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

[cube137]

Quantum Darwinism is about fragments and how different observers should observe the same thing, may I know where the born rule is applied. does the proper mixed state (or one outcome) occurs after the fragments leave the objects, halfway or when the observers received the fragments?

 

[PeterDonis]

Quantum Darwinism

Reference, please?

 

[cube137]

Reference, please?

https://en.wikipedia.org/wiki/Quantum_Darwinism or https://arxiv.org/pdf/0903.5082v1.pdf

 

[PeterDonis]

may I know where the born rule is applied

As far as I can tell, it isn't; the Zurek paper derives all of its key conclusions without ever using the Born rule. The paper also stresses that it is "interpretation free"; there is no claim that collapse occurs or that it doesn't occur. That question is simply not addressed.

 

[cube137]

As far as I can tell, it isn't; the Zurek paper derives all of its key conclusions without ever using the Born rule. The paper also stresses that it is "interpretation free"; there is no claim that collapse occurs or that it doesn't occur. That question is simply not addressed.

The paper quoted below (I'd like to understand what it means exactly by "Quantum states acquire objective existence when reproduced in many copies", I hope seasoned quantumists like Stevendaryl or Simon Phoenix can comment. Peterdonis is a relativist. And also what it means "Consensus between records deposited in fragments of E looks like collapse"):

"Consensus between records deposited in fragments of E looks like collapse". In this sense we have accounted for postulate (iv) using only very quantum postulates (i)- (iii). In particular, in deriving and analyzing Eq. (5) we have not employed Born's rule, axiom (v). We shall be therefore able to use our results as a starting point for such a derivation in the next section.

There was nothing nonunitary above { unitarity was the crux of our argument, and orthogonality of branch
seeds our main result. Relative states of Everett [26, 27, 28] come to mind. One could speculate about reality of
branches with other outcomes. We abstain from this {our discussion is interpretation-free, and this is a virtue.
Indeed, \reality" or \existence" of universal state vector seems problematic. Quantum states acquire objective
existence when reproduced in many copies. Individual states { one might say with Bohr { are mostly information,
too fragile for objective existence. And there is only one copy of the Universe. Treating its state as if it really
existed [26, 27, 28] seems unwarranted and \classical".

 

[PeterDonis]

I'd like to understand what it means exactly by "Quantum states acquire objective existence when reproduced in many copies"

First you have to understand what he means by "objective existence". Basically, it looks to me like he means "repeatability": repeated measurements of a system must give consistent results. The "many copies" is what allows that to happen: each measurement basically takes its information from a different copy. And each measurement transfers the information in the copy to the measuring device, so each copy can only be used once to obtain information about the measured system--that's why you need many copies to have repeated measurements give consistent results.

also what it means "Consensus between records deposited in fragments of E looks like collapse"

I think the key here is "looks like collapse". What he is trying to explain is why, regardless of whether there actually is collapse or not, our observations make it look like there is collapse--where here "look like collapse" basically means the same thing as "objective existence" above, i.e., repeated measurements giving consistent results. And, as above, what allows that to happen is that there are many "records" (copies), each one in a different fragment of E (the environment), and all of them containing the same information about the measured system.

Again, this is all "interpretation free", as the paper says. It doesn't say that collapse "really" happens; it leaves that question open. It just explains why it looks that way to us.

Peterdonis is a relativist.

Just to clarify, I do post most often in the relativity forum, but I'm reasonably conversant with QM and QFT. The main reason I don't post more often in this forum is that it seems far more prone to threads that are more about philosophy than physics (QM is funny that way). But this thread topic does not fall into that category.

 

[cube137]

First you have to understand what he means by "objective existence". Basically, it looks to me like he means "repeatability": repeated measurements of a system must give consistent results. The "many copies" is what allows that to happen: each measurement basically takes its information from a different copy. And each measurement transfers the information in the copy to the measuring device, so each copy can only be used once to obtain information about the measured system--that's why you need many copies to have repeated measurements give consistent results.

Ok. To understand what Zurek was driving at. Supposed there was not "many copies".. but only one copy, what would happen?

This is how I understood Zurek. In the second paragraph he wrote:

"Fragility of states is the second problem with quantumclassical
correspondence: Upon measurement, a general
preexisting quantum state is erased - it "collapses" into
an eigenstate of the measured observable. How is it then
possible that objects we deal with can be safely observed,
even though their basic building blocks are quantum?"

Can we take as example a macroscopic object like a piece of wood. Is this what Zurek refer to as objects where their basic building blocks are quantum? Or was Zurek referring to object as a small isolated quantum system like a buckyball or a particle? Without the "many copies".. did Zurek mean if different people interact with the same wood, they can re-prepare it and the wood can change shape? Or was he referring only to atoms or tiny isolated quantum systems where people can re-prepare it by measuring it? But I don't think he was referring only to electrons, atoms, or other microscopic systems but big objects like wood too. Is that right?

I think the key here is "looks like collapse". What he is trying to explain is why, regardless of whether there actually is collapse or not, our observations make it look like there is collapse--where here "look like collapse" basically means the same thing as "objective existence" above, i.e., repeated measurements giving consistent results. And, as above, what allows that to happen is that there are many "records" (copies), each one in a different fragment of E (the environment), and all of them containing the same information about the measured system.

Again, this is all "interpretation free", as the paper says. It doesn't say that collapse "really" happens; it leaves that question open. It just explains why it looks that way to us.
Just to clarify, I do post most often in the relativity forum, but I'm reasonably conversant with QM and QFT. The main reason I don't post more often in this forum is that it seems far more prone to threads that are more about philosophy than physics (QM is funny that way). But this thread topic does not fall into that category.

