Aspect/Innsbruck Interpretation which respects SR locality

[vanesch]

But let's be clear about what Bell tells us. Suppose we have two detectors A and B with experimenters who are free to choose the settings of the detectors. If Bell's inequalities are violated, then any model which agrees with the results must have something corresponding to information exchange between A and B.

Yes, and that something is the state of the observer which goes from A to B to check the correlations!
The trick is that for the observer at A (the one, for instance who will do the travelling), there IS NO DEFINITE RESULT at B as long as he didn't go there to check. It is when you insist on the definiteness of the remote result, before you can check it, that *something corresponding to information exchange* must travel FTL between A and B. But if all that is done at B remains in a superposition until A checks it (and it is only at that moment that A can verify the correlation), then it is whatever travels from A to B (or from B to A, or from A to X and B to X) that carries with it "the information needed", which corresponds to the "choice of the partial statevector" that has been made when A had to decide on which branch its sentient experience was now going to live (according to Born's rule).

When, however, you insist on the "reality" of the measurement at A and the "reality of the measurement" at B, and you insist on the fact that every correlation must have a causal origin, then yes, "something" must travel from A to B and from B to A forward and backward in time. But as that "reality" cannot be checked by a real transmission of information FTL, that "something" remains very vague! It is then left to the opinion of the interpreter to accept such a "something" which will never have any verifiable influence and call it collapse at a distance, or to accept that there is not such a "something" but that macroscopic objects, such as persons, can exist in superposed states.
My preference goes to the second possibility. Not because I find this exciting or so, but because it introduces the least new elements in the existing theory ; especially the necessary introduction of a "something" which communicates at a distance but in such a way that we will never be able to use it to communicate at a distance, relativity obliging, and which happens or not, according to whether a physical process is called a measurement or an interaction.


cheers,
Patrick.

 

[chronon]

Yes, and that something is the state of the observer which goes from A to B to check the correlations!
The trick is that for the observer at A (the one, for instance who will do the travelling), there IS NO DEFINITE RESULT at B as long as he didn't go there to check.

But what if there are two experimenters one at A and one at B? If they then get together to compare results, you seem to be claiming that the one from A could say to the one from B. "Yes, I realize that you claim to have made a definite measurement at B, but from my point of view that wasn't a real event. In fact you are just part of a superposition which only resolved itself when we met."

You say you think this involves the least new elements, but it still needs "something" which causes our minds to divide when faced with a superposition, which I find far less acceptable than a non-local influence.

 

[vanesch]

You say you think this involves the least new elements, but it still needs "something" which causes our minds to divide when faced with a superposition, which I find far less acceptable than a non-local influence.

This is indeed a matter of personal opinion, and you hit the nail on its head: I think that this is the true content of the physics of the Born rule and that you cannot leave it out and derive it from strictly unitary QM. So you can say that unitary QM is the physics of the universe, and the Born rule is the physics of the mind. It is just a personal, maybe strange, but - I think - coherent viewpoint. But it has one practical consequence, or almost so: if you take that viewpoint, you will not bother trying to make a FTL telephone with entangled states, trying to use "collapse at a distance" in order to force somehow what is proven not to be able to exist in QM

Now if you *really* want my personal opinion, I don't take my opinion very seriously, but take it just as a means to reason in QM, as it stands. It avoids me a lot of traps which are often invited by the smalltalk that goes into EPR and quantum erasure papers, and which come down (IMHO) to applying the Born postulate too early.
One thing is sure in QM: when you apply the Born postulate only COMPLETELY AT THE END of your calculation, for the quantity that you want to plot in your paper, you NEVER make an error. If you apply it earlier, you can get away with it if you are very careful. If you apply it too early, you have an FTL telephone and you made an error against QM theory.

I am completely open on the seriousness with which you have to take all this. If indeed, say, gravity DOES somehow induce the collapse in QM, then this changes completely the picture. But I wasn't talking about an interpretion of _that_ hypothetical theory of which I'm even not aware of the formalism. I'm talking about a way of seeing _current_ QM theory. What *really* happens, I just don't know, and I think that everybody who claims to know is fooling him/herself.


cheers,
Patrick.

 

[vanesch]

But what if there are two experimenters one at A and one at B? If they then get together to compare results, you seem to be claiming that the one from A could say to the one from B. "Yes, I realize that you claim to have made a definite measurement at B, but from my point of view that wasn't a real event. In fact you are just part of a superposition which only resolved itself when we met."

Exactly ! As seen from the "I" experience of A.
The "I" experience seems to require that it can only observe a product state of the self and the rest of the universe ; call it the "self-awareness" or whatever. It seems not to support entanglement with the universe. So that "I" experience HAS to choose when it undergoes a physical interaction which entangles its physical structure with something else, and that choice is dictated by the Born rule.

