Understanding Hilbert Subspace for Two-Particle Entangled Systems

[bluecap]

Where are you getting "perturbing the system without using fragments" from?

Just applying the concepts. Without using fragments of informational copies of observables. We need to mechanically measure the pointer states and this can perturb it. This is why Zurek said fragments are enough because information copies of observables is enough for us.

Can you give other example of perturbing the system without using fragments? When you write with the pencil or break it. It is one example of perturbing the pencil without using fragments.. can you think of others? Just want to be versatile with the idea. Thanks.

 

[PeterDonis]

Without using fragments of informational copies of observables. We need to mechanically measure the pointer states

Zurek's point, as I understand it, is that the "pointer states" can't be measured directly; they are the result of the whole process of decoherence and einselection, which involves interaction with the environment and storing of information about the system in fragments. So there is no way to observe "pointer states" without this process.

Can you give other example of perturbing the system without using fragments?

Anything that changes its state perturbs it. You should be able to think of plenty of examples on your own. But this doesn't have anything to do with the topic of the thread.

 

[bluecap]

Zurek's point, as I understand it, is that the "pointer states" can't be measured directly; they are the result of the whole process of decoherence and einselection, which involves interaction with the environment and storing of information about the system in fragments. So there is no way to observe "pointer states" without this process.

In conventional decoherence, the system is entangled with different subspaces in the environment which destroys the superposition in the system. But note that in Quantum Darwinism, the process is identical. See https://arxiv.org/pdf/1412.5206v1.pdf where it is stated that:

"Decoherence is the loss of phase coherence between preferred states. It occurs when S starts in a superposition of pointer states singled out by the interaction, as in Eq. (1), but now S is ‘measured’ by E, its environment:

$\left( a \vert U \rangle + b \vert D \rangle \right) \vert Eo \rangle \rightarrow HsE \rightarrow a \vert U \rangle \vert E_U \rangle + b \vert D \rangle \vert E_D \rangle = \vert \psi se \rangle$ (4)
"
The equation is Zurek's (I typed U instead of the arrow up because I don't know how to type arrow up). So you see. Zurek's Pointer States is nothing but the classical states of conventional decoherence after it is entangled with the environment. If you think it's not identical. How's Zurek Pointer States not the same as the conventional decoherence classical states?

Also we do measurement prior to decoherence. After decoherence we don't have to measure it. But even after decoherence we can still perturb the system by let's say exposing it to MRI. When you remove the system from the MRI. Would all the spins be back in the original? You said any interaction can perturb a quantum system. So even if the process would be negligible.. a spin or two would be changed by the interaction, right? I think this is what Zurek is saying any measurements can perturb the pointer states (even if negligible enough not to be observed).

Going back to the single atom of the pencil where the atom is interacting with the environment and with itself. When you perturb the atom, wouldn't there be any changes even negligible? If any interaction can perturb the quantum system. What would happen to the single atom, would it increase the energy by the interaction or move its position a bit? Any changes is enough to called it perturbation even if its negligible. Or is there any interaction where the atom isn't change even a single bit?

Anything that changes its state perturbs it. You should be able to think of plenty of examples on your own. But this doesn't have anything to do with the topic of the thread.

 

[PeterDonis]

In conventional decoherence, the system is entangled with different subspaces in the environment which destroys the superposition in the system.

No, that's not correct. The different decoherent branches are not entanglements of the same system with different subspaces in the environment. They are different product terms in the joint quantum state of the same system and the same environment. And these are still in a superposition; decoherence does not change that. (To "destroy a superposition", you would need to collapse the wave function, but collapse is interpretation-dependent, and decoherence is not.)

Zurek's Pointer States is nothing but the classical states of conventional decoherence after it is entangled with the environment

You missed some key points in Zurek's paper. The equation you wrote down, equation (4) in the paper, does not show any loss of phase coherence. The coefficients $a$ and $b$ of the two terms stay the same; those are the relative phases.

In order to derive loss of phase coherence (decoherence), Zurek has to make additional arguments, which he does in the text following equation (4). Those arguments rely on properties of the environment--in particular, that degrees of freedom in the environment that are far away from the measured system can have phase shifts applied to them, and such phase shifts cannot affect the local state of the system (since that would require faster-than-light signaling). I actually find his argument rather hand-waving here, and I think many other quantum physicists do too, which is probably one reason why Zurek's viewpoint is not a majority one among quantum physicists.

However, a more important point is that Zurek's equation (4) is highly schematic. He just waves his hands and assumes that there is some Hamiltonian $H_{SE}$ that entangles the system and the environment. But he can't write down any such Hamiltonian explicitly for any real object interacting with a real environment. Nobody can. Which means that neither he nor anyone else can write down the eigenstates of such a Hamiltonian--the states he labels with up and down arrows--explicitly either. This is very different from a controlled measurement in the lab, where we can in fact write down the explicit Hamiltonian and its eigenstates--for example, for a Stern-Gerlach apparatus for measuring spin.

In fact, in the other Zurek paper we were discussing earlier, Zurek recognizes this by breaking up the process into two stages: first the system and the apparatus interact, then the apparatus and the environment interact. The first interaction is the one that is simple and controlled and we can write it down explicitly. But the second one is the one that we actually observe. What Zurek is doing in equation (4) is writing down the "up" and "down" states of the system and saying they interact with the environment, when what he should really be doing is writing down the "measured up" and "measured down" states of the apparatus and saying that they interact with the environment. Those apparatus states are the "pointer states", and we can't "measure" them directly; we gather information about them from the environment, but we don't control the interaction between the apparatus and the environment that determines which states of the apparatus they are. Whereas we do control the system-apparatus interaction that determines which states of the system correspond to "pointer states" of the apparatus (the ones we are going to observe).

even after decoherence we can still perturb the system by let's say exposing it to MRI. When you remove the system from the MRI. Would all the spins be back in the original?

I don't know what you mean by "back in the original". If you mean, can we "undo" the measurement after it's made and decohered, in the sense of a "quantum eraser" experiment, no, we can't, because to do that it would not be sufficient just to act on the system. We would have to act on the apparatus and the environment, and we can't. (The "environment" might consist of photons which have flown off into space, and there's no way we can catch them, so even in principle we can't act on the environment, let alone in practice.) But of course we can perturb the system further and put it into some other state than the one it was in after the measurement. That's obvious. But that isn't "undoing" the measurement, because we haven't erased all the information about the measurement that is now stored in the environment. We can't do that.

