Observing the observer

A. Neumaier

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What equation governs systems that are among an environment?
Systems in interaction with an environment are either modelled by the Schroedinger equation for the combination system+environment, or by a so-called Lindblad equation, http://en.wikipedia.org/wiki/Lindblad_equation, which eliminates the environment in favor of a (slightly approximate) non-unitary evolution.

For example, realistic quantum optics account for dissipation to the environment
(e.g., energy losses due to imperfections of the experimental set-up) by using Lindblad equations.
 

A. Neumaier

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Do the solutions to the Schroedinger equation change AFTER a measurement has taken place?
I say this because in the book 'Entanglement' the author says that potentialities still exist even if there is actuality. I think a good way to put it is the solution to the equation still exists (i.e. superposition of states), but we are only aware of one of those states [no collapse is postulated]. For can we not find a solution to the Schroedinger equation for anytime in the future, where at that point we can see a definite state?
It is difficult to interpret your question.

A system is at all times in a well-defined state (pure or mixed). ''potential'' are only measurement results - namely before an actual measurement is done.

The Schroedinger equation applies only when the system is fully isolated (and hence unobserved).
 

A. Neumaier

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What if my quantum computer loses coherence, so that the information in qubit on the target line "escapes" into the environment. Does it count as a measurement then?
In order to qualify as a quantum measurement of a system X, the experimental setup must contain a detector describable by nonzero operators M_1,...,M_n (n>1) satisfying
[tex] \sum_i M_i^*M_i = 1,[/tex]
leading to n distinguishable measurement results, such that the measurement transforms a pure state psi of X into the state
[tex] \psi'=M_i \psi/||M_i \psi_i||[/tex]
when result number i is observed and the system X still exists after the measurement.
(Measurements where the state can disappear must be described in a bigger state space describing the system X together with an empty system - called in this context a vacuum state.)

Corresponding to the detector is a POVM http://en.wikipedia.org/wiki/POVM with the operators F_i=M_i^*M_i (in the notation of this reference).

As long as nothing is observed, there is no measurement result, and hence no measurement took place. Observation here means a macroscopic irreversible change, no matter whether in a detector, a piece of equipment recording a result, or in a human brain.
 

Hurkyl

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I guess, but you pretty much just invented a new word to describe what you're talking about, so the fact that I don't disapprove doesn't really tell you anything.
I was trying to be specific in how what I'm talking about differs from what you're talking about. The gist I get is that the only real difference between the CNOT gate and what you're willing to call a measuring device is that the measuring device is too unwieldy to have complete control over or to analyze in complete detail. (both meant jointly with the system being measured)

That difference is something I find rather unimportant -- and it also means that quasi-measurements are good models for real measurements, because we can analyze the quasi-measurement in complete detail, and then work out what would happen if we weren't in control of some aspect of it.
 

Fredrik

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The difference is extremely important, because measurements can test the accuracy of QM, and possibly falsify it. Quasi-measurements can't (unless they're part of a sequence of events that ends with an actual measurement).
 

Hurkyl

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Thus a CNOT gate is not a measuring device but a reversible filter. It is represented by a unitary transformation

...

On the other hand, perfect (projective) measurement devices are represented by irreversible filters orthogonal projectors that are not unitary.
I would like to point out that the interaction of CNOT gate is only represented by a unitary transformation if you consider the joint (control, target) system. On the (control) system, the interaction truly is the non-unitary projection that turns the state represented by the ket [itex]a|0\rangle + b|1\rangle[/itex] into the state represented by the density matrix [itex]|a|^2 |0\rangle\langle 0| + |b|^2 |1\rangle\langle 1|[/itex].
 

Hurkyl

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The difference is extremely important, because measurements can test the accuracy of QM, and possibly falsify it. Quasi-measurements can't (unless they're part of a sequence of events that ends with an actual measurement).
In some sense, your parenthetical is the whole point -- we are considering measuring devices as quantum systems to see if the laws of quantum mechanics is at least plausible in that generality.

I suppose you can break a measurement into two parts: the effect on the measured system, and the "recording" and "viewing" of the results.