So if A and B are two friends, and they get apart, do an EPR experiment and then come together, there's a possibility that the "I" experience of A now talks to the CLONE of B, while B's "I" experience is now in another branch. There's no way for A to find out that he's now separated from his friend forever, and has to deal with a clone

Try to explain that to your wife/girlfriend when cheating on her

cheers,
Patrick.

 

[chronon]

I think you can put it this way:

If the Bell inequalities are still violated when you have two experimenters choosing the detector settings at spacelike separation,
then either
(1) Your model must have something corresponding to FTL communication
or
(2) Your model must include the minds of the experimenters

I agree that this is a matter of personal opinion, but I can think of several arguments against (2)

(a) You are trying to model a system consisting of lasers and detectors. Why should you need to bring in the unrelated subject of mental workings.

(b) It moves towards non-falsifiability. If results disagree with your model you can keep the model, just change what the experimenters minds end up believing.

(c) It can be seen as a delaying tactic. By introducing mental workings, which we are unlikely to understand for several decades, it means we can go on using hand-waving arguments.

you will not bother trying to make a FTL telephone with entangled states, trying to use "collapse at a distance" in order to force somehow what is proven not to be able to exist in QM

I think people will always try to use the results to make an FTL telephone. In case (1) you can think that maybe you can utilise the underlying FTL information transfer, but in case (2) you can think that maybe the recipient gets a random signal but you can arrange things so that you end up in the universe when they get the message you sent.

 

[vanesch]

I think you can put it this way:

If the Bell inequalities are still violated when you have two experimenters choosing the detector settings at spacelike separation,
then either
(1) Your model must have something corresponding to FTL communication
or
(2) Your model must include the minds of the experimenters

I would agree with you if these two options were all there was to it. But it is not!

In (1), you still must explain me why the photodetector cannot be described by the unitary evolution of the schroedinger equation describing the photo detector processes. If the essential process is photo-emission (in a photomultiplier), then we perfectly know how that works, through unitary evolution.
So the "measurement problem" still stands unsolved: what physical processes are "measurements" (and don't follow the Schroedinger equation, but follow the Born rule and the projection postulate), and what physical processes are "interactions" following unitary evolution ? When is the emission of an electron by an impinging photon a measurement (and hence will collapse a wave function through a yet unknown FTL process), and when is it a physical process that could still in principle be used to do further QM with ?
And on top of that you have to invent an FTL transmission in the past in such a way that you cannot use it.

In (2), you can say that you know that already: it is the Born rule.
So in (2), you essentially solve the measurement process issue and you do not need to invent an FTL communication.

So the options are:

(1) Your model must have something corresponding to FTL communication AND you must STILL explain in what a measurement is physically different from an interaction.

(2) Your model of the minds of the experimenters is given by the Born rule.

I think that (2) is much closer to the actual formalism than (1). In fact, (1) expects a NEW theory, with new physics in it. Indeed, whenever the exact physical process responsable for the distinction between a measurement and an interaction is found, it will be possible to determine that experimentally (even if our current technology is maybe not yet up to it), because it will be IMPOSSIBLE in principle to obtain superpositions in that case. An application of the Born rule implies that you fix the basis in which you "measure", while unitary evolution let's you free to work in any basis.

I had a similar discussion in another thread here. Look at your photodetector, say, a PM. You will probably agree with me that it is the electron emission from the photocathode that is the "measurement process". All the rest is amplification.
So, you say, when a photon impinges on a photocathode, there is a relatively high probability that an electron is emitted. But what, in this process, is not unitary ? In what "photon basis" do we now apply the Born rule ? I think it is quite obvious that it is the standard "photon" basis of Fock space (there is no photon, or there is 1 photon, or there are 2 photons...). Does this then mean that in all interactions of light with a metal, we have to work in that basis and apply the Born rule ?
Hell no ! If that were the case, a metal surface wouldn't work as a mirror ! Indeed, to do so, you need to have a coherent light state (a superposition of Fock states) interact with the sea of electrons, in order for them to emit another coherent state which is the reflected beam. So now suddenly, the preferred basis is the basis of coherent states ?
You will then say: no, it is when a photon is "absorbed" that you apply the Born rule in the Fock basis. But isn't the mirror action an absorption and coherent re-emission of the coherent states by the sea of electrons then ?
Ah, you will say: it is when ENERGY is transferred between the EM field and the electrons that you have to apply the Born rule. But (ok, I failed to come up with the correct complete calculation) if that were true, stimulated emission couldn't amplify coherent states in a laser then !

And this is the case each time when you analyse a "measurement device". Each time you think you've found the pivotal process that "does the measurement" you can find situations where very similar interactions are necessarily described by unitary processes and superpositions have to remain so in order to be correct. So why, in some cases, do these processes "collapse" the wavefunction and send out their FTL signals, and not in other cases ?