Going back to the single atom of the pencil...

I can't really say any more about that than I've already said. Basically you keep on making up possible changes and asking if they change something. Of course they do. But talking about all the possible changes that could be done is way too broad a topic for a PF discussion.

 

[bluecap]

No, that's not correct. The different decoherent branches are not entanglements of the same system with different subspaces in the environment. They are different product terms in the joint quantum state of the same system and the same environment. And these are still in a superposition; decoherence does not change that. (To "destroy a superposition", you would need to collapse the wave function, but collapse is interpretation-dependent, and decoherence is not.)

What I meant to say was, the superposition in the system is delocalized to the environment so you can't see the superposition in the system anymore unless you measure the environment too. So "destroy superposition" is bad choice of words.. maybe "delocalized superposition".. I'm following the choice of words Wikipedia where it is stated that "Decoherence can be viewed as the loss of information from a system into the environment (often modeled as a heat bath),[2] since every system is loosely coupled with the energetic state of its surroundings. Viewed in isolation, the system's dynamics are non-unitary (although the combined system plus environment evolves in a unitary fashion).[3] Thus the dynamics of the system alone are irreversible. As with any coupling, entanglements are generated between the system and environment. These have the effect of sharing quantum information with—or transferring it to—the surroundings."

In the following equations:

$\vert \Psi' \rangle \vert E \rangle = \left( a \vert + \rangle \vert U \rangle + b \vert - \rangle \vert D \rangle \right) \vert E \rangle \rightarrow a \vert + \rangle \vert U \rangle \vert E_U \rangle + b \vert - \rangle \vert D \rangle \vert E_D \rangle$

does the coefficients a and b (or the relative phases) of the two terms stay the same? But only loss of phase coherence produce decoherence. I thought whenever the system is entangled with the environment, there is automatically decoherence or loss of phase coherence? It's not automatic? Maybe that's why Kastner emphased the environment must have random phases before decoherence can even occur... and emphasized decoherence can't even take off in quantum Darwinism.. (?)

You missed some key points in Zurek's paper. The equation you wrote down, equation (4) in the paper, does not show any loss of phase coherence. The coefficients $a$ and $b$ of the two terms stay the same; those are the relative phases.

In order to derive loss of phase coherence (decoherence), Zurek has to make additional arguments, which he does in the text following equation (4). Those arguments rely on properties of the environment--in particular, that degrees of freedom in the environment that are far away from the measured system can have phase shifts applied to them, and such phase shifts cannot affect the local state of the system (since that would require faster-than-light signaling). I actually find his argument rather hand-waving here, and I think many other quantum physicists do too, which is probably one reason why Zurek's viewpoint is not a majority one among quantum physicists.

However, a more important point is that Zurek's equation (4) is highly schematic. He just waves his hands and assumes that there is some Hamiltonian $H_{SE}$ that entangles the system and the environment. But he can't write down any such Hamiltonian explicitly for any real object interacting with a real environment. Nobody can. Which means that neither he nor anyone else can write down the eigenstates of such a Hamiltonian--the states he labels with up and down arrows--explicitly either. This is very different from a controlled measurement in the lab, where we can in fact write down the explicit Hamiltonian and its eigenstates--for example, for a Stern-Gerlach apparatus for measuring spin.

In fact, in the other Zurek paper we were discussing earlier, Zurek recognizes this by breaking up the process into two stages: first the system and the apparatus interact, then the apparatus and the environment interact. The first interaction is the one that is simple and controlled and we can write it down explicitly. But the second one is the one that we actually observe. What Zurek is doing in equation (4) is writing down the "up" and "down" states of the system and saying they interact with the environment, when what he should really be doing is writing down the "measured up" and "measured down" states of the apparatus and saying that they interact with the environment. Those apparatus states are the "pointer states", and we can't "measure" them directly; we gather information about them from the environment, but we don't control the interaction between the apparatus and the environment that determines which states of the apparatus they are. Whereas we do control the system-apparatus interaction that determines which states of the system correspond to "pointer states" of the apparatus (the ones we are going to observe).

No one can even write a very very simple Hamiltonian of any real object interacting with a real environment? If you make it so simple like the environment the phonons in the molecules or other even more simple.. why can't no one write the Hamiltonian? (Ping Demystifier, can't you write a Hamiltonian explicitly for any real object interacting with a real environment? I think I read one in your papers before and I forget which it is).

Thanks a lot for the above details Peter. I'll reread all of Zurek papers again including his 1981 classic where he defined his Pointer Basis in the Apparatus.. I think it's very important.

I don't know what you mean by "back in the original". If you mean, can we "undo" the measurement after it's made and decohered, in the sense of a "quantum eraser" experiment, no, we can't, because to do that it would not be sufficient just to act on the system. We would have to act on the apparatus and the environment, and we can't. (The "environment" might consist of photons which have flown off into space, and there's no way we can catch them, so even in principle we can't act on the environment, let alone in practice.) But of course we can perturb the system further and put it into some other state than the one it was in after the measurement. That's obvious. But that isn't "undoing" the measurement, because we haven't erased all the information about the measurement that is now stored in the environment. We can't do that.

I can't really say any more about that than I've already said. Basically you keep on making up possible changes and asking if they change something. Of course they do. But talking about all the possible changes that could be done is way too broad a topic for a PF discussion.

 

[PeterDonis]

does the coefficients a and b (or the relative phases) of the two terms stay the same?

Of course. It's right there in the equation.

I thought whenever the system is entangled with the environment, there is automatically decoherence or loss of phase coherence? It's not automatic?

It's not a question of being "automatic". The loss of phase coherence happens because nobody can keep track of the phases of all the environment degrees of freedom that get entangled with the system and apparatus and thereby store pieces of information about the system. In other words, because we don't know what the exact state of system + apparatus + environment is. If we knew the exact state, we could find a unitary transformation that would exactly reverse the process that entangled system + apparatus + environment, and thereby undo the measurement, the way a "quantum eraser" experiment does. But that's not possible in any practical sense because there are too many degrees of freedom in the environment. (And in the system as well, if the system is a macroscopic object.)

why can't no one write the Hamiltonian?

Because there are too many degrees of freedom in the environment. We can only write explicit Hamiltonians for interactions that involve a very small number of degrees of freedom.