I have always viewed the first part as being the part that was actually interesting, particularly because the initial development of QM had enshrined the conclusion of a no-go theorem (time evolution is unitary, projection from a pure to a mixed state is not) whose crucial hypothesis really isn't ever satisfied.

The latter, on the other hand, sounds more like an engineering problem than a foundational physics problem.



The CNOT is interesting because it has the same effect on the measured qubit as a real measurement would, barring the universe being in an "odd" state that makes the gate behave oddly.
 

Fredrik

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The latter, on the other hand, sounds more like an engineering problem than a foundational physics problem.
It may sound that way, but engineering has nothing to do with it. I would say that this concept of measurement is the single most important detail in the foundations of physics.
 

Hurkyl

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It may sound that way, but engineering has nothing to do with it. I would say that this concept of measurement is the single most important detail in the foundations of physics.
But being important doesn't mean it's interesting or problematic. Is there a theoretical obstacle between "I've done a quasi-measurement and the result is contained in sub-system A" and "The engineering department has built a machine that will display the information contained in sub-system A"?


Or are you thinking now about treating the observer as a quantum system too? That one is interesting and problematic. (unfortunately, IMO, I think a big part of "problematic" is a reluctance to seriously consider (approximately) classical probabilities as reality rather than an ignorance measure)
 

ConradDJ

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I suppose you can break a measurement into two parts: the effect on the measured system, and the "recording" and "viewing" of the results.

I don’t think this statement is supported by QM itself, and there’s a lot of experimental evidence against it – for example, in the various versions of the “quantum eraser”.

To try to put this issue in context –

It seems that there are three empirical findings at the basis of QM. One is that isolated systems need to be described in a peculiar way – as “superpositions” evolving cyclically according to the wave-function. The second is that the state of a system depends on what can be known about it.

For example, when an electron is bound to a proton in a hydrogen atom, we know (without interacting) that its momentum is within a certain range, since otherwise it would no longer be bound. By Heisenberg’s principle this requires a corresponding “uncertainty” in the position of the electron, and I understand that this is the explanation for the size of a hydrogen atom.

The third finding is just an extension of the second. It is that besides the unitary evolution of “isolated” systems, something else happens that (as Fra says) physically “updates” the state of a system when new information about it becomes available, in what we call measurement.

In general, physical interaction as described by QM does not do this. It only “entangles” the two interacting systems, correlating their superposed states. If I understand correctly, your CNOT gate only entangles the input and output channels – so does not model measurement in the sense of QM. There’s a different kind of gate that does model a measurement – see for example http://arxiv.org/abs/0808.1582" [Broken].

There’s a “measurement problem” because QM tells us nothing about the difference between interaction in general and this “updating” interaction. Implicitly it says, any way of obtaining new information about a system – including indirect means that don’t involve any interaction with the system at all – constitute “measurements” that physically affect the system’s state.

Of course any actual measurement involves many different physical interactions. A main point of Von Neumann’s analysis was to show that it’s not relevant to QM which of these interactions is taken to be the “measurement” – the result is the same in any case.

To me, the problem with the various interpretations of Bohr, Heisenberg, Von Neumann and Wigner is that they’re all operating with a conceptual framework in which something is either objectively real in itself, out there in the world, or it’s something in the mind of a conscious observer. This subject/object dichotomy is completely foreign to the structure of QM.

Heisenberg was right in that QM describes the world not as a reality “in itself” but as a structure of information. As http://arxiv.org/abs/quant-ph/9609002" [Broken] says, “Physics concerns the information systems have about each other.” There’s no reason to think this has anything to do with consciousness, other than the fact that we lack a well-developed analysis of how information actually gets defined and communicated in the physical world.

So I agree with Fredrik – “...this concept of measurement is the single most important detail in the foundations of physics. ”
 
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A. Neumaier

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I would like to point out that the interaction of CNOT gate is only represented by a unitary transformation if you consider the joint (control, target) system. On the (control) system, the interaction truly is the non-unitary projection that turns the state represented by the ket [itex]a|0\rangle + b|1\rangle[/itex] into the state represented by the density matrix [itex]|a|^2 |0\rangle\langle 0| + |b|^2 |1\rangle\langle 1|[/itex].
What you write cannot be true since CNOT is an involution, while the projector you describe is not.