So maybe there ARE indeed physical processes that collapse the wavefunction, and maybe there ARE then FTL messages sent out. But you agree with me that that is a whole lot of new physics to be added, so we're not talking about an *interpretation* of QM anymore. It is only in such a setting that (1) makes sense.

cheers,
Patrick.

 

[gptejms]

Have logged in after a couple of days and need to catch up with what has been going on.Firstly what is FTL?Also let me ask what is IMHO---seen this quite commonly used in physicsforums.
Coming to the discussion between vanesch and chronon,there was a thread 'Does decoherence solve the measurent problem?' in s.p.r. which is relevant to the present discussion.Decoherence as you know knocks off the off-diagonal elements of the density matrix.So you are left with the diagonal elements each with its own probability.This probability I think is like a classical probability--for a two state system,you'll find some systems in state |1> and some in state |2>.True you have only one system to make measurement on,but this does not imply that the system continues to be in a superposition of the two states.
Besides,in 'most' decoherence situations,the coherent superposition is short lived because of dissipation and the system mostly ends up in the state of lower energy.So does decoherence not solve the measurement problem?

 

[vanesch]

True you have only one system to make measurement on,but this does not imply that the system continues to be in a superposition of the two states.
Besides,in 'most' decoherence situations,the coherent superposition is short lived because of dissipation and the system mostly ends up in the state of lower energy.So does decoherence not solve the measurement problem?

FTL = Faster Than Light
IMHO = In My Humble Opinion

Concerning decoherence, it solves *part* of the measurement problem, namely the "preferred basis" problem. But it doesn't solve the "projection postulate" problem, however, it gives an interesting property about it.
Note also that in order to apply decoherence theory, you place yourself already in a MWI like situation, otherwise it has no meaning !

In order to apply decoherence theory, you assume that unitary quantum mechanics is strictly correct upto the macroscopic level, so that your macroscopic measurement instruments become entangled with the states of the microsystem under study.

Typically, you consider the begin situation:

System : (a |s1> + b |s2> + c |s3> )

Measurement: |m0> (ignorance state)

Environment: |e0> (not yet interacted).

The physics of the system and the measurement apparatus is then such that the Hamiltonian of it leads to an entanglement:

(a |s1> |m1> + b |s2> |m2> + c |s3> |m3>)

Here, the "m" states are the so-called pointer states of the measurement device.
They are supposed to indicate macroscopically what is the value of the measurement under study. So for example, it could be a spin-z measurement on the system.
But you see the arbitrariness of the procedure: If I would have written my original system state in another basis, which is a linear combination of |s1> |s2> and |s3> then I would find "pointer states" that are linear combinations of m1, m2 and m3, and my "z-spin" measurement apparatus would work just as well as a "y-spin" measurement apparatus.

It is here that decoherence theory comes to rescue, by showing that, when the measurement apparatus interacts with the environment, there will be only one way to write this:
|m1> |e1> + |m2> |e2> + |m3> |e3>
in a way that is stable against remixture (so that the Hamiltonian of the interaction m-e takes on a block-diagonal form in the m-e basis).

As such, the m-basis is NOT arbitrary anymore, but is determined by the Hamiltonian of the interaction measurement system - environment (and corresponds to what we call "classical states").
Once this is fixed, we also see that our trio s-m-e now takes on the form:

a |s1>|m1>|e1> + b |s2> |m2> |e2> + c |s3> |m3> |e3>

This is to where decoherence theory brings you: that the interaction between a macroscopic measurement system and the environment leads to a unitary evolution of the system which can only be written in one way.

But you now still have to apply the Born rule in order to say:

with probability |a|^2 I measured s1 (through the pointer state m1), and now my state is to be considered to be |s1> |m1> |e1>

This last part is still a mystery which is NOT explained by any physical interaction. In fact, it cannot, because all physical interactions are described by unitary transformations which are linear, so you can never pick out one component of a superposition that way.

But decoherence theory is important because it tells you that, IF YOU ARE GOING TO APPLY THE BORN RULE at the end of your calculation, you can just as well apply it in the pointer basis from the moment you will seriously interact with the environment (where the pointer basis comes down to the states that are stable against further mixture with the environment). So it allows you the mathematical shortcut which is always applied: "and we measure the position of the electron |psi(x)|^2 ... " without having to plunge into the details of your measurement apparatus and all that.
So that's why people say that decoherence theory solves the measurement problem FAPP (for all practical purposes). But it doesn't solve it in principle, because it USES it at the end of the calculation.

cheers,
Patrick.

 

[ppnl2]

Pardon a quick question.

Hans talks about a pair of photons that are phase entangled at 90 degrees and polarized at 45 degrees. Is that possible?

Usually when you talk about entangled photons they are randomly polarized. How can you know they are polarized at 45 degree and also entangled at 90 degrees?