 

[secur]

Quantum Darwinism is about fragments and how different observers should observe the same thing, may I know where the born rule is applied. does the proper mixed state (or one outcome) occurs after the fragments leave the objects, halfway or when the observers received the fragments?

Environment is a witness to the state of the system.

... Quantum states acquire objective existence when reproduced in many copies.

... Consensus between records deposited in fragments of E looks like collapse.

... Proliferation of records allows information about S to be extracted from many fragments of E (in the example above, photon E). Thus, E acquires redundant records of S. Now, many observers can nd out the state of S independently, and without perturbing it. This is how preferred states of S become objective.

"Preferred states of S become objective" when "many observers find out the state of S independently". So the answer to your question is the third option: "when the observers received the fragments". The outcome occurs when multiple observers receive consensus information about the state. However, it can occur without humans: i.e. the environment itself can "witness" those multiple fragments.

The fragments concept is not very important in this work, neither is the "Darwinian" ansatz. The key concepts are pointer states, einselection, envariance, equiprobability (symmetry, swap), finegraining (using ancilla C) and a couple others. Some of it's pretty clever but it's not "interpretation-free". Quantum Darwinism is a flavor of MWI.

Supposed there was not "many copies".. but only one copy, what would happen?

More or less nothing. Only one copy would not be enough to allow objective outcome.

Can we take as example a macroscopic object like a piece of wood. Is this what Zurek refer to as objects where their basic building blocks are quantum?

Yes.

did Zurek mean if different people interact with the same wood, they can re-prepare it and the wood can change shape? Or was he referring only to atoms or tiny isolated quantum systems where people can re-prepare it by measuring it? But I don't think he was referring only to electrons, atoms, or other microscopic systems but big objects like wood too. Is that right?

No. "Re-prepare" applies only to quantum objects.

 

[cube137]

"Preferred states of S become objective" when "many observers find out the state of S independently". So the answer to your question is the third option: "when the observers received the fragments". The outcome occurs when multiple observers receive consensus information about the state. However, it can occur without humans: i.e. the environment itself can "witness" those multiple fragments.

The fragments concept is not very important in this work, neither is the "Darwinian" ansatz. The key concepts are pointer states, einselection, envariance, equiprobability (symmetry, swap), finegraining (using ancilla C) and a couple others. Some of it's pretty clever but it's not "interpretation-free". Quantum Darwinism is a flavor of MWI.
More or less nothing. Only one copy would not be enough to allow objective outcome.
Yes.
No. "Re-prepare" applies only to quantum objects.

You agreed that a piece of wood is what Zurek refers to as objects where their basic building blocks are quantum. Hence wood can be considered as quantum objects in quantum Darwinism... hence you can re-prepare a piece of wood in the world of Zurek. But then, maybe let's refer to "Re-prepare" for electrons, photons, and small quantum system, while reserved the word "Transmute" for changing the shape of the wood. If you won't agree. But Zurek just said macroscopic object can considered as quantum object too like buckyball. Remember in the world of quantum decoherence, everything is quantum. Our classical bias only came because of Copenhagen and because of Everett where each branch is a solid classical world hence trapping one to the mindset that there is a classical world when everything is quantum at the core.

 

[PeterDonis]

Supposed there was not "many copies".. but only one copy, what would happen?

If there were only one copy, you would not be observing a macroscopic object and you would not be getting measurement results of the sort we usually associate with such objects. You would be looking at, say, a quantum computing experiment where a single qubit is being manipulated. And the results you get would not look like something with "objective existence" in the sense Zurek is using that term.

Can we take as example a macroscopic object like a piece of wood. Is this what Zurek refer to as objects where their basic building blocks are quantum?

All objects have basic building blocks that are quantum.

Without the "many copies".. did Zurek mean if different people interact with the same wood, they can re-prepare it and the wood can change shape?

No. Zurek is saying that it is impossible for a piece of wood not to have "many copies" of the information about it in the environment, because it is composed of such a huge number of quantum building blocks and it is not at all isolated--each building block is continually interacting with other building blocks and with the environment. To be able to "re-prepare" a piece of wood, you would have to first isolate it from its environment, and then control all the interactions between its quantum building blocks with sufficient accuracy to allow quantum interference effects to be observable. In principle that could be done, but in practice it is impossible.

Or was he referring only to atoms or tiny isolated quantum systems where people can re-prepare it by measuring it?

The reason we can "re-prepare" atoms or tiny isolated quantum systems (like qubits in a quantum computer) by measuring them is that they are tiny isolated quantum systems; we can prevent them from interacting with their environment, and they contain a small enough number of quantum building blocks (just one, in the ideal case) that we can control all their interactions.

 

[PeterDonis]

"Re-prepare" applies only to quantum objects.

I think this is misstated. A correct statement would be that re-preparing is only possible in practical terms for objects containing a small number of quantum building blocks which can be isolated sufficiently from the environment that all of their interactions can be controlled. But there is no magic boundary at which objects stop being quantum. All objects are composed of quantum building blocks. It's just that not all objects meet the practical conditions for us being able to re-prepare them.

 

[PeterDonis]

hence you can re-prepare a piece of wood in the world of Zurek.

Not in practical terms. See my previous posts.

 

[secur]

did Zurek mean if different people interact with the same wood, they can re-prepare it and the wood can change shape? Or was he referring only to atoms or tiny isolated quantum systems where people can re-prepare it by measuring it? But I don't think he was referring only to electrons, atoms, or other microscopic systems but big objects like wood too. Is that right?