Indeed, I haven't seen anything like your conclusion on the page http://en.wikipedia.org/w/index.php?title=Controlled_NOT_gate you had linked to.
On the control system alone, CNOT is undefined given the information on that page, since its definition needs a 4-dimensional vector to act on.

To justify going from the unitary map to the projector, you need already assume decoherence, which happens only if the CNOT gate is significantly coupled to an environment into which information dissipates. Thus the environment must do the observing that you claim the target would do. But in this case, CNOT itself will also be no longer unitary, but turns into a subunitary operator.

The point of quantum computing (and the consideration of CNOT gates), however, is precisely to avoid as much as possible the coupling of the CNOT degrees of freedom to an environment in order to preserve the entanglement that contains the encoded information for quantum computations.
 

A. Neumaier

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What you write cannot be true since CNOT is an involution, while the projector you describe is not.
More specifically: If you put two (ideal) CNOT gates in series, the net effect is nothing: both control and target are what they were before, including all their entanglement if there was any,. This is crucial for its use in quantum computing: There is no loss of entanglement; it is possible to recover the exact input from the output. (Real quantum gates are of course slightly lossy - this is the main reason why it is so difficult to build efficient quantum computers.)

On the other hand, if someone observes the target in between (which means that the experimental arrangement must allow for this observation by some existing interaction with the environment), one gets a different result, expressible in terms of the POVM scenario I had described. The net effect is described by F_1 or F_2 (adding up to the identity), depending whether the first or the second of the two possible results has been observed. Or could have been observed - no human being needs to be there to actually look at how the environment was modified by the observation. Observation (or more general decoherence) is an objective fact (independent of a human observer), which happens because of interactions with the environment that are objectively present in the experimental arrangement.
 
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Hurkyl

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What you write cannot be true since CNOT is an involution, while the projector you describe is not.
Hypothesis:
  • The initial state of the target line is |0>
  • The initial state of the control line is a |0> + b |1>
  • The target and control lines are initially independent. (meaning, for this post, the joint state is the tensor product
  • The joint state undergoes the CNOT interaction.
Conclusion:
  • The final state of the target line has density matrix |a|2|0><0| + |b|2|1><1|
  • The final state of the control line has density matrix |a|2|0><0| + |b|2|1><1|

Proof: the initial joint state is:
a |00> + b|01>​
The final joint state is
a |00> + b|11>​
which has density matrix
|a|2 |00><00| + ab*|00><11| + a*b|11><00| + |b|2|11><11|​
extracting the components (via partial trace) on each subsystem gives
|a|2 |0><0| + |b|2 |1><1|​
 

A. Neumaier

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Hypothesis:
  • The initial state of the target line is |0>
  • The initial state of the control line is a |0> + b |1>
  • The target and control lines are initially independent. (meaning, for this post, the joint state is the tensor product
  • The joint state undergoes the CNOT interaction.
Conclusion:
  • The final state of the target line has density matrix |a|2|0><0| + |b|2|1><1|
  • The final state of the control line has density matrix |a|2|0><0| + |b|2|1><1|
Under the stated assumptions, the conclusion presented is correct, but the application of your argument to measurement meets two difficulties:

1. As my (second) post explained, things do not work for a sequence of two consecutive measurements on the same system, since after the first quasi-measurement (in your terms) your independence assumption no longer applies. A true measurement restores independence because of decoherence through the environment.

2. Your measurement interpretation works only for a single measurement repeated many times with independent inputs. Indeed, when one performs a true (conventional) measurement then, according to the von Neumann/Wigner form of the Copenhagen interpretation, the final state of the target has the density matrix |0><0| or |1><1|, while that of the control has the density matrix |0><0| or |1><1|. One must average over many instances to get your density matrices.

Difficulty 1 is the crucial (and uncurable) problem with your quasi-measurements.

Difficulty 2 is a problem only for those who want to explain why quantum mechanics has something to say about single quantum systems. This wasn't of interest in Born's time but is relevant today, where the experimental possibilities allow one to monitor single quantum systems - such as a particular ion in a particular ion trap -, modeled by Lindblad dynamics for the density matrix.
 