I don't think it affects what he said much. Its just that something about the way he said it seemed wrong. But I get confused easily.

 

[chronon]

In (1), you still must explain me why the photodetector cannot be described by the unitary evolution of the schroedinger equation describing the photo detector processes.

I agree that in (1) we are still left with the (local) measurement problem. However, I think that the real mystery in quantum theory is nonlocality, and that the measurement problem is minor in comparison. Several possibilities have been proposed, and the main reason they are rejected seems to be that they don't also deal with nonlocality. (Actually, the real measurement problem I see is that people have convinced themselves that they can calculate much more than is in fact the case; see http://www.chronon.org/Articles/shut_up_and_calculate.html)

So in (1) we still have some work to do, but I would maintain that this is also the case with (2), as you then have to include the behaviour of the human mind in your theory - not a trivial matter. Even if you have a simplified model of minds splitting, you have to explain why they split in the way they do, essentially giving the same problem as in (1).

 

[gptejms]

Vanesch,let's take a specific example e.g. the double slit experiment with electrons.Before I make any measurement,the electron is in state a|1> + b|2> and there are off-diagonal elements a^*b and ab^*.Now I make a measurement(with a gamma ray microscope) at slit 1 and find that the electron passes thru it.Let me denote my state by |m>---since this is a macroscopic state I don't expect it to change significantlyby detection of an electron(may be some microscopic changes take place which have recorded the fact that the electron has passed).Upon my measurement,the electron decoheres(due to interaction with the gamma photon) from the state a|1>+b|2> to |1> and a becomes equal to 1 and b=0.Not only the off-diagonal elements go to zero,but also only one of the diagonal elements survives.What is your environment in this case--why do you need it all?

What is your version of the above experiment?

 

[vanesch]

I agree that in (1) we are still left with the (local) measurement problem. However, I think that the real mystery in quantum theory is nonlocality, and that the measurement problem is minor in comparison. Several possibilities have been proposed, and the main reason they are rejected seems to be that they don't also deal with nonlocality.

Well, it is hard to accept non-locality which implies non-causality through relativity, especially when it is not strictly necessary (after all, in the QM formalism, there is no real FTL communication (information transmission allowed). But I still claim that the really big problem in QM is the measurement problem. After all, it is not that because you cannot explicitly, without approximations, CALCULATE several measurement processes, that you cannot know that they must be unitary. If they have a Hamiltonian description, then they ARE unitary.

So in (1) we still have some work to do, but I would maintain that this is also the case with (2), as you then have to include the behaviour of the human mind in your theory - not a trivial matter. Even if you have a simplified model of minds splitting, you have to explain why they split in the way they do, essentially giving the same problem as in (1).

No, I really don't, if I say that the essence of their behaviour is given by the Born rule. I don't have to say WHY it follows the Born rule, it is a fundamental postulate, just as the unitary evolution is a fundamental postulate.

Fundamental postulate I: "The universe evolves according to the Schroedinger equation" i hbar d/dt psi = H psi

Fundamental postulate II: "a sentient being gets its subjective experiences of its interactions with the universe through random assignment to one term in the superposition according to the Born rule". Decoherence insures me that this basis is well-defined.

As such, I didn't have to touch at all at the formalism of QM. I just fixed *where* the Born rule had to be applied, and because it doesn't correspond to a physical process but a subjective mental one, I don't have the difficulty of having at the same time a unitary description of it, as would be the problem if ever I fixed the Born rule application earlier in the measurement chain.

I really think it is the minimally invading interpretation in the formalism of QM.
Moreover, I get as a bonus that I do not need any extra non-local stuff.

cheers,
Patrick.

 

[vanesch]

Upon my measurement,the electron decoheres(due to interaction with the gamma photon) from the state a|1>+b|2> to |1> and a becomes equal to 1 and b=0.

This is not a unitary process. After all, a unitary operator is linear:

U (a |1> + b|2> ) = a U |1> + b U |2>

So there's no way to change these a and b through unitary processes. THIS is the central problem of the measurement process. You can make your physical process as complicated as you want, you cannot get away with the fact that the time evolution operator will be a linear operator on the states, and that, by definition, superpositions survive the application of U.

cheers,
Patrick.

 

[gptejms]

I expected you to tell what the environment is,and to give a more detailed explanation where you would include the states of the observer |m1>,|m2> etc.(and possibly also of the environment).
Anyway,coming to your objection.For an atom radiating, the state is a(t)|e> + b(t)|g> and a(t) goes from 1 to 0 and b(t) from 0 to 1---isn't this a unitary process?
Coming back to my double slit experiment,upto what state does decoherence take the initial superposition a|1>+b|2> to?