 

[bluecap]

Of course. It's right there in the equation.
It's not a question of being "automatic". The loss of phase coherence happens because nobody can keep track of the phases of all the environment degrees of freedom that get entangled with the system and apparatus and thereby store pieces of information about the system. In other words, because we don't know what the exact state of system + apparatus + environment is. If we knew the exact state, we could find a unitary transformation that would exactly reverse the process that entangled system + apparatus + environment, and thereby undo the measurement, the way a "quantum eraser" experiment does. But that's not possible in any practical sense because there are too many degrees of freedom in the environment. (And in the system as well, if the system is a macroscopic object.)

Uhm.. since we normally can't keep tract of the phases of all the environment degrees of freedom.. then why can't Zurek equation 4 be written with loss of coherence instead of lack of loss of coherence.. so the equations should be

$\left( a1 \vert U \rangle + b1 \vert D \rangle \right) \vert Eo \rangle \rightarrow HsE \rightarrow a2 \vert U \rangle \vert E_U \rangle + b2 \vert D \rangle \vert E_D \rangle = \vert \psi se \rangle$

Because there are too many degrees of freedom in the environment. We can only write explicit Hamiltonians for interactions that involve a very small number of degrees of freedom.

 

[PeterDonis]

since we normally can't keep tract of the phases of all the environment degrees of freedom.. then why can't Zurek equation 4 be written with loss of coherence instead of lack of loss of coherence

Because you can't write an equation for that at all using state vectors. The equation you wrote does not express a loss of phase coherence, because it still assigns definite phases to the two terms. A loss of phase coherence means you don't know what the phases of the terms are at all; you have to change your whole mathematical formalism to one that ignores the phases altogether. (This is what is being talked about when you see talk about "mixed states", or using density matrices instead of state vectors.)

 

[bluecap]

Because you can't write an equation for that at all using state vectors. The equation you wrote does not express a loss of phase coherence, because it still assigns definite phases to the two terms. A loss of phase coherence means you don't know what the phases of the terms are at all; you have to change your whole mathematical formalism to one that ignores the phases altogether. (This is what is being talked about when you see talk about "mixed states", or using density matrices instead of state vectors.)

So Zurek equation (4) is still a pure state. But then entanglement between system and environment is always in pure state. So we only use mixed state if we don't know the phases due to ignorance.. therefore to apply decoherence to equation (4). You need to write it in mixed state or density matrices?

One of Zurek problems (he mentioned quoted below) is how to define subsystems and also even the MWI folks problems who wrote that nothing happens in many worlds. How do you connect this to the above loss of phase coherence thing. Does it mean to truly have subsystems and decoherence, there must be actual loss of coherence that Kastner kept talking about instead of just ignorance?

"In particular, one issue which has been often taken for granted is looming big, as a foundation of the whole decoherence program. It is the question of what are the “systems” which play such a crucial role in all the discussions of the emergent classicality.(...) [A] compelling explanation of what are the systems - how to define them given, say, the overall Hamiltonian in some suitably large Hilbert space - would be undoubtedly most useful." - Zurek

 

[bluecap]

Zurek's point, as I understand it, is that the "pointer states" can't be measured directly; they are the result of the whole process of decoherence and einselection, which involves interaction with the environment and storing of information about the system in fragments. So there is no way to observe "pointer states" without this process.

I'll reread the rest of the week Maximilian Schlosshaeur textbook "Decoherence and the Quantum To Classical Transition" cover to cover to master the concepts of Decoherence in light of what you shared above that made me realized some past misconceptions. So I won't ask you about Decoherence in general again.

But let me just ask something about the pointer states. You said as you understand it. The "pointer states" can't be measured directly (and hence can't be perturbed at will) and they are the result of the whole process of decoherence and einselection. But if you can control the environment (and hence influence einselection).. you can perturb the pointer states right? For example affecting the phonons as the environment and cooling it near absolute zero from our room temperature (just an example).. won't this affect Einselection and hence perturb the pointer states?

I plan to email Zurek to ask him something. I wonder if anyone has written to Zurek and if he will reply.. so I need to make the questions intelligible in the first place. Thanks.

Anything that changes its state perturbs it. You should be able to think of plenty of examples on your own. But this doesn't have anything to do with the topic of the thread.

 

[PeterDonis]

entanglement between system and environment is always in pure state

Yes. Unitary evolution always takes pure states to pure states, and entanglement happens by unitary evolution.

we only use mixed state if we don't know the phases due to ignorance

Yes.

to apply decoherence to equation (4). You need to write it in mixed state or density matrices?

You need to write the "state" in terms of density matrices, yes. But I put "state" in quotes because the density matrix is not a pure state, it's a way of expressing, mathematically, what you do know about the system if you don't know its exact pure state.

Does it mean to truly have subsystems and decoherence, there must be actual loss of coherence that Kastner kept talking about instead of just ignorance?

I think this is an open question, because we don't know how to look at an arbitrary state of some total system and break it up into subsystems. All of our ways of mathematically modeling such states assume that we already know what the subsystems are. So subsystems are something that we put into the model, not something that we get out of it. That means the model (i.e., quantum mechanics) can't tell us, at least not in its present state of development, what it takes to "truly have subsystems", or even whether that's a meaningful question to ask.

 

[PeterDonis]

The "pointer states" can't be measured directly (and hence can't be perturbed at will)

These aren't the same thing. Perturbing a system is easy--think of examples like writing with the pencil or breaking it. But measuring a pointer state directly is hard--in the case of the pencil, it would require being able to measure the state of every single atom in the pencil. It's not the same as just observing the pencil--observing the pencil in the ordinary way, by looking at light bouncing off of it, touching it, etc., doesn't tell you its exact quantum state.

if you can control the environment (and hence influence einselection).. you can perturb the pointer states right?

Not the way you mean; your notion of "controlling" the environment is much too coarse. See below.

For example affecting the phonons as the environment and cooling it near absolute zero from our room temperature

This doesn't even come close to "controlling" the environment in the sense of precisely preparing its quantum state. All you're doing is affecting one macroscopic variable, the temperature. That's not what you would need to do in order to significantly affect the process of einselection. To do that, you would need to be able to precisely prepare the quantum state of the environment--which is even harder than precisely preparing the quantum state of the pencil, since the environment has many more degrees of freedom even than the pencil does (because the environment is bigger--it potentially could be the entire universe).

 

[bluecap]

These aren't the same thing. Perturbing a system is easy--think of examples like writing with the pencil or breaking it. But measuring a pointer state directly is hard--in the case of the pencil, it would require being able to measure the state of every single atom in the pencil. It's not the same as just observing the pencil--observing the pencil in the ordinary way, by looking at light bouncing off of it, touching it, etc., doesn't tell you its exact quantum state.