No. "Re-prepare" applies only to quantum objects.

You agreed that a piece of wood is what Zurek refers to as objects where their basic building blocks are quantum. Hence wood can be considered as quantum objects in quantum Darwinism... hence you can re-prepare a piece of wood in the world of Zurek. But then, maybe let's refer to "Re-prepare" for electrons, photons, and small quantum system, while reserved the word "Transmute" for changing the shape of the wood.

I think this is misstated. ... It's just that not all objects meet the practical conditions for us being able to re-prepare them.

To back up a little, QD attempts to answer the question why large-scale quantum systems - for instance a block of wood - present one stable state. I.e., why we cannot, in practice, "transmute" or re-prepare them. The answer: decoherence via interaction with the environment causes einselection of pointer states, wherein all phase information is lost. Once that happens we obtain "objective reality". All subsequent measurements will give the same state. You can no longer re-prepare that block of wood, as you can with small quantum objects such as an electron.

Obviously this is a practical program. QD tries to explain why, in practice, a block of wood can't be re-prepared, even though in theory it can. Here's a relevant quote:

We have already noted the special role of the pointer observable. It is stable, and, hence, it leaves behind information-theoretic progeny - multiple imprints, copies of the pointer states - in the environment. By contrast, complementary observables (e.g. the phase between pointer states) are destroyed by the interaction with a single subsystem of E. They can in principle still be accessed, but only when all of the environment is measured. Indeed, because we are dealing with a quantum system, things are much worse than that: the environment must be measured in precisely the right (typically global) basis to allow for such reconstruction. Otherwise, the accumulation of errors over multiple measurements will lead to an incorrect conclusion and re-prepare the state and environment, so that it is no longer a record of the state of S, and phase information is irretrievably lost.

Note, he never defines the term "re-prepare" - anywhere on the net, that I could find. We're all supposing it means a second change, after the initial "collapse". QD usually uses it in that sense. Also he doesn't define "quantum object" but it seems to mean the wavefunction itself. I was using it to mean "small objects"; that might not be exactly consistent with Zurek. Fortunately you both understood what I meant.

Note, I'm only answering the question, "What does QD say?" I don't necessarily agree with it.

This is, indeed, typical "quantum philosophy", contrary to your statement above. It can easily lead to picking terminological nits, attempting to mind-read Zurek, and - in a word - the endless philosophical debate of QM interpretation. So, if you still disagree with my take on "re-prepare", I'll happily concede the point.

OTOH QD is pretty interesting, and I'm up to speed on it now. So that topic is worth pursuing.

 

[PeterDonis]

You can no longer re-prepare that block of wood, as you can with small quantum objects such as an electron.

Yes, small quantum objects. (Which, if unpacked, basically means what I said before, objects made of a small enough number of quantum building blocks that they can be isolated and their interactions controlled.) But your previous post that I objected to didn't include the qualifier "small", which is why I objected and why it appears to have confused the OP.

QD tries to explain why, in practice, a block of wood can't be re-prepared, even though in theory it can.

Or, in terms that the OP seems more comfortable with, it tries to explain why, in practice, only small quantum objects can be re-prepared, even though all objects are quantum objects and in theory any quantum object can be re-prepared.

 

[cube137]

If there were only one copy, you would not be observing a macroscopic object and you would not be getting measurement results of the sort we usually associate with such objects. You would be looking at, say, a quantum computing experiment where a single qubit is being manipulated. And the results you get would not look like something with "objective existence" in the sense Zurek is using that term.

Supposed for sake of discussion there were really only one copy or not even a single copy. So what would the piece of wood look like? (please don't use example of single qubit but a big object like wood). Would the wood become invisible or became a blob in superposition? Or what should the piece of wood look like (just a rough description if you can't describe completely)?

All objects have basic building blocks that are quantum.
No. Zurek is saying that it is impossible for a piece of wood not to have "many copies" of the information about it in the environment, because it is composed of such a huge number of quantum building blocks and it is not at all isolated--each building block is continually interacting with other building blocks and with the environment. To be able to "re-prepare" a piece of wood, you would have to first isolate it from its environment, and then control all the interactions between its quantum building blocks with sufficient accuracy to allow quantum interference effects to be observable. In principle that could be done, but in practice it is impossible.
The reason we can "re-prepare" atoms or tiny isolated quantum systems (like qubits in a quantum computer) by measuring them is that they are tiny isolated quantum systems; we can prevent them from interacting with their environment, and they contain a small enough number of quantum building blocks (just one, in the ideal case) that we can control all their interactions.

 

[PeterDonis]

Supposed for sake of discussion there were really only one copy or not even a single copy. So what would the piece of wood look like?

In order to do this, you would have to completely isolate the piece of wood from its environment. That is, you would have to somehow prevent all interactions between every single quantum building block in the piece of wood and anything else. In practice that is impossible, by many, many orders of magnitude.

But suppose you could, in fact, do the above. You would still have to deal with the fact that the piece of wood is composed of a huge number of quantum building blocks, something like $10^{25}$ of them, and they are interacting with each other, and you can't control the interactions, which means you can't run controlled experiments on the piece of wood to test its quantum state, the way you can for small quantum systems like qubits. Nor can we possibly model the detailed interactions between all the building blocks mathematically.

So I don't think anyone knows what such a piece of wood would look like, nor does it matter since we can't make one anyway.

 

[cube137]

In order to do this, you would have to completely isolate the piece of wood from its environment. That is, you would have to somehow prevent all interactions between every single quantum building block in the piece of wood and anything else. In practice that is impossible, by many, many orders of magnitude.

