Exploring the Limits of Measurement in Quantum Mechanics

[Hurkyl]

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

$$\sum_{i} P(i) \frac{M_i \rho M_i^\dagger}{\mathop{tr}(M_i \rho M_i^\dagger)}$$​

where $M_i = |i \rangle \langle i|$. 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]

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

$$\sum_{i} P(i) \frac{M_i \rho M_i^\dagger}{\mathop{tr}(M_i \rho M_i^\dagger)}$$​

where $M_i = |i \rangle \langle i|$. 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]

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 S_z 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 possesses 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.

 

[QuantumClue]

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 S_z 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]

I
If you remember measuring the S_z 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.

 

[Fredrik]

What I understand of decoherence says nothing about the brain of the scientist.

It says a lot about stable records of the state of the measured system stored in many different places in the environment. Whether a human brain is one of them is of course completely irrelevant. In my reply to Hurkyl, it was just an example of a record of the result.

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?

It makes absolutely no sense to ask me something like that.

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.

That has nothing to do with anything I said.

 

[QuantumClue]

It says a lot about stable records of the state of the measured system stored in many different places in the environment. Whether a human brain is one of them is of course completely irrelevant. In my reply to Hurkyl, it was just an example of a record of the result.


It makes absolutely no sense to ask me something like that.



That has nothing to do with anything I said.

If that has nothing to do with what you said, and both myself and someone else stated close to what we thought you were saying, then it is a matter of you not explaining clear enough yourself.

If that is not what you were saying, maybe it would be good to express yourself a bit clearer for us to understand.

 

[A. Neumaier]

I
That has nothing to do with anything I said.

What is stored in the brain is irrelevant for what happens in an experiment.

For the latter, the only thing that counts is what happens in the detector that amplifies the interaction with the system at the time of the measurement. This happens at the Geiger counter recording a charged particle, at the photodetector or the eye recording a photon emitted by the system, etc., but never in the brain.

 

[Hurkyl]

Then - to avoid confusion - you should not call them by the same name as the established concept.

It was your name, not mine. *shrug* I did misspeak, though; I did mean to replace your phrase with a more generic term, since what I want to model is indistinguishable from what you are calling a binary projective measurement, but without the presumption of a wave-function collapse interpretation.

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.

Er, what's the problem? The interaction results in a mixed state where the photon is absorbed with probability (sin theta)^2, and survives and transmitted in a pure state aligned with the polarizer with probability (cos theta)^2. If you condition the mixture on the hypothesis that the photon is not absorbed, the result is a pure state.

Or, are you talking about something else?

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_?

You mean, you want to know the effect of the operators $|0\rangle\langle 0| \otimes 1$ and $|1\rangle\langle 1| \otimes 1$ on the joint state $a |00\rangle + b |11\rangle$?

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.

I didn't think I was talking about anything unusual, which is why I was somewhat surprised at the opposition...

 

[Fredrik]

If that has nothing to do with what you said, and both myself and someone else stated close to what we thought you were saying, then it is a matter of you not explaining clear enough yourself.

I was wondering if that might be the case, but I've reread my statements several times, and although you might need to read the whole discussion between me and Hurkyl to fully understand the points I was making, it was very clear that I didn't say anything close to what you're suggesting. The kind of questions you're asking makes me a lot less willing to try to explain anything to you.

What is stored in the brain is irrelevant for what happens in an experiment.

For the latter, the only thing that counts is what happens in the detector that amplifies the interaction with the system at the time of the measurement. This happens at the Geiger counter recording a charged particle, at the photodetector or the eye recording a photon emitted by the system, etc., but never in the brain.

You have clearly not understood what I said either. You don't need to explain these things to me.

 

[QuantumClue]

I was wondering if that might be the case, but I've reread my statements several times, and although you might need to read the whole discussion between me and Hurkyl to fully understand the points I was making, it was very clear that I didn't say anything close to what you're suggesting. The kind of questions you're asking makes me a lot less willing to try to explain anything to you.


You have clearly not understood what I said either. You don't need to explain these things to me.

There is no ''might'' about it. Two posters here have been mislead by your post - You clearly demonstrated special knowledge on the memory of the scientist, which is neither here nor there in an experiment.