You stated earlier in message #20: "Yes, but what is the "system"? If we consider the apple, the "system" is not the entire apple. It's just one atom in it. Read his examples more carefully." If the system is one atom, then the pointer state only covers one atom? But you mentioned above regarding the pointer state of the entire pencil. Did you state it because an atom is not like a qubit which is isolated but interacting with the rest of the object so to measure the pointer state of one atom, you need to measure the pointer states of the rest of the object (pencil or apple)? Also even though Einselection is only use for microscopic system like atom.. in an ordinary object like apple. You need to apply System and Pointer State and Einselection to each of the 10^50 atoms one by one meaning you can't define it for one large macroscopic object at the same time but one by one?

Also isn't it when you measure something, you automatically perturb it? This is because of the no-cloning principle so you can't measure a quantum state without changing it.. therefore what do you mean you can measure the pointer state without changing (perturbing) it? Can you please give example of measuring something that doesn't perturb it?

Thanks.

Not the way you mean; your notion of "controlling" the environment is much too coarse. See below.
This doesn't even come close to "controlling" the environment in the sense of precisely preparing its quantum state. All you're doing is affecting one macroscopic variable, the temperature. That's not what you would need to do in order to significantly affect the process of einselection. To do that, you would need to be able to precisely prepare the quantum state of the environment--which is even harder than precisely preparing the quantum state of the pencil, since the environment has many more degrees of freedom even than the pencil does (because the environment is bigger--it potentially could be the entire universe).

 

[bluecap]

No, that's not correct. The different decoherent branches are not entanglements of the same system with different subspaces in the environment. They are different product terms in the joint quantum state of the same system and the same environment. And these are still in a superposition; decoherence does not change that. (To "destroy a superposition", you would need to collapse the wave function, but collapse is interpretation-dependent, and decoherence is not.)

You missed some key points in Zurek's paper. The equation you wrote down, equation (4) in the paper, does not show any loss of phase coherence. The coefficients $a$ and $b$ of the two terms stay the same; those are the relative phases.

In order to derive loss of phase coherence (decoherence), Zurek has to make additional arguments, which he does in the text following equation (4).

You said above (to emphasize) that in order to derive loss of phase coherence (decoherence), Zurek has to make additional arguments. How about conventional decoherence folks. How do they derive loss of phase coherence? Do they do it by relying on reduce density matrices and trace operation which presuppose Born's rule. Can this go around the problem of deriving loss of phase coherence?

Also since Zurek wants to derive the Born's rule from his 3 axiom without collapse, then he has to make additional arguments about loss of phase coherence that ordinary decoherence folks don't worry about? Which is more ad hoc? orthodox decoherence folks using trace operation and assuming born rule or zurek deriving born rule from invariance?

Another critical question below (mentioning this so you won't miss it)

Those arguments rely on properties of the environment--in particular, that degrees of freedom in the environment that are far away from the measured system can have phase shifts applied to them, and such phase shifts cannot affect the local state of the system (since that would require faster-than-light signaling). I actually find his argument rather hand-waving here, and I think many other quantum physicists do too, which is probably one reason why Zurek's viewpoint is not a majority one among quantum physicists.

However, a more important point is that Zurek's equation (4) is highly schematic. He just waves his hands and assumes that there is some Hamiltonian $H_{SE}$ that entangles the system and the environment. But he can't write down any such Hamiltonian explicitly for any real object interacting with a real environment. Nobody can. Which means that neither he nor anyone else can write down the eigenstates of such a Hamiltonian--the states he labels with up and down arrows--explicitly either. This is very different from a controlled measurement in the lab, where we can in fact write down the explicit Hamiltonian and its eigenstates--for example, for a Stern-Gerlach apparatus for measuring spin.

In fact, in the other Zurek paper we were discussing earlier, Zurek recognizes this by breaking up the process into two stages: first the system and the apparatus interact, then the apparatus and the environment interact. The first interaction is the one that is simple and controlled and we can write it down explicitly. But the second one is the one that we actually observe. What Zurek is doing in equation (4) is writing down the "up" and "down" states of the system and saying they interact with the environment, when what he should really be doing is writing down the "measured up" and "measured down" states of the apparatus and saying that they interact with the environment.

Maximilian Schlosshauer in his book seemed to use system and environment only without using the concept of apparatus. The use of apparatus in concept of pointer states is only for illustration and/oronly to conform to traditional methods and it doesn't necessarily mean apparatus is required before the system can get pointer states.. correct?

Thanks a whole lot!

Those apparatus states are the "pointer states", and we can't "measure" them directly; we gather information about them from the environment, but we don't control the interaction between the apparatus and the environment that determines which states of the apparatus they are. Whereas we do control the system-apparatus interaction that determines which states of the system correspond to "pointer states" of the apparatus (the ones we are going to observe).

 

[PeterDonis]

If the system is one atom, then the pointer state only covers one atom?

Yes. But what is the "system" is a matter of choice. It can be one atom, or it can be some small group of atoms, or it can be one atom or a small group of atoms inside a larger object like an apple or a pencil, or it can be a whole apple or pencil. It depends on the scenario and on what questions you are trying to answer.

to measure the pointer state of one atom, you need to measure the pointer states of the rest of the object (pencil or apple)

No. The pointer states are really states of the apparatus, not the system. Go back and read my previous posts again.

even though Einselection is only use for microscopic system like atom

Einselection can be applied--at least if you agree with Zurek's model--to any system, apparatus, and environment. It in no way requires that the system be microscopic.

isn't it when you measure something, you automatically perturb it?

No. When you interact with something, you automatically perturb it. But the size of the perturbation, relative to the size of the something, depends on the something and the perturbation. You can perturb a pencil by bouncing a photon off of it; but the perturbation will be negligible.

This is because of the no-cloning principle so you can't measure a quantum state without changing it

That's not what the no cloning principle says. The no cloning principle says that, if you have an unknown quantum state, there is no way to duplicate it--i.e., to take a system in an unknown quantum state and make a second copy of that exact quantum state in a second system. But measuring a system does not require copying its state, so the no cloning principle says nothing about what you can or can't do with measurement.

Can you please give example of measuring something that doesn't perturb it?

I didn't say measuring things wouldn't perturb them. I said measuring things isn't the same as just perturbing them. Measuring is much harder.

How about conventional decoherence folks. How do they derive loss of phase coherence?