But suppose you could, in fact, do the above. You would still have to deal with the fact that the piece of wood is composed of a huge number of quantum building blocks, something like $10^{25}$ of them, and they are interacting with each other, and you can't control the interactions, which means you can't run controlled experiments on the piece of wood to test its quantum state, the way you can for small quantum systems like qubits. Nor can we possibly model the detailed interactions between all the building blocks mathematically.

So I don't think anyone knows what such a piece of wood would look like, nor does it matter since we can't make one anyway.

If that's true. Then the "objects" he meant in the following are only quantum objects like electrons, photons, and small quantum system and not a block of wood or larger object where it can't be isolated. Correct? But the way he writes it.. it sounds like any objects like cars.

"Fragility of states is the second problem with quantumclassical
correspondence: Upon measurement, a general
preexisting quantum state is erased - it "collapses" into
an eigenstate of the measured observable. How is it then
possible that objects we deal with can be safely observed,
even though their basic building blocks are quantum?"

In the following article. http://www.nature.com/news/2004/041223/full/news041220-12.html They emphasized the "objects" meant macroscopic object like the Buckingham palace:

"If it wasn't for quantum darwinism, the researchers suggest in Physical Review Letters1, the world would be very unpredictable: different people might see very different versions of it. Life itself would then be hard to conduct, because we would not be able to obtain reliable information about our surroundings... it would typically conflict with what others were experiencing.

The difficulty arises because directly finding out something about a quantum system by making a measurement inevitably disturbs it. "After a measurement," say Wojciech Zurek at Los Alamos National Laboratory in New Mexico and his colleagues, "the state will be what the observer finds out it is, but not, in general, what it was before."

Because, as Zurek says, "the Universe is quantum to the core," this property seems to undermine the notion of an objective reality. In this type of situation, every tourist who gazed at Buckingham Palace would change the arrangement of the building's windows, say, merely by the act of looking, so that subsequent tourists would see something slightly different.

So this is very bad example? They should not use example of macroscopic object like Buckingham Palace! But then.. for other mentors.. in case there is a chance PeterDonis may have misunderstood the issue. Please let us know the case. Thanks.

 

[PeterDonis]

If that's true. Then the "objects" he meant in the following are only quantum objects like electrons, photons, and small quantum system and not a block of wood or larger object where it can't be isolated. Correct?

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.

directly finding out something about a quantum system by making a measurement inevitably disturbs it.

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.

So this is very bad example?

No. See above.

 

[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.

Yes. That's what standard decoherence does.. constantly interacting with objects and environment where they are entangled. And the improper mixed state becomes proper mixed state. We accept the proper mixed state which could be Bohmian, branches in Many Worlds, etc.

What Zurek was trying to do (please confirm my understanding) was like trying to derive how improper mixed state becomes proper mixed state. In the objects, they are in improper mixed state with environment. Then decoherence chose the pointer states and there are many fragments in the environment to make up classical world. In Copenhagen, Bohmian, Many worlds.. Classical world is a priori, observation is the primitive so you don't need quantum Darwinism.. but in Zurek quantum Darwinism, quantum state is the primitive. So far my analysis is correct (let's now use the natural language of Density Matrix)? So by this context if there is a way to block the many fragments (just theoretically), then the object would become invisible (because you can't perceive superposition and the eigenstate information flow is blocked). This is just for sake of illustration although I know there is no way to block the fragments.

 

[PeterDonis]

That's what standard decoherence does

Not quite. See below.

the improper mixed state becomes proper mixed state.

I don't think this is correct. Decoherence explains why we can neglect interference terms between different "classical" alternatives, but that means going from a superposition to a mixed state, not from an improper mixed state to a proper mixed state.

What Zurek was trying to do (please confirm my understanding) was like trying to derive how improper mixed state becomes proper mixed state.

No. What Zurek was trying to do is to explain why, when decoherence happens, it always ends up putting systems in mixed states with "classical" alternatives--live cats or dead cats, blocks of wood over here or over there, Buckingham Palace built vs. not built, etc.--instead of mixed states that don't look anything like the classical states we observe--for example, a mixture of ("block of wood here" plus "block of wood there") and ("block of wood here" minus "block of wood there"), which is a perfectly valid mixed state, but not one that describes any block of wood we've ever observed.

In other words, according to Zurek, it's not enough to explain, as standard decoherence theory does, why macroscopic systems end up in mixed states, with interference terms negligible. You have to also explain why they end up in the particular mixed states we actually observe, and never all the other possible mixed states that are possible mathematically. As I understand it, that is what Quantum Darwinism tries to do.

 

[cube137]

Not quite. See below.
I don't think this is correct. Decoherence explains why we can neglect interference terms between different "classical" alternatives, but that means going from a superposition to a mixed state, not from an improper mixed state to a proper mixed state.

Right. So decoherence = pure state (pure superposition) to improper mixed state (entangled state)
Single Outcome = improper mixed state (entangled state) to proper mixed state (one eigenstate or classical state.. for example spin up or spin down and not spin up+spin down (these are called improper mixed state).

No. What Zurek was trying to do is to explain why, when decoherence happens, it always ends up putting systems in mixed states with "classical" alternatives--live cats or dead cats, blocks of wood over here or over there, Buckingham Palace built vs. not built, etc.--instead of mixed states that don't look anything like the classical states we observe--for example, a mixture of ("block of wood here" plus "block of wood there") and ("block of wood here" minus "block of wood there"), which is a perfectly valid mixed state, but not one that describes any block of wood we've ever observed.