 

[vanesch]

I expected you to tell what the environment is,and to give a more detailed explanation where you would include the states of the observer |m1>,|m2> etc.(and possibly also of the environment).
Anyway,coming to your objection.For an atom radiating, the state is a(t)|e> + b(t)|g> and a(t) goes from 1 to 0 and b(t) from 0 to 1---isn't this a unitary process?
Coming back to my double slit experiment,upto what state does decoherence take the initial superposition a|1>+b|2> to?

I'm a bit confused by what you say ; so let's first get tuned :-)
In your previous message, you wrote things I don't understand:

Before I make any measurement,the electron is in state a|1> + b|2> and there are off-diagonal elements a^*b and ab^*.Now I make a measurement(with a gamma ray microscope) at slit 1 and find that the electron passes thru it.Let me denote my state by |m>---since this is a macroscopic state I don't expect it to change significantlyby detection of an electron(may be some microscopic changes take place which have recorded the fact that the electron has passed).

I don't know what you mean with this off-diagonal elements a^*b ... ?

If you "make a measurement and find" you leave already the superposition. Normally, you would say that your gamma ray microscope interacts with your electron, and would get into the state:

a |1> |gammamicroscope_saw_electron> + b |2> |gammamicroscope_didnt_see electron>

Also, your *macroscopic state* after having observed the display of the gamma microscope, is significantly different according to whether you saw or didn't see it. In fact, chances are these are orthogonal states.
Indeed, if you write your state as:
|stateofmyfirstproton>|stateofmysecondproton>...|stateofmylastneutron>
it is sufficient for ONE SINGLE PROTON to be at a slightly different place between two possibilities for your entire states to be orthogonal to each other.

So the end state is:
a |1> |gammamicroscope_saw_electron>|yousawdisplayred> + b |2> |gammamicroscope_didnt_see electron> |yousawdisplaygreen>

cheers,
patrick.

 

[gptejms]

I'm a bit confused by what you say ; so let's first get tuned :-)
In your previous message, you wrote things I don't understand:



I don't know what you mean with this off-diagonal elements a^*b ... ?

|\psi> = a|0> + b|1>

density matrix \rho is
\rho = |\psi&gt;&lt;\psi|,<br /> so &lt;0|\rho|1&gt; = ab^* and &lt;1|\rho|0&gt; = a^*b
These off-diagonal elements go to zero by the act of measurement.Decoherence also leads to decay of the off-diagonal elements--i.e. why it's said to provide a solution to the measurement problem.

So the end state is:
a |1> |gammamicroscope_saw_electron>|yousawdisplayred> + b |2> |gammamicroscope_didnt_see electron> |yousawdisplaygreen>

So do you mean that the off-diagonal elements survive as such in this mega-superposition?

 

[vanesch]

|\psi&gt; = a|0&gt; + b|1&gt;

density matrix \rho is
\rho = |\psi&gt;&lt;\psi|,<br /> so &lt;0|\rho|1&gt; = ab^* and &lt;1|\rho|0&gt; = a^*b
These off-diagonal elements go to zero by the act of measurement.Decoherence also leads to decay of the off-diagonal elements--i.e. why it's said to provide a solution to the measurement problem.

Ah, ok, it was about the components of the density matrix associated with this statevector.

So do you mean that the off-diagonal elements survive as such in this mega-superposition?

Yes, of course they survive, _in the densitymatrix of the entire system_, including the environment. They have to, by unitarity.
What happens is that when you now calculate the LOCAL DENSITY MATRIX, limited to the system, by taking the partial traces, in this LOCAL density matrix the off-diagonal elements become zero.

Let us limit the description to:

a |1> |gammamicroscope_saw_electron>|yousawdisplayred> + b |2> |gammamicroscope_didnt_see electron> |yousawdisplaygreen>, and let us include the gammamicroscope state in the "you saw" state, to simplify notation:

a |1> |yousawelectron> + b |2> |youdidntsee>

The overall densitymatrix, in the basis:

|1>|yousaw> , |1>|youdidntsee>, |1>|yourotherstates...> ...
|2> |yousaw> , |2>|youdidntsee>,|2>|yourotherstates...> ...

takes on the form of 4 blocs:


rho_11 rho_12
rho_21 rho_22

with rho_11 the coefficients of |1>|you..><1|<you...| in |state><state| ;
rho_12 the coefficients of |2>|you...><1|<you...|
etc...

The coefficient a b* appears off-diagonal in rho_21, in the term:
|1>|yousaw><2|<youdidntsee|

The coefficient a^2 appears in rho_11 on the diagonal:
|1>|yousaw><1|<yousaw|

etc...

To get back to the local density matrix, we have to take the traces of these 4 component matrices (that's what partial tracing out means).

So you see that the trace of rho_11 will essentially be a^2,
that the trace of rho_12 and rho_21 will be 0 (because a b* appears off-diagonal) and that the trace of rho_22 will essentially be b^2.