Basically the same way Zurek does, by assuming that you can't keep track of all the degrees of freedom in the environment. Zurek isn't disagreeing with the conventional model of decoherence; he's adding to it, by trying to explain, not just how decoherence happens, but how the states that are left after decoherence somehow always turn out to be the "classical" states that we observe, rather than "Schrodinger's cat" type states. The conventional account of decoherence doesn't really address that (at least, Zurek doesn't think it does). I think it's still an open question at this point how all this is going to turn out.

Maximilian Schlosshauer in his book seemed to use system and environment only without using the concept of apparatus.

I don't have his book so I can't comment on it. I don't know how widespread the three-way split into system, apparatus, and environment is; Zurek is the only place I've seen it, but that doesn't mean he's the only one who uses it.

The use of apparatus in concept of pointer states is only for illustration and/oronly to conform to traditional methods and it doesn't necessarily mean apparatus is required before the system can get pointer states.. correct?

I don't know how widespread Zurek's concept of "pointer states" is either. Lots of other QM texts use that term, but that doesn't mean they mean the same thing by it that Zurek does.

 

[bluecap]

Yes. But what is the "system" is a matter of choice. It can be one atom, or it can be some small group of atoms, or it can be one atom or a small group of atoms inside a larger object like an apple or a pencil, or it can be a whole apple or pencil. It depends on the scenario and on what questions you are trying to answer.

No. The pointer states are really states of the apparatus, not the system. Go back and read my previous posts again.

In this Maximilian paper which is condense of his book. He wrote in page 14 of https://arxiv.org/pdf/quant-ph/0312059.pdf which doesn't mention about apparatrus but directly the system and environment and the pointer states between S and E (pls. comment):

"(1) When the dynamics of the system are dominated
by b HSE, i.e., the interaction with the environment, the pointer states will be eigenstates of b HSE (and thus typically eigenstates of position). This case corresponds to the typical quantum measurement setting; see, for example, the model of Zurek (1981, 1982), which is outlined in Sec. III.D.2 above.
(2) When the interaction with the environment is weak and b HS dominates the evolution of the system (that is, when the environment is “slow” in the above sense), a case that frequently occurs in the microscopic domain, pointer states will arise that are energy eigenstates of b HS (Paz and Zurek, 1999).
(3) In the intermediate case, when the evolution of the system is governed by b HSE and b HS in roughly equal strength, the resulting preferred states will represent a “compromise” between the first two cases; for instance, the frequently studied model of quantum Brownian motion has shown the emergence of pointer states localized in phase space, i.e., in both position and momentum (Eisert, 2004; Joos et al., 2003; Unruh and Zurek, 1989; Zurek, 2003b; Zurek et al., 1993)."

Einselection can be applied--at least if you agree with Zurek's model--to any system, apparatus, and environment. It in no way requires that the system be microscopic.

In your last last message you mentioned about measuring every atom of the pencil to measure the pointer states. But let's say you just take one atom as the system. Can you measure the particular atom to get its pointer states? But how do you handle the other billions of atoms interacting with your particular atom?

No. When you interact with something, you automatically perturb it. But the size of the perturbation, relative to the size of the something, depends on the something and the perturbation. You can perturb a pencil by bouncing a photon off of it; but the perturbation will be negligible.

That's not what the no cloning principle says. The no cloning principle says that, if you have an unknown quantum state, there is no way to duplicate it--i.e., to take a system in an unknown quantum state and make a second copy of that exact quantum state in a second system. But measuring a system does not require copying its state, so the no cloning principle says nothing about what you can or can't do with measurement.

I didn't say measuring things wouldn't perturb them. I said measuring things isn't the same as just perturbing them. Measuring is much harder.

Basically the same way Zurek does, by assuming that you can't keep track of all the degrees of freedom in the environment. Zurek isn't disagreeing with the conventional model of decoherence; he's adding to it, by trying to explain, not just how decoherence happens, but how the states that are left after decoherence somehow always turn out to be the "classical" states that we observe, rather than "Schrodinger's cat" type states. The conventional account of decoherence doesn't really address that (at least, Zurek doesn't think it does). I think it's still an open question at this point how all this is going to turn out.

Kastner seems to be saying that even if you can't keep tract of all the degrees of freedom in the environment, that doesn't mean you can produce a subsystem out of it. You need genuine phase randomization. What do you think?

I don't have his book so I can't comment on it. I don't know how widespread the three-way split into system, apparatus, and environment is; Zurek is the only place I've seen it, but that doesn't mean he's the only one who uses it.

See Maximilian condensed paper above...

I don't know how widespread Zurek's concept of "pointer states" is either. Lots of other QM texts use that term, but that doesn't mean they mean the same thing by it that Zurek does.

Here's a good paper about it called "Understanding the Pointer States" https://arxiv.org/abs/1508.04101

 

[PeterDonis]

how do you handle the other billions of atoms interacting with your particular atom?

Exactly. You just pointed out why it's not practical to measure a single atom in a macroscopic object like a pencil--because you can't isolate it from all the other atoms, the way you have to to measure a single atom in the lab.

 

[PeterDonis]

What do you think?

I think, as I said before, that this is an open area of research and nobody knows what the "right" answers are at this point. So I'm not sure how fruitful further discussion of the various viewpoints is going to be.

 

[bluecap]

Exactly. You just pointed out why it's not practical to measure a single atom in a macroscopic object like a pencil--because you can't isolate it from all the other atoms, the way you have to to measure a single atom in the lab.

But is it correct to say that even if you only want to measure the pointer states of one atom, you need to measure the rest of the 10^30 atoms just to get accurate measurements how the rest affect the pointer states of that one atom? So if you solve for the pure state of the object.. you could in principle understand or get the state of that one atom. Or let's take actual example of a qubit. If you can't isolate the qubit and it is interacting with the rest of the molecules. What happens if you measure the entire molecules (atom by atom).. then the effect is like knowing the state of the single qubit?

However, a more important point is that Zurek's equation (4) is highly schematic. He just waves his hands and assumes that there is some Hamiltonian $H_{SE}$ that entangles the system and the environment. But he can't write down any such Hamiltonian explicitly for any real object interacting with a real environment. Nobody can. Which means that neither he nor anyone else can write down the eigenstates of such a Hamiltonian--the states he labels with up and down arrows--explicitly either. This is very different from a controlled measurement in the lab, where we can in fact write down the explicit Hamiltonian and its eigenstates--for example, for a Stern-Gerlach apparatus for measuring spin.