In other words, according to Zurek, it's not enough to explain, as standard decoherence theory does, why macroscopic systems end up in mixed states, with interference terms negligible. You have to also explain why they end up in the particular mixed states we actually observe, and never all the other possible mixed states that are possible mathematically. As I understand it, that is what Quantum Darwinism tries to do.

What is your definition of improper mixed state to proper mixed state. You only mentioned mixed state, I think the improper mixed state is the mixture of ("block of wood here" plus "block of wood there") and ("block of wood here" minus "block of wood there") while proper mixed state is the classical state or only block of wood here OR block of wood there (just like spin up or spin down). So I'm not wrong to say that Zurek tried to derive how improper mixed state becomes proper mixed state. See http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf (courtesy of Bhobba)

 

[PeterDonis]

decoherence = pure state (pure superposition) to improper mixed state (entangled state)

I see at least two mistakes here. First, whether or not a pure state is a superposition is basis dependent; for every pure state there is a basis in which it is a basis state, not a superposition.

Second, an improper mixed state is not the same as an entangled state. An entangled state is a pure state which is not factorizable, i.e., it is not expressible as a sum of tensor products of pure states of subsystems. At least, that's the terminology I'm familiar with.

Single Outcome = improper mixed state (entangled state) to proper mixed state (one eigenstate or classical state.. for example spin up or spin down and not spin up+spin down (these are called improper mixed state).

I don't think this terminology is correct. It would help if you used actual math instead of words.

Also, Zurek doesn't talk at all about "Single Outcome". That is interpretation dependent (in collapse interpretations, there is a single outcome, but in no-collapse interpretations like the MWI, there isn't). What Zurek is saying is supposed to be interpretation free.

What is your definition of improper mixed state to proper mixed state.

I'm not the one who used those terms, you are; as you yourself say, I only used the term "mixed state" with no adjective in front. So you need to provide definitions for "proper" and "improper" mixed states, not me. As above, it would help if you used math instead of words.

You only mentioned mixed state

Yes. And to take my own advice, I'll use math instead of words and re-express what I was saying that way. I'll use Dirac bra-ket notation since it's what I'm most familiar with.

A pure state is a state that can be written as a ket: for example, $|a>$. Such a state can also be written in density matrix notation as a single term $|a> <a|$. A mixed state is a state that can't be so written; it can only be written as a density matrix $\rho = \Sigma_i c_i |a_i> <a_i|$ with multiple terms.

A pure state can be said to be a superposition if, in the basis being used, it is the sum of more than one basis ket: for example, $\alpha |a> + \beta|b>$. However, as I think I noted in a previous post, this is basis dependent; for any pure state, there will be some basis in which it is a basis state and not a superposition.

A pure state of a composite system is entangled if it cannot be factorized into pure states of its subsystems. For example, a state $|a> |b>$ is not entangled, but a state $|a_1> |b_1> + |a_2> |b_2>$ is.

Note, btw, that the superposition pure state I wrote above, $\alpha |a> + \beta|b>$, is not the same as the density matrix $\alpha |a><a| + \beta |b><b|$. The latter is a mixed state, not a pure state. To write the superposition pure state in density matrix notation, we would have to define a new ket for it: for example, we could define $|S> = \alpha |a> + \beta |b>$, and then we could write this pure state in density matrix notation as $|S> <S|$. But there is no way to "factor" this expression into separate terms with $|a>$ and $|b>$ kets and bras in them. (Similar remarks apply to the entangled pure state I wrote above.)

I think the improper mixed state is the mixture of ("block of wood here" plus "block of wood there") and ("block of wood here" minus "block of wood there") while proper mixed state is the classical state or only block of wood here OR block of wood there (just like spin up or spin down).

No. Section 1.2.3 of the PDF you linked to goes into this. Let me rephrase what that section is saying in terms of the block of wood scenario, using the notation and definitions given above.

The block of wood is not a closed system; it's a subsystem of a larger system that includes both the block of wood and its environment. (For our purposes here we can assume that the "environment" doesn't have to include the entire rest of the universe, just enough of the surroundings of the block of wood to support many "copies" of information about the block of wood, as Zurek describes.) So there will be no way to express the state of the block of wood by itself as a pure state; only states of closed systems (i.e., systems that don't interact with anything else, at least to a good enough approximation for the scenario under discussion) can be expressed as pure states. So the only pure state in this scenario will be the state of the block of wood plus its environment. The state of the block of wood by itself will have to be expressed as a mixed state.

Now, if you look at the three cases described at the top of p. 10 of the PDF, at the start of section 1.2.3, you will see that we've ruled out case 1 (pure state) to describe the block of wood. That leaves case 2 (proper mixture) or case 3 (improper mixture). Let's translate the descriptions of those two cases into the block of wood scenario:

Case 2: We pick a random block of wood from a reservoir of blocks of wood of which half are in state $|H>$ ("here") and half are in state $|T>$ ("there"). This results in that block of wood being in a state:

$$
\frac{1}{2} \left( |H> <H| + |T> <T| \right)
$$

Case 3: We prepare a composite of two systems, the block of wood and its environment, in the superposition state $|S> = |H> |E_H> + |T> |E_T>$ (where the two environment states are just those that record the information "block of wood here" and "block of wood there" as a result of interactions between the block of wood and the environment), and then remove the environment from our control. That leaves the block of wood in the state (obtained by tracing over the environment)

$$
Tr_E |S> <S| = \frac{1}{2} \left( |H> <H| + |T> <T| \right)
$$

You should be wondering what's up at this point, because both of the states I just derived look exactly the same! Yet one is supposed to be a proper mixture and the other is supposed to be an improper mixture. What happened?

