Understanding Hilbert Subspace for Two-Particle Entangled Systems

[bluecap]

I read that if we construct an observable on a two-particle entangled system like the "center of mass" observable, this observable does not pick out a single state of the two-particle system. It only picks out a subspace of the full Hilbert space of all possible states--the subspace that satisfies the constraint that the center of mass position (the average position of the two particles) is equal to the measured value of the center of mass observable.

May I know why it is not possible to pick out a single state of two entangled particle system? What kind of observable where it is possible to pick out a single state of two entangled particle system?

How do you understand Hilbert Subspace?

 

[vanhees71]

$\newcommand{\op}[1]{\hat{#1}}$ I don't understand exactly what you want to say. Formulae say more than 1000 words. I guess you work within non-relativistic QT, i.e., you have a Hamiltonian of the form
$$\hat{H}=\frac{\op{\vec{p}}_1^2}{2m_1} + \frac{\op{\vec{p}}_2^2}{2m_2} + V(\op{\vec{x}}_1-\op{\vec{x}}_2).$$
Then it's easy to show that you can as well use the center-of-mass position and relative coordinates,
$$\op{\vec{R}} = \frac{m_1 \op{\vec{x}}_1 + m_2 \op{\vec{p}}_2}{m_1+m_2}, \quad \op{\vec{r}}=\op{\vec{x}_1}-\op{\vec{x}_2}$$
and write
$$\hat{H}=\frac{\op{\vec{P}}^2}{2(m_1+m_2)} + \frac{\op{\vec{p}}^2}{2 \mu} + V(\op{\vec{r}})$$
with
$$\op{\vec{P}}=\op{\vec{p}}_1+\op{\vec{p}}_2, \quad \op{\vec{p}}=\mu \left (\frac{1}{m_1} \op{\vec{p}}_1 - \frac{1}{m_2 \op{\vec{p}}_2} \right), \quad \mu=\frac{m_1 m_2}{m_1+m_2}.$$
Since $\op{\vec{P}}$ commutes with $\hat{H}$ the problem splits into the free motion of the center of mass and the motion of a "quasiparticle" with mass $\mu$ in an external potential $V$. If $V$ is a central potential you thus can choose a basis of common (generalized) eigenvectors of $\op{\vec{P}}$, $\hat{H}_{\text{rel}}$, $\ell$, and $m$, where
$$\hat{H}_{\text{rel}}=\frac{\op{\vec{p}}^2}{2 \mu} + V(\op{\vec{r}})$$
is the Hamiltonian for the relative motion and $\ell(\ell+1)\hbar^2$ is the eigenvalue of relative orbital angular momentum squared, and $m\hbar$ the eigenvalue of $\op{L}_{\text{rel},z}$ with $$\op{\vec{L}}_{\text{rel}}=\op{\vec{r}} \times \op{\vec{p}}$$.
Of course $\ell \in \{0,1,2,\ldots \}$ and for given $\ell$ one has $m \in \{-\ell,-\ell+1,\ldots,\ell-1,\ell \}$.

 

[bluecap]

$\newcommand{\op}[1]{\hat{#1}}$ I don't understand exactly what you want to say. Formulae say more than 1000 words. I guess you work within non-relativistic QT, i.e., you have a Hamiltonian of the form
$$\hat{H}=\frac{\op{\vec{p}}_1^2}{2m_1} + \frac{\op{\vec{p}}_2^2}{2m_2} + V(\op{\vec{x}}_1-\op{\vec{x}}_2).$$
Then it's easy to show that you can as well use the center-of-mass position and relative coordinates,
$$\op{\vec{R}} = \frac{m_1 \op{\vec{x}}_1 + m_2 \op{\vec{p}}_2}{m_1+m_2}, \quad \op{\vec{r}}=\op{\vec{x}_1}-\op{\vec{x}_2}$$
and write
$$\hat{H}=\frac{\op{\vec{P}}^2}{2(m_1+m_2)} + \frac{\op{\vec{p}}^2}{2 \mu} + V(\op{\vec{r}})$$
with
$$\op{\vec{P}}=\op{\vec{p}}_1+\op{\vec{p}}_2, \quad \op{\vec{p}}=\mu \left (\frac{1}{m_1} \op{\vec{p}}_1 - \frac{1}{m_2 \op{\vec{p}}_2} \right), \quad \mu=\frac{m_1 m_2}{m_1+m_2}.$$
Since $\op{\vec{P}}$ commutes with $\hat{H}$ the problem splits into the free motion of the center of mass and the motion of a "quasiparticle" with mass $\mu$ in an external potential $V$. If $V$ is a central potential you thus can choose a basis of common (generalized) eigenvectors of $\op{\vec{P}}$, $\hat{H}_{\text{rel}}$, $\ell$, and $m$, where
$$\hat{H}_{\text{rel}}=\frac{\op{\vec{p}}^2}{2 \mu} + V(\op{\vec{r}})$$
is the Hamiltonian for the relative motion and $\ell(\ell+1)\hbar^2$ is the eigenvalue of relative orbital angular momentum squared, and $m\hbar$ the eigenvalue of $\op{L}_{\text{rel},z}$ with $$\op{\vec{L}}_{\text{rel}}=\op{\vec{r}} \times \op{\vec{p}}$$.
Of course $\ell \in \{0,1,2,\ldots \}$ and for given $\ell$ one has $m \in \{-\ell,-\ell+1,\ldots,\ell-1,\ell \}$.

