How Do Superposition Preparations Differ for Single vs. Entangled States?

[Edward Wij]

What is the difference in the preparation of superposition for the 2 cases where:

1. The system is in either one state.. for example.. one electron in spin up or spin down. Do these interfere?
2. The system is in superposition of two states (for example 2 entangled electrons that is spin up and spin down).

Are they both called Superposition? What specific term distinguish them?

 

[bhobba]

The system is in either one state.. for example.. one electron in spin up or spin down. Do these interfere?

No.

2. The system is in superposition of two states (for example 2 entangled electrons that is spin up and spin down).

All superposition does is reflect the vector space structure of pure states:
http://www.math.ucla.edu/~tao/resource/general/121.1.00s/vector_axioms.html

Any state is the superposition of tons of other states.

Thanks
Bill

 

[Edward Wij]

No.

Do you generally agree that mixed states occurred because the measuring device is entangled with the system breaking the pure state.. or is it possible for mixed state to occur without entangling with the measurement device or environment? Any example?

All superposition does is reflect the vector space structure of pure states:
http://www.math.ucla.edu/~tao/resource/general/121.1.00s/vector_axioms.html

Any state is the superposition of tons of other states.

Thanks
Bill

 

[vanhees71]

First of all you have to specify, what you mean by "superposition". You have to choose a basis to do so in order to say, a pure state is represented either by a basis vector or is a super position (linear combination) of several such states. Physically a basis in Hilbert space is defined as the complete set of eigenvectors of a self-adjoint operator, representing an observable. More precisely, a pure state (which represents complete knowledge about the system's state within quantum theory) is represented by a ray in Hilbert space (i.e., a non-zero Hilbert-space vector up to a non-zero factor).

Further, a mixture (mixed state) is something different. It represents the description of a system, for which we have not complete knowledge about its state. It is represented by a Statistical operator, which is a positive semidefinite self-adjoint operator with trace 1. Also a pure state can be represented by such an operator, and thus this is the most general (and also most correct) representation of a general state. A state in this sense is pure if and only if the statistical operator is a projection operator, i.e., of the form
$$\hat{R}=\hat{P}_{\psi}=|\psi \rangle \langle \psi|,$$
where $|\psi \rangle$ is a normalized Hilbert-space vector.

 

[bhobba]

Do you generally agree that mixed states occurred because the measuring device is entangled with the system breaking the pure state.. or is it possible for mixed state to occur without entangling with the measurement device or environment? Any example?

Well its not really on topic but what the heck.

Generally in practice mixed states are from entanglement, but that's not the only way.

You can also do it by presenting states to be randomly observed - such mixed states are called proper - otherwise they are improper.

It is the essence of the modern view of the measurement problem that you can't tell the difference between the two.

Thanks
Bill

 

[Edward Wij]

Well its not really on topic but what the heck.

Generally in practice mixed states are from entanglement, but that's not the only way.

You can also do it by presenting states to be randomly observed - such mixed states are called proper - otherwise they are improper.

It is the essence of the modern view of the measurement problem that you can't tell the difference between the two.

Thanks
Bill

Can you give an example where in mixed state.. it is there prior to observation and there is no collapse. Looking at your message archives.. I read you kept saying that. I'd like to know if this your only position or others too? In the stern-gerlach spin equipment, either y or z spin can form.. this means it is not in that state before measurement, this seems to conflict with what you said that in mixed state it is in definite state between measurement (no collapse).. can you please elaborate what the distinctions using the stern-gerlach experiments.. thanks in advanced..

 

[bhobba]

Can you give an example where in mixed state.. it is there prior to observation and there is no collapse.

I gave it above.

Thanks
Bill

 

[vanhees71]

The most common mixed state is given by the canonical statistical operator of a many-body system
$$\hat{R}=\frac{1}{Z} \exp(-\beta \hat{H}), \quad Z=\mathrm{Tr} \exp(-\beta \hat{H}).$$
It describes the many-body system at temperature $T=k_{\text{B}}/\beta$.

 

[Edward Wij]

I gave it above.

Thanks
Bill

I'd like to know if it is just your belief that in proper mixture.. you present states to be randomly observed - hence it is there prior to observation and no need for collapse. For mainstream physicists.. do they also hold this belief? In pure Copenhagen, the spins in stern-gerlach doesn't have definite state before measurements.. you seemed to believe it is because in the final outcome, it is classical.. so you believe it is classical all the way to the quantum state. (Vanhees71, do you also believe that in proper mixture, there is no collapse and the state is there prior to observations, how about others?)