 

[Fredrik]

You weren't misled. You read half a post and made assumptions about the rest. I don't have time for this nonsense anyway.

 

[QuantumClue]

You weren't misled. You read half a post and made assumptions about the rest. I don't have time for this nonsense anyway.

Why mention the scientists brain?

 

[Fredrik]

Why mention the scientists brain?

Because it's an example of a persistent record of the result of a measurement, in a part of the environment that's approximately classical, and because a person who understands that even information storage in a brain can be reversed understands that everything can (in principle) be reversed. Note that I stated explicitly that it's one of many places where the information will be stored.

Hurkyl had been arguing that the "quasi-measurement" performed by a CNOT gate isn't fundamentally different from what the rest of us had been calling a "measurement" (an interaction that creates persistent records of the result in an almost classical part of the environment) and I was telling him that he was right, by saying that even in this extreme case, where enough time had passed to allow for the creation of a persistent record in the physicist's brain, the entire process that created all the information records can in principle be reversed, just as the "quasi-measurement" performed by a CNOT gate. Note that I mentioned time-reversal invariance and said that all the records would be deleted, not just the one in the brain.

 

[A. Neumaier]

Slowly, we seem to converge...

It was your name, not mine. *shrug* I did misspeak, though; I did mean to replace your phrase with a more generic term, since what I want to model is indistinguishable from what you are calling a binary projective measurement, but without the presumption of a wave-function collapse interpretation.

My name was attached (as usual) to an actual measurement, not to your quasi-measurement. The latter doesn't feature a definite measurement result and a corresponding projection of the wave function, but only a probability distribution for this to happen if a separate, fictitious measurement were made.

Er, what's the problem? The interaction results in a mixed state where the photon is absorbed with probability (sin theta)^2, and survives and transmitted in a pure state aligned with the polarizer with probability (cos theta)^2. If you condition the mixture on the hypothesis that the photon is not absorbed, the result is a pure state.

This is the first time in the discussion that you mention the conditioning. Conditioning _is_ the measurement or collapse: accepting that a particular measurement value was obtained, and restricting the ensemble accordingly. As long as the system still is in the mixed state, it is not yet measured since it is still ambiguous which measurement result was obtained, and any measurement result is therefore still possible. After the measurement, it is decided.

In this case, the measurement consists in passing the polarizer - a photon can be observed behind it only if it actually passed, so it is an objective fact that the ensemble has changed from the prepared ensemble to the observed ensemble - providing the projection.
(This is why you can compose two polarizers and get as a result the product of the projections.)

Nothing like that happens in the case of a CNOT gate. To condition subject to a particular ficticious measurement result is a purely subjective act, without any physical basis.
(And composing two CNOT gates gives the identity.)

I didn't think I was talking about anything unusual, which is why I was somewhat surprised at the opposition...

The opposition was due to your very unconventional terminology - calling a (quasi-) measurement what is at best an unperformed measurement.

What you call a (quasi-)measurement is usually called decoherence: The loss of off-diagonal terms in the density matrix due to interaction with the environment. This doesn't tell anything about the achieved measurement result; thus it provides no information.

Whereas a measurement always does: After having measured a quantity, one _knows_ its value, and not only a probability distribution for the possible values.

The two concepts are related, but if one mixes them up, misunderstandings are unavoidable.

 

[Hurkyl]

My name was attached (as usual) to an actual measurement, not to your quasi-measurement. The latter doesn't feature a definite measurement result and a corresponding projection of the wave function, but only a probability distribution for this to happen if a separate, fictitious measurement were made.

I am under the impression that it's rather standard to allow "measurement" to apply to the indefinite case as well.

In any case, your insistence of a definite measurement result is clearly not a useful thing to do in a variety of cases, including:

• You want to say something relevant to a non-collapse based interpretation
• The situation of the opening post -- the consideration of the possibility that a joint "measuring device - measured system" system is governed by the unitary evolution of Quantum Mechanics.

This is the first time in the discussion that you mention the conditioning.

This is the first time anyone asked me to.