Thanks will digest this over the weekend.. but may I know why eigenstates can't be defined for two-particle entangled systems? Just use some words meantime to get a bird eye view first.

 

[vanhees71]

Again I don't understand your question. Can you give some context?

For a very good discussion about position entanglement of the above defined eigenstates considered in terms of the $|\vec{x}_1,\vec{x}_2 \rangle$ basis of the original particles, see

https://arxiv.org/abs/quant-ph/9709052

 

[bluecap]

Again I don't understand your question. Can you give some context?

For a very good discussion about position entanglement of the above defined eigenstates considered in terms of the $|\vec{x}_1,\vec{x}_2 \rangle$ basis of the original particles, see

https://arxiv.org/abs/quant-ph/9709052

My question is.. what really are Hilbert Subspace? Is this not a standard usage or terms?

Also I was trying to solve for an Eigenstate of a pencil in position observable.. but someone said it's not possible and tell me to consider a two-particle entangled system instead of 10^30 entangled particle and how the center of mass observable does not pick out a single state for the system. I just want to know it does not pick out a single state of the system.. and I'd like an example of what it means to pick out a single state of the system.

 

[vanhees71]

A subspace of the Hilbert space is any set of Hilbert-space vectors spanning a vector space.

What do you mean by "pick out a single state of the system"? To uniquely define (up to a constant factor) eigenstates you need to specify a complete set of compatible observables, e.g., as given above $\vec{P}$, $\hat{H}_{\text{rel}}$, $\hat{\vec{L}}_{\text{rel}}^2$, $\hat{L}_{\text{rel},z}$.

 

[bluecap]

A subspace of the Hilbert space is any set of Hilbert-space vectors spanning a vector space.

What do you mean by "pick out a single state of the system"? To uniquely define (up to a constant factor) eigenstates you need to specify a complete set of compatible observables, e.g., as given above $\vec{P}$, $\hat{H}_{\text{rel}}$, $\hat{\vec{L}}_{\text{rel}}^2$, $\hat{L}_{\text{rel},z}$.

Is it possible to solve for Eigenstates for the entire pencil in position observable? Some say not as the center of mass observable doesn't pick out a single state of the system.. but instead it only picks out a superposition of all the possible states that have the measured value for the center of mass position, i.e., that have position values for each of the individual particles that average to the measured center of mass position. And there is no such thing as an "eigenstate" of the center of mass observable.

I'd like to know example of observables or eigenstates where you can use a single state of the system.. is this only for one particle and never for more than one particle?

 

[vanhees71]

Of course, if you only know the center-of mass position of a pencil, already in classical mechanics it's an incomplete information. So how do you expect that it would provide complete information in QT?

 

[bluecap]

Of course, if you only know the center-of mass position of a pencil, already in classical mechanics it's an incomplete information. So how do you expect that it would provide complete information in QT?

Hmm... can you please give an example of what it means to find the Hilbert subspace of a pencil that uses any observable you can think of? I just need some solid example to illustrate the concept of Hilbert subspace.. thank you!

 

[vanhees71]

As I said several times, I don't know, what you mean by the phrase "find the Hilbert subspace of a pencil that uses any observable you can think of." It simply doesn't make any sense to me in the context of standard quantum mechanics. Where have you read this phrase?

 

[bluecap]

As I said several times, I don't know, what you mean by the phrase "find the Hilbert subspace of a pencil that uses any observable you can think of." It simply doesn't make any sense to me in the context of standard quantum mechanics. Where have you read this phrase?