 

[vanhees71]

In my opinion, there is no such thing as collapse at all. For me, the only consistent interpretation of quantum theory which is compatible with Einstein causality is the minimal statistical interpretation.

 

[Edward Wij]

In my opinion, there is no such thing as collapse at all. For me, the only consistent interpretation of quantum theory which is compatible with Einstein causality is the minimal statistical interpretation.

Oh. So both you and Bhobba follow the minimal statistical interpretation. How about those who do not? Or to be interpretation free. Can you give an actual experiment or example where the mixed state is there prior to observation and no collapse? Because I am not sure what experiment you two are thinking about.

 

[bhobba]

Can you give an actual experiment or example where the mixed state is there prior to observation and no collapse?

I gave it bfore. What exactly don't you get about states presented randomly for observation?

And I agree entirely with Vanhees. There is no collapse in the axioms of QM - its simply something some interpretations introduce.

Thanks
Bill

 

[Edward Wij]

I gave it bfore. What exactly don't you get about states presented randomly for observation?

And I agree entirely with Vanhees. There is no collapse in the axioms of QM - its simply something some interpretations introduce.

Thanks
Bill

Let's use the stern-gerlach experiment, there it is shown that the spins of electron don't have definite values before measurements. So how do you apply the mixed state in the stern-gerlach?

 

[bhobba]

I'd like to know if it is just your belief that in proper mixture.. you present states to be randomly observed

Its a definition - you can't really argue with a definition.

You are getting a bit confused about the whole issue. Think of it this way - the minimum statistical interpretation is consistent with a number of interpretations - that's because its minimal. Now one interpretation it's consistent with is BM which has no collapse. Another is MW. An interpretation that has explicit collapse is GRW - its consistent with that. Copenhagen takes a different view - the state is entirely subjective - it has explicit collapse but since it is entirely subjective it matters not.

Thanks
Bill

 

[bhobba]

Let's use the stern-gerlach experiment, there it is shown that the spins of electron don't have definite values before measurements. So how do you apply the mixed state in the stern-gerlach?

Its entangled with the apparatus - you yourself mentioned entanglement creating mixed states.

Its not a proper mixed state - or rather the QM formalism doesn't say it must be, but in some interpretations it is.

Thanks
Bill

 

[Edward Wij]

Its entangled with the apparatus - you yourself mentioned entanglement creating mixed states.

Its not a pure mixed state - or rather the QM formalism doesn't say it must be, but in some interpretations it is.

Thanks
Bill

Ok. My primary question is without the system entangling with the apparatus (or environment) and the system is in pure state, would it be possible to have mixed state inside the pure state by measuring the subsystem of the pure state?

 

[bhobba]

would it be possible to have mixed state inside the pure state by measuring the subsystem of the pure state?

I have no idea what you are trying to say.

Thanks
Bill

 

[Edward Wij]

I have no idea what you are trying to say.

Thanks
Bill

Let's say particle A is entangled with particle B and particle C. If you just measure particle A and B (subsystem of A, B, C). Can A and B form mixed state?

 

[bhobba]

Let's say particle A is entangled with particle B and particle C. If you just measure particle A and B (subsystem of A, B, C). Can A and B form mixed state?

Yes. But I fail to see your issue/point etc.

My suspicion is you are angling for some way a purely quantum process to create a proper mixed state. It can't.

Can you please be more explicit about what exactly you are getting at. If you find that difficult can I suggest you think about it more carefully before posting?

Thanks
Bill

 

[Edward Wij]

Yes. But I fail to see your issue/point etc.

My suspicion is you are angling for some way a purely quantum process to create a proper mixed state. It can't.

Can you please be more explicit about what exactly you are getting at. If you find that difficult can I suggest you think about it more carefully before posting?

Thanks
Bill

Oh. I made a mistake. Let me rephrase it. Let's say particle A is entangled with particle B and particle C. If you just consider (Not Measure) particle A and B (subsystem of A, B, C). Can A and B form mixed state? Without measurement, can A and B as subsystem of A,B,C for mixed state? Please reply again on this exact context (to emphasize the truth).