Of course, the evolution of the states |you...> will make the off-diagonal components wiggle, but if the |you...> space is big enough, they will never gain significant components on the diagonal.

So you see that after tracing out, the LOCAL density matrix is reduced to:

|a|^2 0

0 |b|^2

If the entanglement is perfect, as the state describes. But this is a PARTIALLY TRACED OUT density matrix, and in order for this to be interpreted as probabilities, you in fact USE already the Born rule, saying that you have summed over the probabilities of all the potential exclusive cases of the environment. That's why this local density matrix is called an "improper mixture", because it behaves as a statistical mixture only if:
- we limit ourselves to the local observables
- we have assumed the Born rule for the total system

This is decoherence in a nutshell...

cheers,
Patrick.

 

[vanesch]

I would like to add something, concerning this partial tracing out. Imagine I have 2 systems, one with 2 states |1> and |2> and another one with 3 states, |a>,|b> and |c>.
We construct the tensor basis:

{|1>|a>, |1>|b>,|1>|c>,|2>|a>,|2>|b>,|2>|c>}

Let us assume that we have a pure, but entangled state |psi>, written in this basis, and given by the 6-tupel {u_1a, u_1b...,u_2c} (with u_xx complex numbers, and normalized).

Imagine now that we have an observable which only observes something on system 1. This means that it can be written as: O x 1 (tensor product of operators), and let us imagine that O has as eigenstates |1> and |2>, with eigenvalues o1 and o2. This means that the eigenstates of Ox1 are:
|1>|a>, |1>|b> and |1>|c> with eigenvalue o1
and
|2>|a>, |2>|b> and |2> |c> with eigenvalue o2

The probability of having eigenvalue o1 is the sum of the probabilities of having |1>|a>, |1>|b> and |1>|c> , so this will be |u_1a|^2 + |u_1b|^2 + |u_1c|^2.

And that is nothing else but the trace of the 1-1 block in the overall density matrix |psi><psi| as you can easily verify.
What is very important is that a trace is invariant under a change of basis. So if we would have taken another basis for the H2 system we would find exactly the same trace of the 1-1 block. And this is the proof that the measurement on system 2 (choosing another basis for the second system) has no influence on the local measurement O.

But you also see that in order to give a meaning to this partial trace, we had to apply the Born rule on the 6-state space H1 x H2 ; once these were probabilities, we could then sum them.

The off-diagonal elements in the local density matrix play a role when we have a local observable O which doesn't diagonalize in the |1>, |2> basis. You can work the algebra out if you want to, it is a bit tedious.

cheers,
Patrick.

 

[gptejms]

Excellent posts---cheers Patrick!
So is this your conclusion:-decoherence 'theory'(I see some people call it a theory) assumes Born's rule in its derivation,so it really does not explain much.Use Born's rule to get at a local/reduced density matrix that does not have off-diagonal elements;then say since you now have only a statistical mixture you have solved the measurement problem--the argument is flawed.Is this what you are saying?
Because of interactions with measuring device/environment,the phase information gets dispersed,but is never lost.Superpositions stay---so MWI kind of thing is needed(?).But my problem is:-once you have included the measuring device as well as the environment(which includes one who wrote down the wavefunction,plus everyone else) into your wavefunction,who is left to make a measurement?

 

[vanesch]

Excellent posts---cheers Patrick!

Thanks

So is this your conclusion:-decoherence theory(I see some people call it that way) assumes Born's rule in its derivation,so it really does not explain much.Use Born's rule to get at a local/reduced density matrix that does not have off-diagonal elements;then say since you now have only a statistical mixture you have solved the measurement problem--the argument is flawed.Is this what you are saying?

No, decoherence people, like Zeh, realize this and say this also. Decoherence DOES show us something, namely the "preferred basis", the one in which the product states remain product states that way under time evolution ; this is determined by the character of the interaction between the system and the environment, and always leads to a basis of states which "looks classical" (like position states for particles ; or coherent field states for EM fields). THAT is the real contribution of decoherence.
It allows you to make the shortcut of applying the Born rule on the system level instead of having to work out the complicated QM of the interaction with the measurement instrument.
But, as you say, considering that it solves the measurement problem is based upon circular reasoning.

Because of interactions with measuring device/environment,the phase information gets dispersed,but is never lost.Superpositions stay---so MWI kind of thing is needed(?).But my problem is:-once you have included the measuring device as well as the environment into your wavefunction,who is left to make a measurement,collapse the wavefunction?

Hehe, you're beginning to see the issue ! Who's left ? My way to solve it is:
my consciousness is left :-) But you got to the gist of the problem I think.

cheers,
Patrick.

 

[gptejms]

Put your consciousness also into the wavefunction--what's left now?In fact when you put yourself into the wavefunction,you can't ask your consciousness to stay away!
I realize one can go on and on discussing this issue but never reach a conclusive conclusion.