As long as the system still is in the mixed state,

Unless you are invoking a collapse-based interpretation of QM, the system is always in the mixed state. Otherwise, collapse is a mathematical trick -- e.g. for studying mixtures -- and has nothing to do with the evolution of the state under study.
Nothing like that happens in the case of a CNOT gate. To condition subject to a particular ficticious measurement result is a purely subjective act, without any physical basis.
(And composing two CNOT gates gives the identity.)

Whereas a measurement always does: After having measured a quantity, one _knows_ its value, and not only a probability distribution for the possible values.

Only if you make the metaphysical choice to insist on definite outcomes. Otherwise, both the "quantity measured" and "the value you 'know'" both remain indeterminate (but equal) variables.

 

[yuiop]

The quantum Zeno effect is often described by the causal informal analogy that "A watched pot really never boils in QM". Although it sounds technical the quantum Zeno effect can be very simply demonstrated. Take a light source with random linear polarisation and pass it through a horizontal linear polariser. Now pass the polarised light through a polarisation rotator that rotates the light by 15 degrees followed by another horizontal linear polariser. Repeat this 6 times like so:

Now after being rotated 15 degrees six times the light should be rotated by 90 degrees and should have zero probability of passing through the final horizontal polariser, but in fact the probability of passing through the final polariser is none zero ( 100*cos(pi/12)^2)^6 = approx 66%). It is said that because the photon is being "observed" between successive rotations it does not rotate as much as it normally would when "unobserved". It becomes clear from this experiment that what "observe" means passing a photon through a polariser. This qualifies as a measurement (and this is demonstrated in many other experiments). The use of the word "observe" for passing a particle through a polariser or a Stern Gerlach magnet is a bit misleading as it implies that a sentient observer is required and causes people to rush off and write rubbish like the "The Tao Of Physics". As long as we accept that a polarising filter like the one you attach to the front of your SLR camera is not a sentient being, then we need not consider that sentient observers are required.

 

[A. Neumaier]

I am under the impression that it's rather standard to allow "measurement" to apply to the indefinite case as well.

I cannot see how one could meaningfully call something a measurement which doesn't produce a measurement result but only a distribution of possibilities.

Could you please substantiate your impression by quoting a standard textbook? The book by Asher Peres is the most thorough I know of, and discusses the meaning of the term in Sections 1-5, 3-6, and the whole of Chapter 12. (I haven't seen Schlosshauers book, which should also be good, given his excellent survey article in Rev.Mod.Phys.76:1267-1305,2004 arXiv:quant-ph/0312059 )

In any case, your insistence of a definite measurement result is clearly not a useful thing to do in a variety of cases, including:

• You want to say something relevant to a non-collapse based interpretation
• The situation of the opening post -- the consideration of the possibility that a joint "measuring device - measured system" system is governed by the unitary evolution of Quantum Mechanics.

An interpretation of quantum mechanics that is unable to account for the fact that performing a real measurement in a real-life situation yields real measurement results is an incomplete interpretation. This holds independent of what sort of assumptions an interpretation makes, so it must be possible to talk about it also in a non-collapse based interpretation, if the latter is complete in this sense.

I don't understand your second point. The situation of the opening post was discussed by von Neumann and by Wigner _assuming_ the existence of definite measurement results.

Unless you are invoking a collapse-based interpretation of QM, the system is always in the mixed state. Otherwise, collapse is a mathematical trick -- e.g. for studying mixtures -- and has nothing to do with the evolution of the state under study.
Nothing like that happens in the case of a CNOT gate. To condition subject to a particular ficticious measurement result is a purely subjective act, without any physical basis.
(And composing two CNOT gates gives the identity.)

Was the repetition of the last sentences (which were my words) intended?
I'll reply to this after this is clarified.

Only if you make the metaphysical choice to insist on definite outcomes. Otherwise, both the "quantity measured" and "the value you 'know'" both remain indeterminate (but equal) variables.

I don't understand why you consider definite outcomes a metaphysical choice. It is the most basic observation in any experiment that measurement outcomes are definite, and not a mathematical trick. All trained observers agree (for measurements of non-integer numbers, within a small error margin) on which value was measured, something that any complete interpretation must be able to account for.