About this environment very complex and difficult to write the Hamiltonian of real object interacting with a real environment.. Is it not like adding all the Hamiltonian together like Hse1 + Hse2 + Hse3 + Hse4 + Hse5 + Hse6 + Hse7 + Hse8 + Hse9 + Hse10 and so on? Is the reason we can't write or solve for it is due to insufficient computer processing power? Is it not like solving for the Schrodinger Equations of all the atoms of the molecules in quantum chemistry? But at least we have idea of how the dynamics or how solve for it. Just lack of computing powers. Why is the Hamiltonian of the environment and object any different?

 

[PeterDonis]

is it correct to say that even if you only want to measure the pointer states of one atom, you need to measure the rest of the 10^30 atoms just to get accurate measurements how the rest affect the pointer states of that one atom?

If the atom is interacting with $10^{30}$ other atoms, because they're all part of the same macroscopic object, I'm not sure the concept of "pointer states" of that one atom even makes sense. (I'm not sure it makes sense for one atom even if the atom is isolated--that's why I said that to me, the pointer states really belong to the apparatus, not the system.)

What happens if you measure the entire molecules (atom by atom).. then the effect is like knowing the state of the single qubit?

No.

Is it not like adding all the Hamiltonian together

Hamiltonians don't just add that way. The Hamiltonian of a system of $10^{30}$ atoms is not just the sum of $10^{30}$ individual atom Hamiltonians. It also includes all the interactions, and we don't know what all those interactions are. We know the fundamentals--basically, interactions between atoms are based on electromagnetic forces--but that doesn't mean you can just put in, say, a Coulomb potential term for each pair of atoms. There are all kinds of collective effects involved, which even with quantum field theory we can only approximate.

at least we have idea of how the dynamics

No, we don't. For that, we would have to know the Hamiltonian for the entire system, and we don't, not even conceptually. See above.

 

[bluecap]

If the atom is interacting with $10^{30}$ other atoms, because they're all part of the same macroscopic object, I'm not sure the concept of "pointer states" of that one atom even makes sense. (I'm not sure it makes sense for one atom even if the atom is isolated--that's why I said that to me, the pointer states really belong to the apparatus, not the system.)

Ok. I believe you that the pointer states has to do with the apparatus. This paper is a valid source since it is summary of Zurek and others idea about Pointer States. https://arxiv.org/abs/1508.04101 But something puzzles me. The environment is not just entangled to the apparatus.. it is also entangled to the systems. So what is the side effect of this direct system-environment entanglement. In the paper in page 10:

"4.1 Example
In Sec. 3 we analysed the process of pre-measurement and, at this stage, we considered only the main system S and the measurement apparatus A. Let us now approach the complete process, from the initial evolution of the system until the measurement, step-by-step. Our analysis involves:

• the initial evolution of the system S and the measurement apparatus A, without interaction between them;
• the pre-measurement process, with the interaction between S and A;
• the beginning of the measurement process, with the introduction of the environment B, interacting with A;
• the determination of the evolution under the effects of the environment;
• the average of the environmental effects;
• the end of the measurement process, with analysis of the final S + A state after a long time."

and a few paragraphs prior, the author said:

"We will follow here the Zurek’s work [27]. If ˆ PA is the observable we wish to measure, an ideal apparatus will leave the system in one of the eigenstates of ˆ PA, not any relative state, but we have already seen this is not a simple task. In introducing the environment in the description, Zurek [27] imposed some conditions that had to be satisfied. In his original article, he admits these conditions are stronger than necessary, and for this reason here we will keep only two of them:
1. The environment does not interact with the system (i.e. ˆ HSB = 0). Otherwise, the state of the system would keep suffering environmental interference after the end of the measurement. (This could mean two repeated measurements of the same observable could give different results, which is against the tenets of quantum mechanics.)
2. The system-observer interaction is well-localized in time."

Or let's say you don't think the paper is correct.

What is the effect of this direct interaction between the environment and system? The trinity of system, apparatus, environment assumes the interaction is between system and apparatus.. then between apparatus and environment.. but surely the system is directly interacting with the environment too.. is it not? If you know of example of the apparatus being part of the objects or molecules.. please share it because I'm assuming the apparatus is separate from the object... but still environmental decoherence both engage the system and apparatus.. not just the apparatus.

No.
Hamiltonians don't just add that way. The Hamiltonian of a system of $10^{30}$ atoms is not just the sum of $10^{30}$ individual atom Hamiltonians. It also includes all the interactions, and we don't know what all those interactions are. We know the fundamentals--basically, interactions between atoms are based on electromagnetic forces--but that doesn't mean you can just put in, say, a Coulomb potential term for each pair of atoms. There are all kinds of collective effects involved, which even with quantum field theory we can only approximate.
No, we don't. For that, we would have to know the Hamiltonian for the entire system, and we don't, not even conceptually. See above.

 

[PeterDonis]

The environment is not just entangled to the apparatus.. it is also entangled to the systems.

It's entangled with the system because of its entanglement with the apparatus. The environment and the system don't interact directly--at least not in a typical "lab" measurement where we take care to isolate the system. The quote you give appears to agree with that; it says the environment B interacts with A (the apparatus). It doesn't say B interacts with S (the system); only A does.

What is the effect of this direct interaction between the environment and system?

There doesn't seem to be any such thing in any of the quotes you give.

 

[bluecap]

It's entangled with the system because of its entanglement with the apparatus. The environment and the system don't interact directly--at least not in a typical "lab" measurement where we take care to isolate the system. The quote you give appears to agree with that; it says the environment B interacts with A (the apparatus). It doesn't say B interacts with S (the system); only A does.
There doesn't seem to be any such thing in any of the quotes you give.

I'm referring to decoherence and open systems where the system is not isolated or not the typical lab measurement. Zurek concept of Pointer States occur for macroscopic object where it is not isolated.. So think of the environment causing loss of coherence in the pointer states of the apparatus that prevents the cat alive and dead superposition (or other superpositions), the same environment can also cause loss of superpositions of the system so the system can no longer have dead and alive cats.. but won't this be redundant the environment affect both system and apparatus at same time?

 

[PeterDonis]

think of the environment causing loss of coherence in the pointer states of the apparatus that prevents the cat alive and dead superposition (or other superpositions), the same environment can also cause loss of superpositions of the system so the system can no longer have dead and alive cats

Once again, I don't see anything in what you've quoted from the papers that says the environment interacts with the system directly; it interacts with the apparatus, and that interaction with the apparatus, since the apparatus is already entangled with the system, entangles the environment with the system.