A. Neumaier

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IOn the (control) system, the interaction truly is the non-unitary projection that turns the state represented by the ket [itex]a|0\rangle + b|1\rangle[/itex] into the state represented by the density matrix [itex]|a|^2 |0\rangle\langle 0| + |b|^2 |1\rangle\langle 1|[/itex].
Note also that the mapping you describe is not a projection operator on the Hilbert space, but the latter is what the traditional theory of projective quantum measurements assumes.
 

Hurkyl

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Under the stated assumptions, the conclusion presented is correct, but the application of your argument to measurement meets two difficulties:

1. As my (second) post explained, things do not work for a sequence of two consecutive measurements on the same system, since after the first quasi-measurement (in your terms) your independence assumption no longer applies. A true measurement restores independence because of decoherence through the environment.

...

Difficulty 1 is the crucial (and uncurable) problem with your quasi-measurements.
As I said to Fredrick, I don't find this a problem at all -- it's nothing more than a matter of how much fine control we have over the system. I'm pretty sure Maxwell's Demon could use his talents to make anything you would consider a measuring device behave badly.

I'll also ask you a question I asked Fredrick:
What if my quantum computer loses coherence, so that the information in qubit on the target line "escapes" into the environment. Does it count as a measurement then?​




2. Your measurement interpretation works only for a single measurement repeated many times with independent inputs.
:confused:

Indeed, when one performs a true (conventional) measurement then, according to the von Neumann/Wigner form of the Copenhagen interpretation, the final state of the target has the density matrix |0><0| or |1><1|, while that of the control has the density matrix |0><0| or |1><1|. One must average over many instances to get your density matrices.
This objection has reached the level of purely classical probability. I'm pretty sure that, even in theory, there is no experiment you could perform to demonstrate reality is not in a mixed state.

Even if we insist that measurements must result in collapse, I will consider some other notion that is indistinguishable from measurement to be good enough.
 

Fredrik

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As I said to Fredrick, I don't find this a problem at all -- it's nothing more than a matter of how much fine control we have over the system. I'm pretty sure Maxwell's Demon could use his talents to make anything you would consider a measuring device behave badly.
You seem to be saying that even a measurement is reversible in principle, and if that's what you meant to say, you're right. It's at least conceivable that an interaction records the result of a measurement in the brain of a physicist and many other places as well, and after some time deletes all those records. It will never happen in the real world of course, but it's a part of QM due to the time-reversal invariance of the Schrödinger equation. Of course, now someone is going to mention that T (the time reversal operator) isn't preserved in QFTs, but CPT (the composition of charge conjugation, parity and time reversal) is, so I'll just say right away that I don't know what that implies about what I just said.

I'll also ask you a question I asked Fredrick:
What if my quantum computer loses coherence, so that the information in qubit on the target line "escapes" into the environment. Does it count as a measurement then?​
Since you're not saying anything about his reply in #53, I'm guessing that you missed that post.
 

Hurkyl

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You seem to be saying that even a measurement is reversible in principle, and if that's what you meant to say, you're right. It's at least conceivable that an interaction records the result of a measurement in the brain of a physicist and many other places as well, and after some time deletes all those records. It will never happen in the real world of course, but it's a part of QM due to the time-reversal invariance of the Schrödinger equation.
Yes, this is a good way of putting it.

Requiring a model of measurement not to be reversible, even in principle, is an unduly strict requirement. Instead, what we need is to have an idea of the basic interactions that are going on, and then see what happens when thermodynamics takes over.

I always like to use the kinetic theory of gas as an analogy. In this case, the analogy I want is:
Quasi-measurement is to measurement as
"A set of particles bouncing off another set of particles" is to "pressure exerted by a gas against a surface"​


Since you're not saying anything about his reply in #53, I'm guessing that you missed that post.
I did miss it. I guess whether he means yes or no boils down to whether or not a subtle change in a macroscopic system counts as macroscopic change. ("subtle change" referring to the fact that the information in the escaped photon affect change everywhere through repeated interactions so the effect cannot be localized to a microscopic system (at least, not a 'normal' one))
 
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You seem to be saying that even a measurement is reversible in principle, and if that's what you meant to say, you're right. It's at least conceivable that an interaction records the result of a measurement in the brain of a physicist and many other places as well, and after some time deletes all those records. It will never happen in the real world of course, but it's a part of QM due to the time-reversal invariance of the Schrödinger equation. Of course, now someone is going to mention that T (the time reversal operator) isn't preserved in QFTs, but CPT (the composition of charge conjugation, parity and time reversal) is, so I'll just say right away that I don't know what that implies about what I just said.