What happened is that I skimped on notation. In case 2, the block of wood is implicitly assumed to be the entire system--there is no environment. (Notice that I didn't have to trace over the environment in case 2.) But we already said that wasn't true. So case 2 is ruled out, and we're only left with case 3; and I really should have put subscripts on the bras and kets in case 3 to reflect the fact that they refer only to states of a subsystem, not the full system (notice that the PDF does this for its case 3). So the state of the block of wood is an improper mixture, in the terminology of the PDF.

In other words, the difference between a proper mixture and an improper mixture is that a proper mixture is a mixed state of the entire system, while an improper mixture is a mixed state of a subsystem only, obtained by tracing over the rest of the system (the parts that aren't measured), with the system as a whole being in a pure state (notice that in case 3 the state of the whole system is pure--it's a state in which the block of wood is entangled with its environment). We get a proper mixture when we have a closed system which is in some pure state but we don't know which (in case 2 above the lack of knowledge is due to the random selection). We get an improper mixture when we can only measure a subsystem which is entangled with the rest of the system, and we want to express the state of the subsystem.

I'm not wrong to say that Zurek tried to derive how improper mixed state becomes proper mixed state.

I disagree. In Zurek's paper, he is always talking about measurements on subsystems that are entangled with their environment, so all mixtures he is dealing with are improper mixtures, as should be obvious from the above. What he is really trying to explain are why the mixtures we get are always of the form

$$
Tr_E |S> <S| = \frac{1}{2} \left( |H> <H| + |T> <T| \right)
$$

i.e., expressing lack of knowledge about whether the block of wood is here or there, and never of the form

$$
Tr_E |Z> <Z| = \frac{1}{2} \left( |X> <X| + |Y> <Y| \right)
$$

where $|X> = |H> + |T>$ and $|Y> = |H> - |T>$, and the mixture is obtained by tracing over the environment in the whole system state $|Z> = |X> |E_X> + |Y> |E_Y>$. This improper mixture expresses lack of knowledge about whether the block of wood is in a superposition of "here plus there" or "here minus there". Mathematically, this improper mixture is perfectly well defined, and the states $|X>$ and $|Y>$ are perfectly good states of the block of wood subsystem (because they are just superpositions of the "here" and "there" states). Zurek's argument is that the state $|Z>$ is not stable (and nor are all the other possible mixtures derived by forming superpositions of the "here" and "there" states, entangling them appropriately with the environment, then tracing over the environment), while the state $|S>$ is, and that is why we always get mixtures of the first form but not the second.

One other note: in the above I was basically assuming that the block of wood had been through some process that could have resulted in its being either here or there (for example, say it passed down a ramp with a shunt that could route it to one place or the other, and the shunt's position was controlled by the random decay of a radioactive atom), and we have not yet observed the block ourselves, so we don't know which position it ended up at. But, according to Zurek, the environment has observed (measured) the block, and the resulting state of the system is $|S>$ (and not, say, $|Z>$) because that is the stable state that can result from the whole process.

 

[Simon Phoenix]

Hi Cube,

Peter has given you some very nice responses - and I don't think I can usefully add to them, but since you asked me to reply I'll have a go.

First off I haven't really read the paper in a great deal of detail - and I probably should. I tend to get a bit turned off by the term 'pointer basis' in discussions of decoherence because it immediately conjures up an association with "solutions of the measurement problem" in my mind. Some physicists think there is no measurement problem in QM; I'm not one of them. I also don't believe decoherence provides a solution either, but some physicists do. Having said that, Zurek has done some really wonderful and important work, so it would be wrong to dismiss this paper because of my own personal prejudices

I would class this paper as part of Zurek's programme to give insights into why the world appears 'classical' - why do some superpositions survive the environmental maelstrom, but not others? In this paper he's focusing, I think, on the part that is often neglected in discussions of decoherence; the environment itself. So, having shown that there is an effect of the environment on our system of interest in previous work, he's now focusing on the information about that system of interest that lives in the environment (because of the entanglement between the system and the environment).

It would seem that an emergent classicality can only happen if there is considerable redundancy of this information in the environment - so 'copies' of the information exist in these so-called environmental fragments. I'll have to look at this in much more detail and that will take me a while. I'm not convinced at the moment by this approach because if I have a 2 state system coupled to an environment (so that the combined system is in a pure state) then the entropy of the entire environment can, at most, be 1 bit. I don't know yet how to reconcile this with the notion that there are multiple (redundant) copies of the same information distributed throughout the environment - but I need to read the paper properly because I'm clearly missing something from my brief initial reading (and Zurek wouldn't make an elementary mistake).

I'm also not overly impressed with the term 'Darwinism' applied here - but I guess it had to be called something. I think over the past couple of decades there has been a tendency to attach 'quantum' to everything in sight; physicists have been scouring the scientific lexicon and attaching 'quantum' to any concept that looks like it might cause a sensation, or create a new bandwagon, or look good on a grant application - but I'm just getting cynical in my old age.

 

[cube137]

I see at least two mistakes here. First, whether or not a pure state is a superposition is basis dependent; for every pure state there is a basis in which it is a basis state, not a superposition.

Second, an improper mixed state is not the same as an entangled state. An entangled state is a pure state which is not factorizable, i.e., it is not expressible as a sum of tensor products of pure states of subsystems. At least, that's the terminology I'm familiar with.
I don't think this terminology is correct. It would help if you used actual math instead of words.