 

[bhobba]

Oh. I made a mistake. Let me rephrase it. Let's say particle A is entangled with particle B and particle C. If you just consider (Not Measure) particle A and B (subsystem of A, B, C). Can A and B form mixed state? Without measurement, can A and B as subsystem of A,B,C for mixed state? Please reply again on this exact context (to emphasize the truth).

I don't think so - entanglement is really something associated with pure states by the definition of entanglement - mixed states can't come into it. But others may be able to think of some way - I cant. For example in EPR one particle becomes entangled with the apparatus, becomes a mixed state, and breaks entanglement.

Thanks
Bill

 

[Edward Wij]

I don't think so - entanglement is really something associated with pure states by the definition of entanglement - mixed states can't come into it. But others may be able to think of some way - I cant. For example in EPR one particle becomes entangled with the apparatus, becomes a mixed state, and breaks entanglement.

Thanks
Bill

Ok. At least it's clear. But I heard in Many worlds, the mixed states exist even within pure states because the mixed states are all real. What do you make of this?

 

[bhobba]

Ok. At least it's clear. But I heard in Many worlds, the mixed states exist even within pure states because the mixed states are all real. What do you make of this?

You heard wrong.

In MW the INTERPRETATION is the components of the mixed state after decoherence is a separate world. No collapse - everything simply keeps evolving.

Thanks
Bill

 

[Edward Wij]

I don't think so - entanglement is really something associated with pure states by the definition of entanglement - mixed states can't come into it. But others may be able to think of some way - I cant. For example in EPR one particle becomes entangled with the apparatus, becomes a mixed state, and breaks entanglement.

Thanks
Bill

But wait here. In EPR, if one particle become entangled with the apparatus, then they form global entanglement of EPR and apparatus. So they are all still in pure state, so how do you get mixed state out of it? This goes right back to my example.

 

[bhobba]

But wait here. In EPR, if one particle become entangled with the apparatus, then they form global entanglement of EPR and apparatus. So they are all still in pure state, so how do you get mixed state out of it? This goes right back to my example.

Come again? Remember what I said - it breaks entanglement. You have one particle not entangled with anything and the other entangled with the apparatus.

I think you need to go and think about this a bit.

Thanks
Bill

 

[Edward Wij]

Come again? Remember what I said - it breaks entanglement. You have one particle not entangled with anything and the other entangled with the apparatus.

I think you need to go and think about this a bit.

Thanks
Bill

But in decoherence, the system and environment still form one big superposition, so analogy wise, I thought the pair should still have superposition with the apparatus. Our ever reliable guide wiki referenced:

"Decoherence does not generate actual wave function collapse. It only provides an explanation for the observation of wave function collapse, as the quantum nature of the system "leaks" into the environment. That is, components of the wavefunction are decoupled from a coherent system, and acquire phases from their immediate surroundings. A total superposition of the global or universal wavefunction still exists (and remains coherent at the global level), but its ultimate fate remains an interpretational issue."

Here total superposition still occurs. So in the EPR pair and apparatus, can't they still form superposition and hence in pure state?

 

[bhobba]

No. Entanglement is broken.

Thanks
Bill

 

[Edward Wij]

No. Entanglement is broken.

Thanks
Bill

I think entanglement of positions is broken. But if the EPR pair and apparatus were isolated in a box. They like form Hilbert space and there is aspect of it still in superposition (although not in the position basis).

 

[bhobba]

I think entanglement of positions is broken. But if the EPR pair and apparatus were isolated in a box. They like form Hilbert space and there is aspect of it still in superposition (although not in the position basis).

In a vector space like a Hilbert space any element can be broken down into the sum of other vectors in innumerable ways - as I explained right at the start. Of course it's in superposition. 9 = 5+ 4 = 8+1 = all sorts of other things. So?

Thanks
Bill

 

[Edward Wij]

In a vector space like a Hilbert space any element can be broken down into the sum of other vectors in innumerable ways - as I explained right at the start. Of course it's in superposition. 9 = 5+ 4 = 8+1 = all sorts of other things. So?

Thanks
Bill

So if the EPR pair and apparatus is still in superposition in some vectors.. and yet they are in mixed state as the particles entanglement broken. Then it means in a system where particle A, B, C are entangled, the A and B as subsystem can form mixed state (pure state can become mixed). Unless you are saying the EPR pair and apparatus can be in superposition but not entangled?

 

[bhobba]

So if the EPR pair and apparatus is still in superposition in some vectors.. and yet they are in mixed state as the particles entanglement broken. Then it means in a system where particle A, B, C are entangled,

It doesn't follow.