Last November 2016 I created a thread that discussed it where you also participated.. see https://www.physicsforums.com/threads/eigenstate-probability.892724/page-2

For 9 months I have toiled and I tried to understand it reading all sorts of textbooks but there are some points I want to seek clarification in that thread.. specifically I queried Peterdonis "When you want to solve a single Eigenstate for the entire apple in position observable.." and his reply is that:

"There is no such thing. Read my previous post again. Even for the two-particle quantum system I described there (let alone for an apple with something like 10^25 particles), the center of mass observable does not pick out a single state for the system. It only picks out a superposition of all the possible states that have the measured value for the center of mass position, i.e., that have position values for each of the individual particles that average to the measured center of mass position. So there is no such thing as an "eigenstate" of the center of mass observable."

I just need to know for now that he means "to pick up a single state of the system"... I want an example of something that can pick out a single state of the system. Is he saying it is not possible to do with when entangled particles are 2 or more.. and why it that exactly since one can define wave function for the entire entangled system (but not for subsystem).. so why can't you use it to pick up a single state of the entangled system?

 

[vanhees71]

Ok, then let's wait for #PeterDonis to explain it. I don't know, what he means with that phrase.

 

[bluecap]

Ok, then let's wait for #PeterDonis to explain it. I don't know, what he means with that phrase.

To avoid being redundant since PeterDonis has spent time answering that thread a year ago like no others can.. let me home in on the following details:

1. For nearly a year I have toiled and I tried to understand it reading all sorts of textbooks but there are some points I want to seek clarification in that thread.. specifically I mentioned "When you want to solve a single Eigenstate for the entire apple in position observable.." and Peterdonis comment was "There is no such thing. Read my previous post again. Even for the two-particle quantum system I described there (let alone for an apple with something like 10^25 particles), the center of mass observable does not pick out a single state for the system." Ok. I'd like to know what kind of observable can pick out a single state for the system, is there one? Can the "joint position" observable do that? What other observable in addition to it that can do that where it can pick out a single state for the system? Or if not possible, why is that not possible since an entangled system is supposed to be one single state?

2. I'm interested in subspaces because of this paper by Zurek titled "Pointer Basis of Quantum Apparatus: Into What Mixture does the Wave Package Collapse" and quoting "In contrast to this simplied model, setups of real-world apparatuses are much more complicated and demand extensive product spaces to allow for a complete description. Out of this vast Hilbert space we have singled out just one subspace, claiming it describes the "pointer," and hence epitomizes the apparatus itself." Therefore Pointer States are indeed these Hilbert Subspaces. Now consider an apple. Something so called Predictability Sieve can sort out the Hilbert Space looking for these Subspaces that are the most classical in order to avoid superpositions. So I need an actual example (or close to it) of what subspaces in the apple can produce superpositions that can interfere with itself. Decoherence with the environment can make it lost phase coherence as it entangles with the different subspaces in the environment.. but in the apple.. what subspaces or what observables (is it more than one at same time?) of the subspaces can cause superposition of these subspaces (or Pointer States)?

This is a question of critical importance. Thank you.

 

[PeterDonis]

I want an example of something that can pick out a single state of the system.

Picking out a single state of the system means the same thing as @vanhees71 described in post #6: you need to specify exact values for a complete set of observables. For example, if your pencil contains $10^{25}$ atoms, then picking out a single state of the system would mean specifying an exact position, exact spin, etc. for every single one of those $10^{25}$ atoms. Obviously you can only specify a lot less than that if all you know is the center of mass position of the pencil as a whole.

 

[bluecap]

Picking out a single state of the system means the same thing as @vanhees71 described in post #6: you need to specify exact values for a complete set of observables. For example, if your pencil contains $10^{25}$ atoms, then picking out a single state of the system would mean specifying an exact position, exact spin, etc. for every single one of those $10^{25}$ atoms. Obviously you can only specify a lot less than that if all you know is the center of mass position of the pencil as a whole.

Hilbert subspaces are not rare thing. In fact, in everyday life.. objects we interact with are actually Hilbert subspaces.. in https://arxiv.org/pdf/quant-ph/0105127.pdf

"Predictability sieve sifts all of the Hilbert space, ordering states according to their predictability. The top of the list will be the most classical.."
"Predictability sieve can be generalized to situations where the initial states are mixed (Paraonanu, 2002). Often whole subspaces emerge from the predictability sieve, naturally leading to “decoherence-free subspaces”

Let's take our apple. A real apple before decoherence is in superposition of top and down, left and right. But decoherences choose subspace that is the most classical. I'd like to know what observables of the classicality it usually or has chosen.. is it many observables at the same time or just one? What is it? Just some hint if you can't specify the entire subspaces of the apple.