Also, Zurek doesn't talk at all about "Single Outcome". That is interpretation dependent (in collapse interpretations, there is a single outcome, but in no-collapse interpretations like the MWI, there isn't). What Zurek is saying is supposed to be interpretation free.
I'm not the one who used those terms, you are; as you yourself say, I only used the term "mixed state" with no adjective in front. So you need to provide definitions for "proper" and "improper" mixed states, not me. As above, it would help if you used math instead of words.
Yes. And to take my own advice, I'll use math instead of words and re-express what I was saying that way. I'll use Dirac bra-ket notation since it's what I'm most familiar with.

A pure state is a state that can be written as a ket: for example, $|a>$. Such a state can also be written in density matrix notation as a single term $|a> <a|$. A mixed state is a state that can't be so written; it can only be written as a density matrix $\rho = \Sigma_i c_i |a_i> <a_i|$ with multiple terms.

A pure state can be said to be a superposition if, in the basis being used, it is the sum of more than one basis ket: for example, $\alpha |a> + \beta|b>$. However, as I think I noted in a previous post, this is basis dependent; for any pure state, there will be some basis in which it is a basis state and not a superposition.

A pure state of a composite system is entangled if it cannot be factorized into pure states of its subsystems. For example, a state $|a> |b>$ is not entangled, but a state $|a_1> |b_1> + |a_2> |b_2>$ is.

Note, btw, that the superposition pure state I wrote above, $\alpha |a> + \beta|b>$, is not the same as the density matrix $\alpha |a><a| + \beta |b><b|$. The latter is a mixed state, not a pure state. To write the superposition pure state in density matrix notation, we would have to define a new ket for it: for example, we could define $|S> = \alpha |a> + \beta |b>$, and then we could write this pure state in density matrix notation as $|S> <S|$. But there is no way to "factor" this expression into separate terms with $|a>$ and $|b>$ kets and bras in them. (Similar remarks apply to the entangled pure state I wrote above.)
No. Section 1.2.3 of the PDF you linked to goes into this. Let me rephrase what that section is saying in terms of the block of wood scenario, using the notation and definitions given above.

The block of wood is not a closed system; it's a subsystem of a larger system that includes both the block of wood and its environment. (For our purposes here we can assume that the "environment" doesn't have to include the entire rest of the universe, just enough of the surroundings of the block of wood to support many "copies" of information about the block of wood, as Zurek describes.) So there will be no way to express the state of the block of wood by itself as a pure state; only states of closed systems (i.e., systems that don't interact with anything else, at least to a good enough approximation for the scenario under discussion) can be expressed as pure states. So the only pure state in this scenario will be the state of the block of wood plus its environment. The state of the block of wood by itself will have to be expressed as a mixed state.

Now, if you look at the three cases described at the top of p. 10 of the PDF, at the start of section 1.2.3, you will see that we've ruled out case 1 (pure state) to describe the block of wood. That leaves case 2 (proper mixture) or case 3 (improper mixture). Let's translate the descriptions of those two cases into the block of wood scenario:

Case 2: We pick a random block of wood from a reservoir of blocks of wood of which half are in state $|H>$ ("here") and half are in state $|T>$ ("there"). This results in that block of wood being in a state:

$$
\frac{1}{2} \left( |H> <H| + |T> <T| \right)
$$

Case 3: We prepare a composite of two systems, the block of wood and its environment, in the superposition state $|S> = |H> |E_H> + |T> |E_T>$ (where the two environment states are just those that record the information "block of wood here" and "block of wood there" as a result of interactions between the block of wood and the environment), and then remove the environment from our control. That leaves the block of wood in the state (obtained by tracing over the environment)

$$
Tr_E |S> <S| = \frac{1}{2} \left( |H> <H| + |T> <T| \right)
$$

You should be wondering what's up at this point, because both of the states I just derived look exactly the same! Yet one is supposed to be a proper mixture and the other is supposed to be an improper mixture. What happened?

What happened is that I skimped on notation. In case 2, the block of wood is implicitly assumed to be the entire system--there is no environment. (Notice that I didn't have to trace over the environment in case 2.) But we already said that wasn't true. So case 2 is ruled out, and we're only left with case 3; and I really should have put subscripts on the bras and kets in case 3 to reflect the fact that they refer only to states of a subsystem, not the full system (notice that the PDF does this for its case 3). So the state of the block of wood is an improper mixture, in the terminology of the PDF.

In other words, the difference between a proper mixture and an improper mixture is that a proper mixture is a mixed state of the entire system, while an improper mixture is a mixed state of a subsystem only, obtained by tracing over the rest of the system (the parts that aren't measured), with the system as a whole being in a pure state (notice that in case 3 the state of the whole system is pure--it's a state in which the block of wood is entangled with its environment). We get a proper mixture when we have a closed system which is in some pure state but we don't know which (in case 2 above the lack of knowledge is due to the random selection). We get an improper mixture when we can only measure a subsystem which is entangled with the rest of the system, and we want to express the state of the subsystem.
I disagree. In Zurek's paper, he is always talking about measurements on subsystems that are entangled with their environment, so all mixtures he is dealing with are improper mixtures, as should be obvious from the above. What he is really trying to explain are why the mixtures we get are always of the form

$$
Tr_E |S> <S| = \frac{1}{2} \left( |H> <H| + |T> <T| \right)
$$

i.e., expressing lack of knowledge about whether the block of wood is here or there, and never of the form

$$
Tr_E |Z> <Z| = \frac{1}{2} \left( |X> <X| + |Y> <Y| \right)
$$

where $|X> = |H> + |T>$ and $|Y> = |H> - |T>$, and the mixture is obtained by tracing over the environment in the whole system state $|Z> = |X> |E_X> + |Y> |E_Y>$. This improper mixture expresses lack of knowledge about whether the block of wood is in a superposition of "here plus there" or "here minus there". Mathematically, this improper mixture is perfectly well defined, and the states $|X>$ and $|Y>$ are perfectly good states of the block of wood subsystem (because they are just superpositions of the "here" and "there" states). Zurek's argument is that the state $|Z>$ is not stable (and nor are all the other possible mixtures derived by forming superpositions of the "here" and "there" states, entangling them appropriately with the environment, then tracing over the environment), while the state $|S>$ is, and that is why we always get mixtures of the first form but not the second.