 

[vanesch]

Put your consciousness also into the wavefunction--what's left now?In fact when you put yourself into the wavefunction,you can't ask your consciousness to stay away!

Well, first I would like to point out that the idea is to find an interpretation that goes with a formalism, so I'm not saying here "what is real", I'm saying only what is a possibly consistent "story" that goes with the usual rules of QM, *if you insist on having such a story*. You can also say that QM is just a theory which describes statistical properties of observations without any interpretation of "what happens physically". I don't really like that approach - although it is a logically possible one - because when you are modelling you QM calculations and so on, you like to think of "physical things out there" you're dealing with, and not some abstract "statistics generator".

When you say "you put yourself in the wavefunction", I can argue that I put my body in the wavefunction, with my brain and everything. But if you consider that a consciousness is a non-material item, then it has no physical description consisting of particles or fields, but is only _associated with a physical system_. And I simply say that *that* non-material item, call it "mind" or "soul" or "consciousness" or whatever, is NOT described by the unitary evolution that describes all matter in the universe, but subscribes to the OTHER postulate of QM, namely the Born rule.
You can write a hamiltonian of your brain, but you cannot write a hamiltonian of your consciousness.

I realize one can go on and on discussing this issue but never reach a conclusive conclusion.

I know, that's the danger. But the nice thing of this approach is that
1) it doesn't contain any inconsistencies, as does the "measurement" versus "physical process" in the "standard" Copenhagen description (namely, why should a physical process sometimes apply the "measurement" postulate, and sometimes the "unitary" postulate ?)
2) you don't need to change anything to the ordinary rules of QM, in that you still have the Born rule, you still have unitary evolution for all physical processes. True Everettians try to _deduce_ the Born rule from the unitary part, but they always have to introduce extra assumptions. And the Copenhagen view should add a theory on why certain physical processes "collapse" the wavefunction, and don't simply work with their hamiltonian.

Ok, the price to pay is that we have to consider now that there is some special physics associated with "mind" and so on, and that we have to consider that what our mind observes is only part of what is "reality" and not all of it.

But there's an extra bonus which I also like: in a MWI like scenario as the one I present, there is no "spooky action at a distance" in the real world, it is just something that is induced by the part of reality which is observed by your mind, and as your mind is attached to a physical structure (say, your brain), it cannot move FTL. As such, there's no reason for any underlying non-locality in physics.

So again, the point is not that I want to talk about what exactly is consciousness or so. It is that it is a story that goes with the standard QM formalism, without any additions, and which permits me to "demystify" a lot of quantum fuzz, such as EPR situations, quantum erasers and so on.
I don't know of any OTHER way of seeing QM where you 1) are not somehow introducing "extra unknown physics you will have to work out later" or 2) some arbitrariness in the concepts you introduce (such as "measurement").

cheers,
Patrick.

 

[caribou]

You can also say that QM is just a theory which describes statistical properties of observations without any interpretation of "what happens physically". I don't really like that approach - although it is a logically possible one - because when you are modelling you QM calculations and so on, you like to think of "physical things out there" you're dealing with, and not some abstract "statistics generator".

As I've been learning a little about the decoherent histories approach to quantum theory, I've been reading that the finest-grained description of a series events in quantum theory almost always has interference terms between all the different possible histories. As a result, I'm not entirely sure what the "physical things out there" actually are anymore or even in the theory.

It seems almost everything is doing the equivalent of the "going through both slits at the same time" trick of the two slit experiment... but a whole lot worse.

 

[gptejms]

When you say "you put yourself in the wavefunction", I can argue that I put my body in the wavefunction, with my brain and everything. But if you consider that a consciousness is a non-material item, then it has no physical description consisting of particles or fields, but is only _associated with a physical system_. And I simply say that *that* non-material item, call it "mind" or "soul" or "consciousness" or whatever, is NOT described by the unitary evolution that describes all matter in the universe, but subscribes to the OTHER postulate of QM, namely the Born rule.

I am very skeptical about this consciousness thing,but I don't want to enter into a discussion on it right now.I repeat my earlier question which went unanswered.An atom goes from an excited state |e> to the ground state |g>---its state while in transit is a(t)|e> + b(t) |g>,where a(0)=1,b(0)=0.So the atom goes into a superposition(from the initial state |e>) and then leaves the superposition---all via a unitary transformation.So your argument(earlier post) that a unitary transformation can't break a superposition(which though looks reasonable) doesen't seem to hold---kindly comment.

 

[vanesch]

I am very skeptical about this consciousness thing,but I don't want to enter into a discussion on it right now.