 

[A. Neumaier]

(I haven't seen Schlosshauers book, which should also be good, given his excellent survey article in Rev.Mod.Phys.76:1267-1305,2004 arXiv:quant-ph/0312059 )

After reading the reviews posted at the author's home page http://www.nbi.dk/~schlossh/ , I am less optimistic about his book. All four reviews highly recommend the book as a source to learn about decoherence; two reviews are wholly favorable. But the review by Zeilinger (in Nature) explains why the arguments given there against the Copenhagen interpretation are not convincing, and that by Landsman (in Stud. Hist. Phil. Mod. Phys.) emphasizes conceptual shortcomings, and refers (among others) to http://plato.stanford.edu/archives/win2004/entries/qm-decoherence for a more balanced discussion of the merits of decoherence.

 

[Fredrik]

My main complaint is that it's too wordy, and not mathematical enough. But I think it's still a good (possibly the best) place to start. (I haven't read the whole book, so I wasn't even aware that he argues against Copenhagen).

 

[Hurkyl]

I don't understand why you consider definite outcomes a metaphysical choice. It is the most basic observation in any experiment that measurement outcomes are definite, and not a mathematical trick. All trained observers agree (for measurements of non-integer numbers, within a small error margin) on which value was measured, something that any complete interpretation must be able to account for.

Let X be the variable denoting what trained observer #1 sees.
Let Y be the variable denoting what trained observer #2 sees.

You are asserting that X and Y both have definite, equal values. That is stronger than what the bold part implies, which is merely that the two variables are equal.

 

[A. Neumaier]

Let X be the variable denoting what trained observer #1 sees.
Let Y be the variable denoting what trained observer #2 sees.

You are asserting that X and Y both have definite, equal values. That is stronger than what the bold part implies, which is merely that the two variables are equal.

But even the latter must be shown, and not merely assumed.

So please tell me how to augment the CNOT gate by two observers #1 and #2 to a combined quantum system (control,target,#1,#2) in a way that the variables X and Y are well-defined. In order to meaningfully interpret them as expressing what you wrote above, X must be defined on the system of #1 alone, and Y must be defined on the system of #2 alone.

 

[Fra]

I'm not sure what the discussion really is about but...

In any case, your insistence of a definite measurement result is clearly not a useful thing to do ...
The situation of the opening post -- the consideration of the possibility that a joint "measuring device - measured system" system is governed by the unitary evolution of Quantum Mechanics.

As far as I see, there is no difference between the Observer A's expectations of how Observers B + S evolves (including B making measurements on S) and the normal "unitary" evolution of S' if we define S' = observer B+S.

Then the "expected evolution" in between A's measurement on S' should be unitary. Meaning that collapses vs unitarity is just a matter of perspective.

Ie. one can see the "collapse" as a form of "naked description", but once renormalized into an external observer, there is no way to observer the naked observer, one just sees a screened complex of observer + environment. So observer A's can not observe the naked action that constitues the B's observation process of S.

However, I think a proper measurement, is only defined relative to the correct observer. When one observer, "observes" the "measurement act" of another observer, it's not the same thing.

?

/Fredrik

 

[A. Neumaier]

My main complaint is that it's too wordy, and not mathematical enough. But I think it's still a good (possibly the best) place to start. (I haven't read the whole book, so I wasn't even aware that he argues against Copenhagen).

My main complaint is that Schlosshauer takes sides with a particular interpretation, the ''many minds interpretation'', whereas in his 2003 survey he was impartial.

 

[Hurkyl]

But even the latter must be shown, and not merely assumed.

So please tell me how to augment the CNOT gate by two observers #1 and #2 to a combined quantum system (control,target,#1,#2) in a way that the variables X and Y are well-defined. In order to meaningfully interpret them as expressing what you wrote above, X must be defined on the system of #1 alone, and Y must be defined on the system of #2 alone.

Do you really find that unclear?


At the level of observables, it's easy.

Observable X acts on control by sending |m> to m|m>
Observable Y acts on target by sending |m> to m|m>

And it's easy to see that the final joint state a|00>+b|11> is an eigenvector of X-Y with eigenvalue 0.