 

[PeterDonis]

Hilbert subspaces are not rare thing. In fact, in everyday life.. objects we interact with are actually Hilbert subspaces

You really should think twice before making categorical statements about a concept which, only a few posts ago, you were asking for the meaning of.

The paper by Zurek that you linked to is not mainstream QM; it is simply one proposal for how to apply the phenomenon of decoherence to explain how it is that we observe a world that appears classical. (And any serious discussion of that proposal requires an "A" level thread; we can't really do justice to it at the "I" level.) And it does not say anything that can be reasonably interpreted as "in everyday life, objects we interact with are actually Hilbert subspaces". If you think it says that, you are misunderstanding something.

A real apple before decoherence

There is no such thing in any practical sense. A real apple is continuously decohering itself; it contains something like $10^{25}$ atoms which are interacting in all kinds of ways, and there is never any time at which any part of it can be viewed as "not having decohered yet". The apple doesn't even have to interact with anything else for that to be true; the interactions of its atoms alone are already more than enough.

I'd like to know what observables of the classicality it usually or has chosen

This question doesn't really have a meaningful answer beyond the obvious one that the classical "observables" (things like center of mass position) will have reasonably definite values. If you're looking for something like an observable applied to every single one of the $10^{25}$ atoms in the apple, AFAIK nobody knows of one or has any reason to believe that there is one. I certainly don't see anything in Zurek's paper that looks like he's proposing one.

 

[bluecap]

You really should think twice before making categorical statements about a concept which, only a few posts ago, you were asking for the meaning of.

The paper by Zurek that you linked to is not mainstream QM; it is simply one proposal for how to apply the phenomenon of decoherence to explain how it is that we observe a world that appears classical. (And any serious discussion of that proposal requires an "A" level thread; we can't really do justice to it at the "I" level.) And it does not say anything that can be reasonably interpreted as "in everyday life, objects we interact with are actually Hilbert subspaces". If you think it says that, you are misunderstanding something.
There is no such thing in any practical sense. A real apple is continuously decohering itself; it contains something like $10^{25}$ atoms which are interacting in all kinds of ways, and there is never any time at which any part of it can be viewed as "not having decohered yet". The apple doesn't even have to interact with anything else for that to be true; the interactions of its atoms alone are already more than enough.
This question doesn't really have a meaningful answer beyond the obvious one that the classical "observables" (things like center of mass position) will have reasonably definite values. If you're looking for something like an observable applied to every single one of the $10^{25}$ atoms in the apple, AFAIK nobody knows of one or has any reason to believe that there is one. I certainly don't see anything in Zurek's paper that looks like he's proposing one.

But in the first paragraph of the paper quoting:

"Decoherence is caused by the interaction with the environment which in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the Universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal “Schrödinger cat” states."

It means without decoherence... you have mostly the flagrantly nolocal "Schrödinger cat" states". While you were right an object such as an a real apple is " continuously decohering itself; it contains something like 1025'>10251025 10^{25} atoms which are interacting in all kinds of ways, and there is never any time at which any part of it can be viewed as "not having decohered yet""

But Zurek clearly stated you need Einselection to avoid the coherence between the pointer states (subspaces) of the apple. I think what he is saying is this. The apple is having unitary evolution in all branches.. so without the environment, the apple is doing all sorts of histories.. now the environment simply choose one classical state. This description is right isn't it. Now I'd just want to know what observable does this einselected states choose.. It couldn't be just the center of mass. What are the other observables Einselection use?

 

[PeterDonis]

The apple is having unitary evolution in all branches.. so without the environment, the apple is doing all sorts of histories.. now the environment simply choose one classical state. This description is right isn't it.

No. In Zurek's terminology, the apple is its own "environment". (More precisely, you can pick anyone of the $10^{25}$ atoms in the apple, and the rest of the apple will be that atom's "environment" in Zurek's sense.)

Also, decoherence and einselection (the process Zurek describes) doesn't pick out "one classical state". If we are using the MWI, then there are still multiple branches in which the apple has different classical states. All decoherence and einselection can do, according to Zurek's model, is to rule out the non-classical states--the "Schrödinger's cat" type states. It can't pick out a single classical state from among all of the different possible branches.

 

[bluecap]

No. In Zurek's terminology, the apple is its own "environment". (More precisely, you can pick anyone of the $10^{25}$ atoms in the apple, and the rest of the apple will be that atom's "environment" in Zurek's sense.)