You need to learn what superposition and entanglement is.

Before proceeding can you define entanglement concisely?

Thanks
Bill

 

[Edward Wij]

It doesn't follow.

You need to learn what superposition and entanglement is.

Before proceeding can you define entanglement concisely?

Thanks
Bill

This is precisely why I wrote this thread.. trying to inquire how to differentiate (or mathematically specify) between superposition with and without entanglement. In message #1 I stated:

"What is the difference in the preparation of superposition for the 2 cases where:

1. The system is in either one state.. for example.. one electron in spin up or spin down. Do these interfere?
2. The system is in superposition of two states (for example 2 entangled electrons that is spin up and spin down).

Are they both called Superposition? What specific term distinguish them?"

In your first reply to it and succeeding. You didn't specify what distinguish superposition without and with entanglement. That's what I was asking. Again thanks in advanced.

 

[bhobba]

Are they both called Superposition? What specific term distinguish them?"

This is basic stuff.

I thought about a longish post defining this stuff, but really you need to go through it in a systematic way:
http://quantum.phys.cmu.edu/CQT/index.html

Thanks
Bill

 

[Edward Wij]

This is basic stuff.

I thought about a longish post defining this stuff, but really you need to go through it in a systematic way:
http://quantum.phys.cmu.edu/CQT/index.html

Thanks
Bill

Verbally The dead and alive cat is in superposition but not in entanglement.
But when you have two cats. You can entangle them.
So when you have 2 or more copies in the system, you can entangle them.
I think this is the big difference.
Ey. I just want to visualize the concept.

In decoherence, system and environment can be in superposition. But then system and environment are also entangled. Here I concluded the EPR pair can be entangled with the apparatus because system and environment can be entangled. I wonder what erroneous thinking have I made.

 

[bhobba]

Verbally The dead and alive cat is in superposition but not in entanglement.

The dead and alive cat are never in superposition. The superposition is before the particle detector in the nucleus that emits the particle.

Please read the link I gave.

I will be taking my leave of this thread - you need to become acquainted with basic stuff.

Thanks
Bill

 

[Edward Wij]

The dead and alive cat are never in superposition. The superposition is before the particle detector in the nucleus that emits the particle.

Please read the link I gave.

I will be taking my leave of this thread - you need to become acquainted with basic stuff.

Thanks
Bill

The link includes words like Linear Algebra in Dirac Notation. How do you expect us who know only how to add, subtract, multiply divide to comprehend them.
Hope others would give other easier reference with mostly verbal summaries..
Anyway. I think I understood the difference between superposition with and without entanglement.
Here's the summary:

1. Entanglement only works for subatomic particles in pure state, you can't entangle system already in mixed states.
2. Superposition is when the entanglement for instance takes all positions (say in position observable) .
3. The EPR pair can't be in entanglement with the apparatus because the apparatus is in mixed state.. which destroyed the EPR pair in pure state.

If the above are all correct. Then I understood it and end of thread. Hope someone can verify what I learned so far from the exchanges with Mr. Hobba. Thanks to him I grasped more.

 

[bhobba]

The link includes words like Linear Algebra in Dirac Notation. How do you expect us who know only how to add, subtract, multiply divide to comprehend them.

Some things can be explained in English, but physics is written in the language of math, and you soon run into where it must be used. You have ventured into an area that only can be explained in math. Or at least I can't explain it without math.

1. is basically correct. 2. is wrong and 3. is partially correct.

Thanks
Bill

 

[Nugatory]

The link includes words like Linear Algebra in Dirac Notation. How do you expect us who know only how to add, subtract, multiply divide to comprehend them.
Hope others would give other easier reference with mostly verbal summaries..

I understand why you want a mostly verbal summary, but you're posing questions that cannot be answered without more powerful tools than that.

You're understand adding, subtracting, multiplying, and dividing... But how would you explain these concepts to someone who wanted "a verbal summary" that didn't require knowing about numbers and counting?

 

[vanhees71]

There is no other way to understand quantum theory than to learn the appropriate language in which it can be formulated, and this is the language of (rigged) Hilbert spaces. It is impossible to understand quantum theory otherwise. The reason is simple: We are used to experience with macroscopic bodies, and these behave according to classical physics, which in fact is understood nowadays as an effective description following from the quantum behavior of interacting many-body systems.