Since you're not saying anything about his reply in #53, I'm guessing that you missed that post.
When did quantum mechanics care about the brain of the scientist, or the records of events? Just because the brain is a system which thinks and recreates the outside world in our holograph-like projection of reality does not assume that the outside is somehow dependant on the observer. Our interpretation of the outside is almost certainly dependant on reality but not the other way around.

It is not concievable therefore to assume the interaction of events are somehow stored in the brain with a realization that perhaps this means there is some physical connection even after the system has collapsed. That believe it or not, was a very old theory which first was speculated among physicists when undergoing an understanding of the wave function, and it turned out it was not projecting reality in our brains as a mere way to keep up with results but rather was a physical property of all matter and energy.

The brain and how it collects a memory is certainly not needed to understand the collapse hypothesis. Nor is any notion of turning a system back on itself required the idea that human brains possess memory. Memory does not make an event happen, no more than erasing the memory reverse the event.
 

A. Neumaier

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Requiring a model of measurement not to be reversible, even in principle, is an unduly strict requirement. [...]

I did miss it. I guess whether he means yes or no boils down to whether or not a subtle change in a macroscopic system counts as macroscopic change.
Subtle changes in the environment caused by the interaction with a system have a decoherence effect on the latter. They count as a quantum measurement in the established sense of the word if and only if their effect on the system is to turn the pure state psi of the system into the state
[tex] \psi'=M_i \psi/||M_i \psi_i||[/tex]
when result number i is observed and the system X still exists after the measurement, in a way that a sequence of observations (i_1,...,i_n) of the same system under repeated measurements turns the pure state psi of the system into the state
[tex] \psi'=M_{i_n}...M_{i_1} \psi/||M_{i_n}...M_{i_1}\psi_i||[/tex]
(the observation sequence being impossible if the divisor is zero).. The latter is the experimentally verifiable, actually observed behavior under quantum measurements.
This strong form of irreversibilty is one of the most well established facts of physics.

In many textbook presentations, one even identifies quantum measurements with the much more special case where the M_i are mutually orthogonal orthogonal projectors. (Today this special case is usually called a projective quantum measurement, to distinguish them from the more realistic quantum measurements typically performed today on microscopic systems in a routine way.)

Your quasi-measurements do not satisfy this composition law, and slight imperfections in the CNOT gate (due to residual interactions with the environment) do not improve the situation. Hence your quasi-measurements resemble only superficially quantum measurements in the established sense of the word.

Apparently you also missed my comment #65, where I pointed out that what you called a projection is not even an operator on the Hilbert space of wave functions, while traditional binary projective measurements that you apparently want to model with CNOT act as projectors on wave functions.
 

Hurkyl

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Apparently you also missed my comment #65, where I pointed out that what you called a projection is not even an operator on the Hilbert space of wave functions, while traditional binary projective measurements that you apparently want to model with CNOT act as projectors on wave functions.
I thought #65 was merely informative; there was nothing of contention there.

I admit that "projection" was not the word I originally meant to use, but I decided to leave it as appropriate: not only is it an idempotent transformation of the state space* of the qubit, but it even acts as orthogonal projection onto the axis through |0> and |1>!

1: by this I mean Bloch sphere along with its interior, rather than the two-dimensional Hilbert space containing the pure states.



The binary projective measurements I want to model are not projections on Hilbert space. If you can arrange things so that unitary evolution can reliably result in such a thing on a subsystem, I would be interested -- but I'm under the impression that the no-go theorem does still apply here.

If you want to assume wave-function collapse happens after the interaction is completed, that's your business. I, however, am perfectly content with a model of measurement that results in the system being measured transitioning to a mixed state weighted correctly. My post-measurement state for the CNOT is
[tex]\sum_{i} P(i) \frac{M_i \rho M_i^\dagger}{\mathop{tr}(M_i \rho M_i^\dagger)}[/tex]​
where [itex]M_i = |i \rangle \langle i|[/itex]. If you decide to apply a wave-function collapse to my post-measurement state, you'll get projection onto |i> with probability P(i).
 