I can understand that. It isn't really essential in the discussion, in fact. Replace it by the "observing experience" or something of the kind. Indeed, in a relative-state view, there are no objective observations, and what is experienced as such depends on what entity is supposed to experience the observation. In the rawest state, it is a kind of memory of experienced events.

I repeat my earlier question which went unanswered.An atom goes from an excited state |e> to the ground state |g>---its state while in transit is a(t)|e> + b(t) |g>,where a(0)=1,b(0)=0.So the atom goes into a superposition(from the initial state |e>) and then leaves the superposition---all via a unitary transformation.So your argument(earlier post) that a unitary transformation can't break a superposition(which though looks reasonable) doesen't seem to hold---kindly comment.

That is because you view the atom as a system, while it is interacting with something else. Indeed, you cannot have that an atom in isolation goes from an excited state to a ground state. By definition, a stationary state (such as an excited state) remains forever in that state.
You get transitions because of the coupling to the EM quantum field, and what YOU are describing looks like the local density matrix description, limited to the atom, of the bigger system "atom + EM".

So, "in the beginning" you need at least a photon in your EM field, and an atom in the ground state. What I will write here is "in principle". Nobody (knows how to) do that this way, and uses approximations to do the calculations.

|start> = |1 photon> |g>

Note that this is not a stationary state of the coupled system, otherwise there would be no interaction, and hence no excitation or decay.

So the above state must be rewritten in stationary states which are entanglements between the atom and the field (because you have to diagonalize your total hamiltonian: the |start> state is not an eigenstate of the total hamiltonian of atom + EM + interaction).

So we rewrite this as:

|start> = A |energy 1> + B |energy 2> + C |energy 3> + ...

Where |energy 1> is an entangled state of all the |n photon> states and all the ground and excited states of the atom, and |energy 2> also (but in different proportions)

A = <start| energy 1> ; B = <start | energy 2> ...

This can then evolve into:

|intermediate t> = A exp (i t E1) |energy 1> + B exp(i t E2) |energy 2> ...

and it is NOT (as you seem to suggest) necessarily factorizable in a product of an atom state (such as u |g> + v |e>) and an EM field state. I'm pretty sure you remain in a kind of entangled state, but if you wait long enough, and if the excited state is "rather" stable, you might hope that this evolves into something close to:

|intermediate 2> ~ |0 photon> |e> + small terms.

Now the atom is "excited".

The desexcitation will then be simply a further evolution of the state:
A exp (i t E1) |energy 1> + B exp(i t E2) |energy 2> ...

and for very long times, you will end up again in a state that is close to:

|1 photon> |g>

But note that the whole time, we've never left the superposition:

A exp (i t E1) |energy 1> + B exp(i t E2) |energy 2> ...


If I can give you a rough analogy: if you consider a pulse, which is Fourier transformed, it looks a bit as if you're saying: hey, a pulse cannot be composed of sine waves, because a long while before it, I have no signal, so where are your sine waves, and a long while after the pulse, the same. Nevertheless, during my pulse, I can understand that you build it up with sine waves. Well, here it is the same. The sine waves are the stationary states of the overall system, and their composition, all along time, remain the same. Your "initial, intermediate and final" views are particular stretches of your function in the time domain. It can be flat (state |g> or |e>) or a pulse ("superposition a|g> + b|e>"). But all this is just a result of the different phase factors in front of the stationary states of the overall system.

cheers,
Patrick.

 

[gptejms]

Again wonderfully put and very convincing.Can you do an actual calculation for the full system evolution assuming some model (bath of harmonic oscillators coupled to a particle) and show that the wavefunction for the total system actually oscillates in time between different particle states?Even if you can't, nevermind---it looks like the only reasonable thing that can happen.

For the reduced description,do we conclude that the evolution is non-unitary?

 

[gptejms]

<br /> Let me try to answer my own question:-is the reduced description/evolution unitary?The reduced density matrix description is equivalent to the following quantum Langevin equation:-<br /> \begin{equation}<br /> \frac{dp}{dt} = -V^{\prime}(x) + F(t) - \gamma p,<br /> \end{equation}<br /> <br /> where p is the momentum of the particle in potential V(x) coupled to a bath of harmonic oscillators whose fluctuations in the equilibrium state(i.e. when the bath was decoupled from the particle) are encoded in F(t), and the last term gives the average friction the bath exerts on the particle(assuming the memory function to be a delta function).<br /> If F(t) were zero then the commutator [x,p] would decay off in the manner<br /> \begin{equation}<br /> [ x,p ] = \iota \hbar exp(-\gamma t),<br /> \end{equation}<br />

once the coupling between the particle and bath were switched on.So this evolution would be non-unitary.But the term F(t) ensures that the overall evolution is unitary.

In the case of an atom interacting with e.m. field,the constant friction term is its interaction with vacuum(spontaneous decay).F(t), I guess, should be fluctuations of the e.m. field over and above the vacuum.