If you insist on the observer being part of the system, I would just model them as CNOT gates:

                           +--------+
obs #1 --------------------|t       |--\
                           |   CNOT |   \
                        /--|c       |    \
           +-------+   /   +--------+     \    +--------+
device ----|t      |--/                    \---|t       |-----
           |  CNOT |                           |   CNOT |
system ----|c      |--\                    /---|c       |
           +-------+   \   +--------+     /    +--------+
                        \--|c       |    /
                           |   CNOT |   /
obs #2 --------------------|t       |--/
                           +--------+

(c,t denote control and target). If the device, obs#1, and obs#2 start in |0>, then at the end, the obs #1 and obs #2 are in the mixture of |00> and |11> with weights |a|² and |b|² respectively.

In the diagram above, I went further and attached a circuit that performs the measurement to compare what the two observers' observations, by adding them. It's not hard to see that the final state will be a pure |0>.

(in the diagram, I've suppressed the lines that are no longer relevant)



If none of this resembles what you asked for, could you be somewhat more precise?

 

[A. Neumaier]

Do you really find that unclear?

Yes; it is not obvious how to do it.

If you insist on the observer being part of the system,

Yes, I had asked for that.

If none of this resembles what you asked for, could you be somewhat more precise?

In the displayed version, the diagram looks garbled. I deciphered it by copy-pasting it to an editor with constant width characters; this may help others to understand your arrangement.

But the diagram doesn't yet do what the discussion requires: ''All trained observers agree on which value was measured''. Thus #1 and #2 don't look at the control measured but they both look (perhaps later, at different times, and the control might no longer exist) at the measurement device, i.e., the target, where they infer (by ''seeing'' it - which is another measurement) a common measurement value.

 

[Hurkyl]

But even the latter must be shown, and not merely assumed.

So please tell me how to augment the CNOT gate by two observers #1 and #2 to a combined quantum system (control,target,#1,#2) in a way that the variables X and Y are well-defined. In order to meaningfully interpret them as expressing what you wrote above, X must be defined on the system of #1 alone, and Y must be defined on the system of #2 alone.

Now that I think of it, answering your challenge isn't actually the right response.

Because all of the interactions and observables involved are operating in the |0> - |1> basis, relative phase is irrelevant -- everything projects down to mixtures of 0-1 basis states.

After making the projection, the analysis is not merely analogous to ordinary probability theory -- it is identical.

 

[Hurkyl]

In the displayed version, the diagram looks garbled. I deciphered it by copy-pasting it to an editor with constant width characters; this may help others to understand your arrangement.

Argh. The [ code ] block is supposed to be a constant width font. I didn't know some browsers would opt to display it otherwise.

Thus #1 and #2 don't look at the control measured but they both look (perhaps later, at different times, and the control might no longer exist) at the measurement device, i.e., the target, where they infer (by ''seeing'' it - which is another measurement) a common measurement value.

In the diagram I drew, observer #1 is observing the output of the CNOT gate that corresponds to the readout of the measuring device (the control line of the top CNOT gate is the target line of the left CNOT gate), and observer #2 is observing the output of the CNOT gate that corresponds to the system that was observed (the control line of the bottom CNOT gate is the control line of the left CNOT gate).

For some reason, I thought that's what you were asking, since the alternative is even more trivial. New diagram coming right up...

                          +-------+
obs #2 -------------------|t      |----------------------\
                          |  CNOT |                       \   +-------+
           +-------+   /--|c      |--\                     \--|t      |-----
device ----|t      |--/   +-------+   \                       |  CNOT |
           |  CNOT |                   \                   /--|c      |
system ----|c      |                    \   +-------+     /   +-------+
           +-------+                     \--|c      |    /
                                            |  CNOT |   /
obs #1 -------------------------------------|t      |--/
                                            +-------+

 

[A. Neumaier]

Argh. The [ code ] block is supposed to be a constant width font. I didn't know some browsers would opt to display it otherwise.

I am using Konqueror Version 4.2.2 (KDE 4.2.2) on a Linux platform.

the alternative is even more trivial. New diagram coming right up...

OK. This satisfies the requirements of objectivity (in the sense of intersubjectivity).

Nice, thanks! I wasn't aware of this. Where did you learn this from? Where is it discussed most clearly?

Indeed, I checked that with CNOT gates one can completely reproduce the ancilla simulating an arbitrary sequence of binary projective measurements. Thus your quasi-measurements behave more like true measurement than what I had imagined.

