy of spin up and spin down or can superposition of it be simply quantum information and is this compatible with all data so far??)

 

[secur]

is there really a copy of spin up and spin down or can superposition of it be simply quantum information and is this compatible with all data so far?

It could be "simply quantum information" or both copies could "really exist". Neither of those statements is well-defined (since they're philosophy not science) but they're both possible, and so are other interpretations.

What counts is the physics and math. That starts with mainstream QM math, interpretation-free. But also "What does Zurek (or whoever) mean here"? "Is his math consistent"? "Does it really work"? - are good questions. My advice, avoid the question "Is it true?" - leave that to philosophy forums.

It's also not worth debating "what is Zurek's stance" but FWIW I think it's MWI.

There is no need to attribute reality to all the branches.

That doesn't sound like MWI.

unitarity precludes fundamental collapse ...

But that does. FWIW I think he's an MWI believer making a commendable effort to kick the habit, knowing such belief is scientifically unjustifiable. But it doesn't matter, of course.

My advice: don't either believe or disbelieve a particular interpretation. Also, don't care what someone else's beliefs are. It's worth discussing interpretations, but only with the goal of making sure they really are valid. That brings me back to your question. Yes, the superpositions could be "really real" or they could be "quantum information" - whatever those two phrases mean. Both are vague enough to be compatible with the facts of QM.

And that's my last word - literally - concerning the "truth" of interpretations!

 

[cube137]

Is the reason many of you guys dislike interpretations is because you can still turn it the other way around and say that the classical world is all there is and everything is just observation effect and Zurek stuff is just all imagination? Prior to the 1970 before the time of Zeh there was no discussion about decoherence. So can those pre-Zeh, pre decoherence period still be correct in that in the buckyball experiment where different thermal condition can affect the interference. It is really all classical world with the observation effect. This means in spite of all evidences of decoherences.. it doesn't prove that Zurek is correct but Bohr could still be correct about the classical and quantum world?

How many of physicists believe in Bohr and believe in Zurek?
For those who only want to focus on the mathematics like Peterdonis, does it mean either Bohr or Zurek could still be correct and all the evidences of decoherences don't prove (or disprove) any of them?

 

[PeterDonis]

the reason many of you guys dislike interpretations

The reason interpretations are not good topics for discussion here on PF is that they are experimentally indistinguishable; they all make exactly the same predictions for the results of all experiments. So any difference between them is not a matter of physics; it's a matter of "philosophy", or whatever you want to call it, but it not being a matter of physics is what makes it an unsuitable topic for PF discussion (at least in the physics forums).

does it mean either Bohr or Zurek could still be correct and all the evidences of decoherences don't prove (or disprove) any of them?

No evidence can "prove" or "disprove" any interpretation relative to any other, since, as above, they all make exactly the same predictions for all experimental results.