One other note: in the above I was basically assuming that the block of wood had been through some process that could have resulted in its being either here or there (for example, say it passed down a ramp with a shunt that could route it to one place or the other, and the shunt's position was controlled by the random decay of a radioactive atom), and we have not yet observed the block ourselves, so we don't know which position it ended up at. But, according to Zurek, the environment has observed (measured) the block, and the resulting state of the system is $|S>$ (and not, say, $|Z>$) because that is the stable state that can result from the whole process.

Thanks a lot for this masterpiece explanation. So proper mixed state and improper mixed state are not standard usage. So Witten or Stephen Hawking won't necessarily be familiar with the terms? Roughly how many percentage of physicists are familiar with the terms? Perhaps these terms are only found inside Physicsforums. Can physicists make a vote whether the terms proper and improper mixed states must be standardized. In garden variety mixed states application of the density matrix. What is more often the case.. A mixed state that is a "definite deterministic physical state" or a mixed state as subsystem of an ensemble of pure state. If the latter is more often used in density matrix.. maybe there must be interchanged.. the subsystem of an ensemble of pure state being the proper mixed state instead of improper. Or maybe they should not be used at all. Can we vote.

 

[secur]

It would seem that an emergent classicality can only happen if there is considerable redundancy of this information in the environment - so 'copies' of the information exist in these so-called environmental fragments.

The simplest way to envision these copies is, they're photons bouncing off the quantum system - e.g., a block of wood. The information conveyed might typically be position: here, or there. At first glance one thinks he's talking about clones of the quantum state, but no, they're "copies of observables".

Quantum Darwinism leads to appearance, in the environment, of multiple copies of the state of the system. However, the no-cloning theorem prohibits copying of unknown quantum states. If cloning is outlawed, how can redundancy seen in Fig. 2 be possible? Quick answer is that cloning refers to (unknown) quantum states. So, copying of observables evades the theorem.

When all these fragments convey the same information to many observers, we have "objective reality". That can only happen when a pointer state is reached. In more conventional picture, the wavefunction has "collapsed". The "copies of observables" are more conventionally known as measurements of the observable and of course the measurements all agree after "collapse". He doesn't use "collapse", of course: this is MWI.

I'm also not overly impressed with the term 'Darwinism' applied here - but I guess it had to be called something. I think over the past couple of decades there has been a tendency to attach 'quantum' to everything in sight; ...

The same tendency has been seen with 'Darwinism', over the last century and a half.

 

[Simon Phoenix]

Thanks for this Secur,

I must read this stuff in more detail. On first glance I wasn't convinced it was telling me much more than the 'standard' decoherence/pointer basis stuff. But you've convinced me I should look at it more carefully.

 

[PeterDonis]

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

I have not seen either of these terms enough in what literature I've read to know. I am using the definitions given for those terms in the PDF that you linked to. Other sources might use other definitions. That's why, as I said before, it's better to use math instead of words; math is unambiguous.

Can we vote.

I don't see the point of such a vote, since, as I've said several times now, if you want to be precise, you use math, not words. That's what physicists do in their actual published papers and textbooks. They might, for brevity, define certain words to refer to certain particular mathematical expressions; but they are doing so not because of any "standard usage", but because they think it makes the presentation clearer in the particular paper or textbook they're writing. And you should interpret the words based not on any "standard usage" but on the precise definitions given in the particular paper or textbook you are reading.

 

[PeterDonis]

if I have a 2 state system coupled to an environment (so that the combined system is in a pure state) then the entropy of the entire environment can, at most, be 1 bit. I don't know yet how to reconcile this with the notion that there are multiple (redundant) copies of the same information distributed throughout the environment

I think Zurek would say that in the case of a 2 state system, for example a qubit, you would not have multiple redundant copies of the same information about it distributed throughout the environment, for exactly the reason you state. Such a system would not be capable of having "objective reality" by Zurek's definition. Only systems that are composed of a large number of quantum building blocks, like the block of wood with $10^{25}$ atoms in it, are capable of that.

 

[secur]

I must read this stuff in more detail. On first glance I wasn't convinced it was telling me much more than the 'standard' decoherence/pointer basis stuff. But you've convinced me I should look at it more carefully.

If you (or anyone) get to it here's the part that confuses me. In the "BOX" of page 8, on the 6th line, he says:

The state |+> E = (|0> E +|1> E) / √2 ...

I don't see the justification for this step. In particular why can't the state |2> E be expanded as a 2-dimensional subspace, just as well as |+> E ? I found on the net some further information (concerning the ancilla C) but not a satisfactory answer to this question. It's critical for his derivation of Born rule. Apart from this one thing, the rest of QD has some interesting aspects, and some rather trivial, but all seem to work Ok.

 

[PeterDonis]

why can't the state |2> E be expanded as a 2-dimensional subspace

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").

 

[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]