Let us now inquire to which extend CNOT gates can serve as measurement devices. The whole ansatz works only if your assumption ''The initial state of the target line is |0>'' from post #64 holds for each CNOT gate.

So please tell me how - in your world of quasi-measurements without what you call ''metaphysical choices'' - the target can be objectively prepared in that state.

 

[A. Neumaier]

Let us now inquire to which extend CNOT gates can serve as measurement devices. The whole ansatz works only if your assumption ''The initial state of the target line is |0>'' from post #64 holds for each CNOT gate.

So please tell me how - in your world of quasi-measurements without what you call ''metaphysical choices'' - the target can be objectively prepared in that state.

In http://arxiv.org/pdf/quant-ph/0612216, Mermin demonstrates (though it is not quite a proof) that - and why - this is not possible without a true measurement. He shows that what you call ''metaphysical choices'' is an important (and necessary) part of any quantum computation.

Therefore I'd like to suggest that you retract your pejorative labeling of my act of claiming that the result of a measurement is something definite.

 

[Hurkyl]

Nice, thanks! I wasn't aware of this. Where did you learn this from? Where is it discussed most clearly?

My learning style is rather erratic -- I can't usually point to someplace and say "I learned from here".

I do know that a short course on quantum computing solidified most of the meager understanding I had of quantum mechanics before then. Learning about programming such a computer means setting up circuits to compute function of some input qubits and adding the result into an qubit in some fashion (or some other invertible manipulation of the output bits) -- and that is where I got the idea of such a thing being analogous to a measurement.

I'm sure that at least some of what I have subsequently read about decoherence, particularly involving decoherence-based interpretations of QM, had similar ideas in mind. I couldn't really point to anything specific, except for one.

Rovelli's paper on Relational Quantum Mechanics was the next most significant thing I encountered. It wasn't the interpretation that impressed me, but the treatment of the situation where Alice is observing Bob observe a system.


Bob, in his analysis, places the von Neumann cut between himself and the system; he does his measurement, sees the result, then continues his study as if the system has collapsed into the corresponding state.

Alice, however, places the von Neumann cut between herself and Bob. Alice does her analysis by treating Bob+System as Bob does the measurement as a quantum system, evolving according to Schrödinger's equation. She may eventually perform a measurement herself to collapse Bob+System into a definite state.

(Alas, the discussion in section II doesn't take the next step to apply decoherence or anything of the sort)


My impression is that this shows the way you can have your cake and eat it too, regarding interpretations. We know that, so long as something unusual is going on, it doesn't matter where you place the von Neumann cut between the quantum and classical world.

From Alice's point of view, we see the consistency between treating Bob+System as if it collapses when Bob makes a measurement, and treating Bob+System as if it continues to evolve a là Schrödinger.

This is made even clearer if you suppose decoherence occurs in the latter treatment, (or you partial trace to extract Bob's state from the joint system), because the quantum state is now a mixture of all the possible collapsed states.

Let us now inquire to which extend CNOT gates can serve as measurement devices. The whole ansatz works only if your assumption ''The initial state of the target line is |0>'' from post #64 holds for each CNOT gate.

So please tell me how - in your world of quasi-measurements without what you call ''metaphysical choices'' - the target can be objectively prepared in that state.

This I don't know. But then, I don't know how it can happen in classical mechanics either, so I'm content that QM is no more lacking than classical mechanics in that regard.

 

[A. Neumaier]

Thanks for the explanation of your learning process. I accept it as your personal history, but the scientific content raises more problems than it answers.

a short course on quantum computing solidified most of the meager understanding I had of quantum mechanics before then. Learning about programming such a computer means setting up circuits to compute function of some input qubits and adding the result into an qubit in some fashion (or some other invertible manipulation of the output bits) -- and that is where I got the idea of such a thing being analogous to a measurement.

But in quantum computing they clearly distinguish between measurements and quantum circuits of the kind we discussed, so there must have been a misunderstanding since you wrote:

I am under the impression that it's rather standard to allow "measurement" to apply to the indefinite case as well.