A. Neumaier

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I thought #65 was merely informative; there was nothing of contention there.

I admit that "projection" was not the word I originally meant to use, but I decided to leave it as appropriate: not only is it an idempotent transformation of the state space* of the qubit, but it even acts as orthogonal projection onto the axis through |0> and |1>!

1: by this I mean Bloch sphere along with its interior, rather than the two-dimensional Hilbert space containing the pure states.
It is a projector in a Hilbert space of linear operators, but this is very different from the use of projectors in traditional measurement theory.

The binary projective measurements I want to model are not projections on Hilbert space.
Then - to avoid confusion - you should not call them by the same name as the established concept.

If you can arrange things so that unitary evolution can reliably result in such a thing on a subsystem, I would be interested -- but I'm under the impression that the no-go theorem does still apply here.

If you want to assume wave-function collapse happens after the interaction is completed, that's your business. I, however, am perfectly content with a model of measurement that results in the system being measured transitioning to a mixed state weighted correctly. .
Then how does your measurement concept explain the basic experiments with polarized light (described in the introductory part of Sakurai's book)? After passing a polarizer, the photon is not in a mixture but in a pure state - described by a projection characterized by the orientation of the polarizer. (In this case, Born's law is nothing but the Malus law from 1809.)
It is this sort of experiments that gave rise to von Neumann's measurement theory.

My post-measurement state for the CNOT is
[tex]\sum_{i} P(i) \frac{M_i \rho M_i^\dagger}{\mathop{tr}(M_i \rho M_i^\dagger)}[/tex]​
where [itex]M_i = |i \rangle \langle i|[/itex]. If you decide to apply a wave-function collapse to my post-measurement state, you'll get projection onto |i> with probability P(i).
When you - much later - decide to observe (i.e., do a measurement on) the _target_, how does this collapse the projected post-measurement state of the _control_?

Actually, what you try to do looks to me in some way similar to the time-honored ancilla approach to quantum measurement (which reconstructs a unitary dynamics in a bigger space that explains measurement results in a particular sense). It is well-described in Sections 9-5 and 9-6 of the (mostly excellent) book ' Quantum theory: concepts and methods'' 'by Asher Peres.
 

Fredrik

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When did quantum mechanics care about the brain of the scientist, or the records of events?
It's an important concept in decoherence theory, but more importantly, it's a part of the (theory-independent) concept of "measurement".

It is not concievable therefore to assume the interaction of events are somehow stored in the brain with a realization that perhaps this means there is some physical connection even after the system has collapsed.
Huh? If you remember measuring the Sz of a silver atom to be +1/2, then the result has been stored in your brain.

Just because the brain is a system which thinks and recreates the outside world in our holograph-like projection of reality does not assume that the outside is somehow dependant on the observer.
Nor is any notion of turning a system back on itself required the idea that human brains possess memory. Memory does not make an event happen, no more than erasing the memory reverse the event.
I have no idea why you think the things you're saying have anything to do with the things I said.
 
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It's an important concept in decoherence theory, but more importantly, it's a part of the (theory-independent) concept of "measurement".


Huh? If you remember measuring the Sz of a silver atom to be +1/2, then the result has been stored in your brain.


I have no idea why you think the things you're saying have anything to do with the things I said.
What I understand of decoherence says nothing about the brain of the scientist. The reason why I said what I said, was because you said:

''It's at least conceivable that an interaction records the result of a measurement in the brain of a physicist and many other places as well, and after some time deletes all those records.''

Delete what records, memory? And if one deletes a record of memory from the brain, do you think this effects the outside world, the experiment to be more precise after the transaction has occurred?
 

A. Neumaier

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If you remember measuring the Sz of a silver atom to be +1/2, then the result has been stored in your brain..
This sort of memory has nothing to do with the measuring process.

Your memory might fail you because you were reading too many readings at the same time and mixed two of them up. This may affect your subjective interpretation of the experiment, bus doesn't matter at all for what actually happened in the measurement - the true result of the observation will not change because of your mistake.
 

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