 

[bluecap]

Once again, I don't see anything in what you've quoted from the papers that says the environment interacts with the system directly; it interacts with the apparatus, and that interaction with the apparatus, since the apparatus is already entangled with the system, entangles the environment with the system.

or see directly this Zurek paper: https://arxiv.org/pdf/0903.5082.pdf

"Interactions that depend on a certain observable correlate it with the environment, so its eigenstates are singled out, and phase relations between such pointer states are lost [6]. "
"Negative selection due to decoherence is the essence of environment-induced superselection, or einselection [7]: Under scrutiny of the environment, only pointer states remain unchanged. Other states decohere into mixtures of stable pointer states that can persist, and, in this sense, exist: They are einselected."

Think of this setup. You have an apple on the table and an apparatus to measure it's position. The environment is exposed to both the system and apparatus. In decoherence, the system is open.. it is not isolated. so it's exposed to the environment same as the apparatus.

 

[bluecap]

Once again, I don't see anything in what you've quoted from the papers that says the environment interacts with the system directly; it interacts with the apparatus, and that interaction with the apparatus, since the apparatus is already entangled with the system, entangles the environment with the system.

I think when system is directly exposed to the environment. We can bypass apparatus which is only used if system is isolated to environment so apparatus acts as interface between them. In the same Zurek paper, the following is clear that he only deals with system and environment and omits any apparatus:

"These ideas can be made precise. The basic tool is the reduced density matrix ρS. It represents the state of the system that obtains from the composite state ΨSE of S and E by tracing out the environment E: ρS = TrE|ΨSEihΨSE| . (1) Evolution of ρS reveals preferred states: It is most predictable when the system starts in a pointer state. To quantify this one can use (von Neumann) entropy, HS = H(ρS) = −TrρS lgρS, as a function of time. Pointer states result in smallest entropy increase. By contrast, their superpositions produce entropy rapidly, at decoherence rates, especially when S is macroscopic."

So "when S is macroscopic" and directly exposed to environment. We can use pointer states directly on the system.. right?

 

[PeterDonis]

So "when S is macroscopic" and directly exposed to environment. We can use pointer states directly on the system.. right?

I'm not sure. For example, if you "measure" an apple's position by looking at it (and perhaps comparing it with a ruler or a grid of coordinates marked on the table it's sitting on), the "apparatus" is photons bouncing off the apple (and the ruler/grid/table, etc.) and coming into your eyes (and perhaps your eyes themselves). The "system" is the apple, and the "environment" is everything else. You are correct that both the system and the apparatus interact with the environment, but the environment interaction that counts is with the apparatus. Basically, multiple people can look at photons bouncing off the apple and agree on its position, because there are so many photons. But there are so many photons because the environment is basically an inexhaustible source of them, and the information that each person receives through their eyes gets stored in other environment degrees of freedom like their brains, the sound waves they emit when they describe what they see, etc. None of this involves any interaction with the system--the only interaction with the system that made any difference was the photons bouncing off the apple.

Zurek's formulation in the paper you link to here, which doesn't include an "apparatus", is unclear to me in an example like the above. Would he include the photons bouncing off the apple in the "system" or the "environment"? Neither one seems right to me; the photons seem like an "apparatus", which isn't the same as the system (since the photons don't stay localized at the apple), but isn't the same as the environment either (because other degrees of freedom don't interact with the apple, at least not in a way that makes any difference in the scenario above).

But as I said before, all this is an open area of research, so I'm not sure how much weight to put on any particular description of it.

 

[bluecap]

I'm not sure. For example, if you "measure" an apple's position by looking at it (and perhaps comparing it with a ruler or a grid of coordinates marked on the table it's sitting on), the "apparatus" is photons bouncing off the apple (and the ruler/grid/table, etc.) and coming into your eyes (and perhaps your eyes themselves). The "system" is the apple, and the "environment" is everything else. You are correct that both the system and the apparatus interact with the environment, but the environment interaction that counts is with the apparatus. Basically, multiple people can look at photons bouncing off the apple and agree on its position, because there are so many photons. But there are so many photons because the environment is basically an inexhaustible source of them, and the information that each person receives through their eyes gets stored in other environment degrees of freedom like their brains, the sound waves they emit when they describe what they see, etc. None of this involves any interaction with the system--the only interaction with the system that made any difference was the photons bouncing off the apple.

Zurek's formulation in the paper you link to here, which doesn't include an "apparatus", is unclear to me in an example like the above. Would he include the photons bouncing off the apple in the "system" or the "environment"? Neither one seems right to me; the photons seem like an "apparatus", which isn't the same as the system (since the photons don't stay localized at the apple), but isn't the same as the environment either (because other degrees of freedom don't interact with the apple, at least not in a way that makes any difference in the scenario above).

But as I said before, all this is an open area of research, so I'm not sure how much weight to put on any particular description of it.

I'm trying to relate what you mentioned above about the scattering photons being apparatus to what this Maximilian book is saying. In page 74:
"For the simple example of the system-environment interaction given by (2.78), we immediately see that a system that starts out in either one of the states |psi1> and |psi2> will not get entangled with the environment: The final composite system-environment state will at all times remain in the separable product forms|psi1>|E1> and |psi2>|E2> respectively. Thus the environment can be regarded as carrying out a quantum nondemolition measurement on the system. i.e. a measurement that does not disturb the state of the system. In the case of the scattering of environment particles discussed above, |psi1> and |psi2> represent spatially well-localized states, and accordingly the environment-superselected preferred observable is the position of the system. Conversely, if the system is described by the superposition (2.70) of |psi1> and |psi2> , it will become maximally entangled with the environment, leading to decoherence in the {|psi1> , psi2> } basis. The system observable corresponding to the measurement of a superposition of positions would there be "difficult" to measure, in the sense that the spatial interference terms in the reduced matrix corresponding to a measurement of this observable decay rapidly due to the interaction with the environment."

How do you use the concept of maximal entanglement here (monogamy?). How come if the system is described by the superposition (2.70) of |psi1> and |psi2> will they be maximally entangled with the environment while if |psi1> and |psi2> are separated, it will remain in the separable product forms |psi1>|E1> and |psi2>|E2> respectively (the comment in page 71 described this "We motivated this form of the interaction by referring to the fact that the states |psi1> and |psi2> correspond to sufficiently distinct physical states of the system (such as different spatial positions and orientations of a body), and this difference is resolved by the environment continuously monitoring the system.".. and this refers to page 67 "Supposed the system is described by a coherent superposition of two quantum states |psi1> and |psi2> representing localization around two different positions x1 and x2 (in the case of the double-slit experiment, the corresponding position space wave functions psi1(x) and psi2(x) would represent the partial waves at the slits)."