What passage did you read this? The following clearly says the environment is outside of the system: https://arxiv.org/pdf/1412.5206.pdf
"Decoherence selects preferred pointer states that survive interaction with the environment. They are localized and effectively classical. They persist while their superpositions decohere. Decoherence marks the border between quantum and classical, alleviating concern about flagrant and manifestations of quantumness in the macroscopic domain. Here we consider emergence of ‘the classical’ starting at a more fundamental pre-decoherence level, tracing the origin of preferred pointer states and deducing their probabilities from the core quantum postulates. We also explore role of the environment as a medium through which observers acquire information. This mode of information transfer leads to perception of objective classical reality."

It is clearly mentioned above that the environment which is a medium which observers acquire information is not the system or apple itself.

Also, decoherence and einselection (the process Zurek describes) doesn't pick out "one classical state". If we are using the MWI, then there are still multiple branches in which the apple has different classical states. All decoherence and einselection can do, according to Zurek's model, is to rule out the non-classical states--the "Schrödinger's cat" type states. It can't pick out a single classical state from among all of the different possible branches.

I wanted to ask you something about this since months ago. in this thread https://www.physicsforums.com/threads/zureks-existential-interpretation.724690/
atyy wrote that:

"By Zurek's approach do you mean decoherence and quantum darwinism? I think in more recent work, he's tended to say that decoherence does not lead to apparent collapse, and that quantum darwinism is needed for apparent collapse, and apparent collapse is the only collapse that happens (ie. apparent collapse with decoherence and quantum darwinism = collapse in Copenhagen).

I'm thinking of his terminology in http://arxiv.org/abs/0707.2832 and http://arxiv.org/abs/0903.5082."

so apparent collapse with decoherence and quantum darwinism = collapse in Copenhagen

apparent collapse is when states are in improper mixed state.. add quantum Darwinism, the implication is it can explain how improper mixed state becomes proper mixed state.
Also in the last paragraph of the paper shared at start of this message.. it was written:
"Our proofs of Hermiticity, 4a and of Born’s rule, 5 are straightforward. They fit into the picture based on decoherence process that amplifies and disseminates information about selected (pointer) observables throughout the environment. Quantum Darwinism shows why only such redundantly recorded pointer states are accessible to observers—it can account for perception of ‘quantum jumps’. However, full account of collapse involves ‘consciousness’, and may have go beyond just mathematics or physics. Good questions are valuable. It may yet turn out that residual worries about collapse lead to a good question"

Hence it can indeed pick out a single classical state from among all of the different possible branches. I think Zurek has two goals.. one is what you mentioned that "decoherence and einselection can do, according to Zurek's model, is to rule out the non-classical states--the "Schrödinger's cat" type states". In addition to that.. Zurek also tries to make it indeed pick out a single classical state from among all of the different possible branches.

Also why did he mentioned about consciousness which can explain full account of collapse in the passage above?

 

[PeterDonis]

The following clearly says the environment is outside of the system

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.

apparent collapse with decoherence and quantum darwinism = collapse in Copenhagen

This part of what he's saying is certainly not mainstream QM; it's a speculative proposal on his part. It's not even necessarily implied by decoherence and einselection by themselves; those are perfectly compatible with the MWI.

why did he mentioned about consciousness

That's even more speculative on his part, since he admits, in the passage you quote, that it goes "beyond" math or physics. So that part is really off topic here.

 

[bluecap]

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.

I'm understanding it for years that his system are macroscopic objects like balls and telephones. This is the passage that says it all:

https://arxiv.org/pdf/quant-ph/0105127.pdf

"An observer perceiving the Universe from within is in a very different position than an experimental physicist studying a state vector of a quantum system. In a laboratory, Hilbert space of the investigated system is typically tiny. Such systems can be isolated, so that often the information loss to the environment can be prevented. Then the evolution is unitary. The experimentalist can know everything there is to know about it. Common criticisms of the approch advocated in this paper are based on an unjustified extrapolation of the above laboratory situation to the case of the observer who is a part of the Universe. Critics of decoherence often note that the differences between the laboratory example above and the case of the rest of the Universe are ‘merely quantitative’ – the system under investigation is bigger, etc. So why cannot one analyze – they ask – interactions of the observer and the rest of the Universe as before, for a small isolated quantum system?

In the context of the existential interpretation the analogy with the laboratory is, in effect, turned “upside down”: For, now the observer (or the apparatus, or anything effectively classical) is continuously monitored by the rest of the Universe. Its state is repeatedly collapsed – forced into the einselected states – and very well (very redundantly) ‘known’ to the rest of the Universe."

For Zurek, systems are macroscopic objects like apparatus, us, balls or others.. why did you understand it as otherwise.. what do you mean he has examples where the system is an atom of the apple instead of the whole apple? reference please?

