I 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 “Schrodinger cat” states."

It means without decoherence... you have mostly the flagrantly nolocal "Schrodinger 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 "Schrodinger'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 "Schrodinger'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 "Schrodinger'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..

 

[PeterDonis]

If there is no way to perturb microscopic object. Then Zurek shouldn't mention about it occurring if observers don't use fragments.

I don't understand where you're getting this from. It seems like you're misunderstanding Zurek's model. It does not require observers to perturb the system. It only requires observers to interact with fragments in the environment that store information they originally obtained by interacting with the system.

 

[bluecap]

I don't understand where you're getting this from. It seems like you're misunderstanding Zurek's model. It does not require observers to perturb the system. It only requires observers to interact with fragments in the environment that store information they originally obtained by interacting with the system.

I know. I'm just asking if it is possible to perturb the system and how. Zurek seems to be saying it is possible. I just want to know one example of how to do it. Anyone got any ideas? Unless you meant Zurek was saying it was impossible to perturb the system? But in his papers. He seemed to be saying it was possible. I just want a model of how to do it.

 

[PeterDonis]

I'm just asking if it is possible to perturb the system and how.

Any interaction with a system will perturb it to some extent. Roughly speaking, the more energetic the interaction, the greater the perturbation. For example, if the system is a pencil, you can perturb it by writing with it--a small amount of graphite gets transferred from the pencil to the paper. Or you can perturb it more strongly by applying enough force to break it.

 

[bluecap]

Any interaction with a system will perturb it to some extent. Roughly speaking, the more energetic the interaction, the greater the perturbation. For example, if the system is a pencil, you can perturb it by writing with it--a small amount of graphite gets transferred from the pencil to the paper. Or you can perturb it more strongly by applying enough force to break it.

Hmm.. Perhaps what Zurek meant was that if we don't use photons (fragments) to map the positions of objects.. Then we need to touch the objects (perhaps a blind person) so as to know the shape of the object and this perturbing can cause his fingerprint to be transferred to the systems. Maybe this is what is meant by perturbing the system without using fragments.. isn't it?

 

[PeterDonis]

Maybe this is what is meant by perturbing the system without using fragments

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

 

[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.