Thanks!

 

[PeterDonis]

How do you use the concept of maximal entanglement here (monogamy?)

Maximal entanglement isn't the same thing as monogamy of entanglement, though they are related. Monogamy of entanglement basically says that a subsystem can't be maximally entangled with more than one other subsystem.

I'm not sure what you mean by "how do you use the concept".

How come if the system is described by the superposition (2.70) of |psi1> and |psi2> will they be maximally entangled with the environment while if |psi1> and |psi2> are separated, it will remain in the separable product forms |psi1>|E1> and |psi2>|E2> respectively

Because that's how the interaction with the environment works. It's just unitary evolution. It's no different in principle from putting a spin up or spin down electron through a vertically oriented Stern Gerlach device, vs. putting a spin left electron (equal superposition of spin up and spin down) through it. At least, that's the type of model that it looks like Maximilian is using. The only difference is that here the "system" (as well as the apparatus) is made up of many quantum particles, not just one.

 

[bluecap]

Maximal entanglement isn't the same thing as monogamy of entanglement, though they are related. Monogamy of entanglement basically says that a subsystem can't be maximally entangled with more than one other subsystem.

I'm not sure what you mean by "how do you use the concept".
Because that's how the interaction with the environment works. It's just unitary evolution. It's no different in principle from putting a spin up or spin down electron through a vertically oriented Stern Gerlach device, vs. putting a spin left electron (equal superposition of spin up and spin down) through it. At least, that's the type of model that it looks like Maximilian is using. The only difference is that here the "system" (as well as the apparatus) is made up of many quantum particles, not just one.

My question really was. If the system is described by the superposition of |psi1> and |psi2> , and it will become maximally entangled with the environment, and lead to decoherence in the {|psi1> , psi2> } basis. How come the system observable corresponding to the measurement of a superposition of positions would there be "difficult" to measure (in the sense that the spatial interference terms in the reduced matrix corresponding to a measurement of this observable decay rapidly due to the interaction with the environment). You earlier just described the photons reflecting off the object. But now we have to consider it directly interacting with the system. Do you have example of system being maximally entangled with the environment and the setup difficulty getting the preferred observable or basis? Contrast this with the case where the final composite system-environment state will at all times remain in the separable product forms|psi1>|E1> and |psi2>|E2>respectively. Here how come "the environment can be regarded as carrying out a quantum nondemolition measurement on the system. i.e. a measurement that does not disturb the state of the system.". In this message we are.not just describing photons reflecting from object but interacting with the object. Thanks!

 

[PeterDonis]

My question really was

Instead of trying to answer this as you ask it, I'm going to try to rephrase it to simplify things.

The environment's interaction with the system picks out a "preferred" basis of states, which in this example are position eigenstates. (Technically those aren't actually states since they aren't normalizable, and you have to do a fair bit of mathematical work to make all this rigorous, but we'll ignore that here.)

If we use that basis, it is obvious that if the system is in a position eigenstate, measuring its position doesn't change its state; that is what he is talking about when he describes this case as a "quantum nondemolition" measurement.

If we use the position basis, it is also obvious that if the system is in a superposition of two position eigenstates, measuring its position, which is what the environment is doing (that's why the environment picks out the position basis), will entangle the system with the measuring device (which in this case is the environment), so you will end up with two branches, each corresponding to one of the position eigenstates and the corresponding "measured that position" state of the environment (i.e., the environment state that stores the information that that particular position was measured). The branches will then decohere.

 

[bluecap]

Instead of trying to answer this as you ask it, I'm going to try to rephrase it to simplify things.

The environment's interaction with the system picks out a "preferred" basis of states, which in this example are position eigenstates. (Technically those aren't actually states since they aren't normalizable, and you have to do a fair bit of mathematical work to make all this rigorous, but we'll ignore that here.)

If we use that basis, it is obvious that if the system is in a position eigenstate, measuring its position doesn't change its state; that is what he is talking about when he describes this case as a "quantum nondemolition" measurement.

If we use the position basis, it is also obvious that if the system is in a superposition of two position eigenstates, measuring its position, which is what the environment is doing (that's why the environment picks out the position basis), will entangle the system with the measuring device (which in this case is the environment), so you will end up with two branches, each corresponding to one of the position eigenstates and the corresponding "measured that position" state of the environment (i.e., the environment state that stores the information that that particular position was measured). The branches will then decohere.

Thanks..in an apple, how many percentage of the particles are in approx eigenstates of position and how many percentage are in superposition of approx position eigenstates? (I mentioned approx bec I'm aware that HUP disallows exact position of eigenstates). What must you do to the apple to change the percentage of each of the above (between approx eigenstates of position vs superpositions of approx eigenstates of position)?

Anyway I believe you now that the pointer states (or preferred states) are a result of joint system environment (or even apparatus) processes so observer can't perturb the pointer states directly. After 5 years. I finally understood it (thanks to you).. but it's bittersweet knowing the limitation of the model of Einselection and even quantum darwinism.

 

[vanhees71]

If we use the position basis, it is also obvious that if the system is in a superposition of two position eigenstates, measuring its position, which is what the environment is doing (that's why the environment picks out the position basis), will entangle the system with the measuring device (which in this case is the environment), so you will end up with two branches, each corresponding to one of the position eigenstates and the corresponding "measured that position" state of the environment (i.e., the environment state that stores the information that that particular position was measured). The branches will then decohere.

I thought you discuss about photons. How can you have a "position basis" then? Photons don't have a position. You cannot define a position operator for massless quanta with spin $\geq 1$!

 

[PeterDonis]

in an apple, how many percentage of the particles are in approx eigenstates of position and how many percentage are in superposition of approx position eigenstates?

The atoms are all entangled so they don't have definite states by themselves. There isn't really any difference from one atom to another, so it doesn't really make sense to ask "what percentage" are in one state (or one kind of entanglement) vs. another.

 

[PeterDonis]

I thought you discuss about photons.

We've been bouncing around among different examples. You're correct that there is no "position basis" for photons, but there is for, e.g, atoms in an apple (with the caveats I mentioned earlier).