I think what Zurek is proposing is this. An object like apple is in many different branches and undergoing all evolutions on its own. And one branch is simple a subspace (or pointer states). Mathematically is this possible where an MWI branch is the subspace itself. If yes, then this is it. What Zurek is proposing.

This part of what he's saying is certainly not mainstream QM; it's a speculative proposal on his part. It's not even necessarily implied by decoherence and einselection by themselves; those are perfectly compatible with the MWI.
That's even more speculative on his part, since he admits, in the passage you quote, that it goes "beyond" math or physics. So that part is really off topic here.

 

[PeterDonis]

I'm understanding it for years that his system are macroscopic objects like balls and telephones.

You're understanding incorrectly. He gives an explicit example in his paper (the first one you linked to) where the "system" is one qubit, the "apparatus" is another qubit, and the "environment" is a third qubit.

I think what Zurek is proposing is this. An object like apple is in many different branches and undergoing all evolutions on its own. And one branch is simple a subspace (or pointer states).

This is not what Zurek is proposing. You have misunderstood his model.

Mathematically is this possible where an MWI branch is the subspace itself.

No.

 

[PeterDonis]

what do you mean he has examples where the system is an atom of the apple instead of the whole apple? reference please?

The key passage in the paper is on p. 14, the first sentence of the second paragraph of section IV:

"Environments can be external (such as particles of the air or photons that scatter off, say, the apparatus pointer) or internal (collections of phonons or other internal excitations)."

His "internal" example is not quite the same as mine--he's not picking out certain atoms as the "system", but rather certain degrees of freedom internal to the object (phonons are internal vibrational degrees of freedom). But that's really a better way of putting it anyway, since it's more general ("atoms" is really a name for particular degrees of freedom). The key point is that the "environment" is internal to the object (the apple, pencil, whatever); it's not a matter of the object interacting with anything else, but of different internal degrees of freedom of the object interacting with each other. Which degrees of freedom you call "system", "apparatus", or "environment" is in general an arbitrary choice; it depends on what you're trying to do with your model.

 

[bluecap]

The key passage in the paper is on p. 14, the first sentence of the second paragraph of section IV:

"Environments can be external (such as particles of the air or photons that scatter off, say, the apparatus pointer) or internal (collections of phonons or other internal excitations)."

His "internal" example is not quite the same as mine--he's not picking out certain atoms as the "system", but rather certain degrees of freedom internal to the object (phonons are internal vibrational degrees of freedom). But that's really a better way of putting it anyway, since it's more general ("atoms" is really a name for particular degrees of freedom). The key point is that the "environment" is internal to the object (the apple, pencil, whatever); it's not a matter of the object interacting with anything else, but of different internal degrees of freedom of the object interacting with each other. Which degrees of freedom you call "system", "apparatus", or "environment" is in general an arbitrary choice; it depends on what you're trying to do with your model.

I believed you that the system is tiny degrees of freedom and not the entire object.

I reread all of Zurek papers over the weekend and some others in light of that fact and pondering on it continuously. So this message is written after careful considerations to avoid redundant thoughts. One thing that made me think a long time is his stuff about fragments.

You see. While it is true that a macroscopic object is measured by the environment via decoherence and all those 10^25 atoms in the pencils are interacting amongst themselves and with the environments. But it is not impossible to design a device that can scan micron by micron the pencil and perturb the quantum state to reset it (or format it).

In the Orthodox or Copenhagen. Observations is the primitive but in Quantum Darwinism, Zurek's attempt is to reduce QM to the first two axioms he talks about in his paper. Since that doesn't include the concept of observation he has to give a fully quantum account of it, which he does via his idea of observing fragments.

However, note that observing fragments is not enough. Because of the no cloning mechanism, if you observe fragments, it can still perturb the system, hence more important is the concept of progeny of observables.. or as Zurek put it in his Quantum Darwinism paper:

“To obtain information about S from E one can then measure fragments F of the environment – non-overlapping collections of subsystems of E, (c). there are many copies of the information about S in E – “progeny” of the “fittest observable” that survived monitoring by E proliferates throughout E. This proliferation of the multiple informational offspring defines Quantum Darwinism. The environment becomes a witness with redundant copies of information about the preferred observable. This leads to the objective existence of pointer states: Many can find out the state of the system independently, without prior information, and they can do it indirectly, without perturbing S.”

“Redundancy allows for objective existence of the state of S: It can be found out indirectly, so there is no danger of perturbing S with a measurement. Error correction allowed by redundancy is also important: Fragility of quantum states means that copies in F’s are damaged by measurements (we destroy photons!), and may be measured in a “wrong” basis. One cannot access records inE without endangering their existence. But with many (Rδ) copies, state of S can be found out by ∼ Rδ observers who can get their information independently, and without prior knowledge about S. Consensus between copies suggests objective existence of the state of S.”

Peter, not only Zurek mentions this in all his papers but so many Ph.D. physicists such as this doctorate thesis http://tuprints.ulb.tu-darmstadt.de/5148/1/Balaneskovic_Dissertationstext_Pflichtexemplar.pdf

Point is. They all know that macroscopic objects have over 10^50 atoms and these interact amongst themselves and with the environment or in other words, they know macroscopic objects are constantly being monitored by the environment, hence the need or nature of the fragments is not merely because the macroscopic objects transmit many copies of the informations to the environment.. maybe one can always set up experiments that can perturb this object micron by micron.. what is critical is the fragments can produce progeny to defeat the “no cloning” mechanism and can be read by many observers without perturbing the system.

The following is my questions to organize my thoughts.

1. Outside Quantum Darwinism, why do physicists not bothered about perturbing the system? In conventional decoherence. Decoherence is an interaction between systems, usually the environment, that transforms a pure state into a mixed state in a particular basis (like position… i.e. macro objects are decohered to be in eigenstates of position). And that basis is said to be stable meaning it does not change as the interaction evolves. In conventional decoherence. Doesn’t it really change? For those who don’t subscribe to Quantum Darwinism, why are they not bothered that when objects are decohered to be in eigenstates of position, one can still perturb it by measuring it again with say the Momentum basis where the wave function will turn into a spread out wave?

2. I’m thinking up of one experiment that can illustrate Zurek idea. Can you think of one? Why can’t you scan the macroscopic object using devices like laser for example micron by micron and reset the quantum state of each particle (or your atom or other pointer states). After many days of such scanning.. the macroscopic object should be perturbed. Again note Zurek doesn’t write dozens of papers and other Ph.D. doctorates about the need for quantum darwnism to avoid perturbing the macroscopic system if they know it’s not important. They surely know macroscopic object has 10^50 atoms and these interact among themselves and with the environment transmitting many copies. These are NOT the reasons for introducing the concepts of fragments and progeny. It seems to be separate reason.. maybe because observations are derived from Zurek first two axioms and macroscopic object can really be perturbed by this mechanism that can’t by Copenhagen with the classical and quantum divide?

Thanks so much for sharing.

 

[PeterDonis]

it is not impossible to design a device that can scan micron by micron the pencil and perturb the quantum state to reset it (or format it)

What is your basis for this statement?

the fragments can produce progeny to defeat the “no cloning” mechanism

No, that's not correct. The fragments do not store copies of the entire quantum state of the original system. They only store "copies" of a portion of it. The no cloning mechanism cannot be "defeated".

Note carefully this statement in what you quoted:

The environment becomes a witness with redundant copies of information about the preferred observable.

"Information about the preferred observable" is not the same as "a complete cloned copy of the entire quantum state".

For those who don’t subscribe to Quantum Darwinism, why are they not bothered that when objects are decohered to be in eigenstates of position, one can still perturb it by measuring it again with say the Momentum basis where the wave function will turn into a spread out wave?

Why should they be bothered by this? It's possible in principle, but it never happens with ordinary objects, so what's the problem?

Why can’t you scan the macroscopic object using devices like laser for example micron by micron and reset the quantum state of each particle (or your atom or other pointer states).

Why do you think this is possible? When we do this in the lab with individual qubits, we have to take great care to make sure the qubits aren't interacting with anything else; otherwise "resetting the quantum state" doesn't work. But if you are trying to "scan" an individual atom in a macroscopic object, the atom is interacting with all of the other atoms--there's no way to isolate it the way we isolate qubits in the lab. So why would you expect to be able to manipulate that atom the way qubits in the lab are manipulated?

 

[bluecap]

What is your basis for this statement?
No, that's not correct. The fragments do not store copies of the entire quantum state of the original system. They only store "copies" of a portion of it. The no cloning mechanism cannot be "defeated".

Note carefully this statement in what you quoted:
"Information about the preferred observable" is not the same as "a complete cloned copy of the entire quantum state".

What I meant was the progeny are informational offspring.. so as to go around the cloning mechanism - not to violate it. This is clearly stated in:

"This insight captures the essence of Quantum Darwinism: Only states that produce multiple informational offspring – multiple imprints on the environment – can be found out from small fragments of E. The origin of the emergent classicality is then not just survival of the fittest states (the idea already captured by einselection), but their ability to “procreate”, to deposit multiple records – copies of themselves – throughout E."

Why should they be bothered by this? It's possible in principle, but it never happens with ordinary objects, so what's the problem?

How do you know it never happens with ordinary objects. There are so many stuff that physicists ignore without even investigating. That is why some of us has to come out and tie up the loose ends.

Why do you think this is possible? When we do this in the lab with individual qubits, we have to take great care to make sure the qubits aren't interacting with anything else; otherwise "resetting the quantum state" doesn't work. But if you are trying to "scan" an individual atom in a macroscopic object, the atom is interacting with all of the other atoms--there's no way to isolate it the way we isolate qubits in the lab. So why would you expect to be able to manipulate that atom the way qubits in the lab are manipulated?

If Zurek knew this. Why does he have to cook up the entire idea of Quantum Darwinism? He kept reasoning the idea was so they observers can't perturb the macroscopic system. Maybe he just mentions it to attract followers when his main purpose is how to derive the Born Rule (Invariance) only from his first two axioms which is: (i) States are represented by vectors in Hilbert space, and; (ii) Evolutions are unitary?

If that's his purpose. He could just focus on the Born Rule deriviations but he and others just kept repeating the idea that the Fragments purpose is so the observers can't perturb the system or pointer states directly. If this is true. He has to produce an experimental setup where the macroscopic object can be perturbed. If he couldn't think of one and others. Then why did he believe it is possible?? This is what puzzles me the whole weekend.

 

[PeterDonis]

What I meant was the progeny are informational offspring.. so as to go around the cloning mechanism

What does "go around" mean? The no cloning theorem always applies.

How do you know it never happens with ordinary objects.

Because we observe ordinary objects to behave classically.

There are so many stuff that physicists ignore without even investigating. That is why some of us has to come out and tie up the loose ends.

This is verging on personal speculation and is off topic here. Please review the PF rules.

He kept reasoning the idea was so they observers can't perturb the macroscopic system.

No, that's not his key point. His key point is that observers can obtain information about the system without needing to perturb it, by getting the information from the fragments in the environment. That is important because it explains how multiple observers can all obtain the same information about a macroscopic system while the system itself stays in the same state.

He has to produce an experimental setup where the macroscopic object can be perturbed.

I don't understand why you think this. See above.

 

[bluecap]

What does "go around" mean? The no cloning theorem always applies.

I know. I don't know the exact terms to use. We know gravity always applies. To go around gravity, we need to use airplane.. so it's not violating gravity but going around it. Btw.. if the fragments are just copies of the observable information, can you please give an example of how to copy observable information.. how do the photons (fragments) just copy the observable information? is it in the frequency.. or what?

Because we observe ordinary objects to behave classically.
This is verging on personal speculation and is off topic here. Please review the PF rules.
No, that's not his key point. His key point is that observers can obtain information about the system without needing to perturb it, by getting the information from the fragments in the environment. That is important because it explains how multiple observers can all obtain the same information about a macroscopic system while the system itself stays in the same state.
I don't understand why you think this. See above.

Let's say observers need to perturb the system. Any idea of one experimental setup of how it can occur? If we can't perturb qubits when it's not isolated or can't perturb the atoms because it is in interaction with the rest of the molecules (thermal) and photons. Does it mean you can only perturb by doing it to the entire macroscopic at once? for example you have a porcelain figure that you want to bend. If you perturb it with lasers simultaneously every molecules and every atoms. Can you make the porcelain figure bend? Or other examples you can think of where you perturb the entire object at once? In principle can this perturb the system?

 

[PeterDonis]

if the fragments are just copies of the observable information, can you please give an example of how to copy observable information

It depends on the specific scenario. If you are getting information about an object by looking at photons that bounced off of it, the information is going to be stored in the frequencies (energies) of the photons and the directions they are coming from.

Let's say observers need to perturb the system

Why?

 

[bluecap]

It depends on the specific scenario. If you are getting information about an object by looking at photons that bounced off of it, the information is going to be stored in the frequencies (energies) of the photons and the directions they are coming from.
Why?

I can't understand this logic.. or this logic seems not to be right. If there is no way to perturb microscopic object. Then Zurek shouldn't mention about it occurring if observers don't use fragments. Can you give other examples of this logic in other areas of life where this reasoning is used by others? For those taking science of logic. What is this logic called where:

1. it's not possible to perturb macroscopic object..
2. observers use fragments so as not to perturb macroscopic object..

the logic doesn't seem to be correct.. hope others can help me verbalize this in other ways of this seeming illogic in the logic..