Rovelli's paper on Relational Quantum Mechanics was the next most significant thing I encountered. It wasn't the interpretation that impressed me, but the treatment of the situation where Alice is observing Bob observe a system.

Bob, in his analysis, places the von Neumann cut between himself and the system; he does his measurement, sees the result, then continues his study as if the system has collapsed into the corresponding state.

Alice, however, places the von Neumann cut between herself and Bob. Alice does her analysis by treating Bob+System as Bob does the measurement as a quantum system, evolving according to Schrödinger's equation. She may eventually perform a measurement herself to collapse Bob+System into a definite state.

This is essentially a replay of the analysis von Neumann gave in 1932, phrased in more modern terminology, and simplified because only binary signals are considered.

My impression is that this shows the way you can have your cake and eat it too, regarding interpretations. We know that, so long as something unusual is going on, it doesn't matter where you place the von Neumann cut between the quantum and classical world.

The latter was von Neumann's conclusion, too. But nevertheless, he postulated two different processes, since he knew that one cannot have the cake and eat it too.

You got the opposite impression because you forgot to analyze the starting point, how observers #1 and #2 can check that the CNOT gates are properly initialized. In a world without collapse, they cannot! Thus in such a world, they never know whether or not they made a measurement according to your rules. (The many world interpretation does not help, since even in this interpretation, there is a collapse in the world actually observed, and the postulated unobserved worlds don't explain anything but only introduce problems of their own.)

You just shifted the burden of the interpretation from the measurement apparatus to the preparation apparatus. (In measurement theory, the two are often seen to be two sides of the same coin - a perfect measurement preparing an eigenstate.)

In fact, as Mermin points out in http://arxiv.org/pdf/quant-ph/0612216 , one needs proper measurements not only for preparing the input of quantum gates but also for error correction (without which all serious quantum computing would have to remain a dream forever).

This is made even clearer if you suppose decoherence occurs in the latter treatment, (or you partial trace to extract Bob's state from the joint system), because the quantum state is now a mixture of all the possible collapsed states.

Perhaps you realize now that what you call a quasi-meaurement is nothing else than what others call decoherence: loss of off-diagonal entries in the density matrix.

Let us now inquire to which extend CNOT gates can serve as measurement devices. The whole ansatz works only if your assumption ''The initial state of the target line is |0>'' from post #64 holds for each CNOT gate.

So please tell me how - in your world of quasi-measurements without what you call ''metaphysical choices'' - the target can be objectively prepared in that state.

This I don't know.

I find it inconsistent that you feel entitled to assume that the input to a gate is fully determined, while you belittle my insistence on definite outcomes, denouncing it as a metaphysical choice:

Only if you make the metaphysical choice to insist on definite outcomes. Otherwise, both the "quantity measured" and "the value you 'know'" both remain indeterminate (but equal) variables.

But then, I don't know how it can happen in classical mechanics either, so I'm content that QM is no more lacking than classical mechanics in that regard.

I don't understand why there should be a problem is in preparing a zero input state in classical circuit design. You measure an arbitrary state, and negate the result in case it happens to be 1.

One can do the same in the quantum case, but only if one accepts that a measurement has a definite outcome and leaves the measured system in an eigenstate. A quantum measurement gate indeed does this - so what you call a metaphysical choice is a well established empirical fact.

 

[A. Neumaier]

in quantum computing they clearly distinguish between measurements and quantum circuits of the kind we discussed, so there must have been a misunderstanding since you wrote:

I am under the impression that it's rather standard to allow "measurement" to apply to the indefinite case as well.

Since you apparently quit the discussion just at the point where the crucial gap in your argument had been identified, let me summarize the findings of our extended discussion:

Our CNOT gate discussion started with your claim that it is a measurement device:

Let's start with something possibly very simple. I consider a CNOT gate (wikipedia link) a measuring device. It measures the qubit on its control line, and records the result of measurement by adding it to the target line.

Our discussion revealed that if each target line is initialized with a definite zero state, CNOT gates can be used to construct an ancilla for a sequence of quasi-measurements, such that the reduced density matrix on the output control line is decohered, i.e., diagonal. Therefore different observers see the same result, conditioned on a particular measurement result for one observer.

But this doesn't hold for CNOT gates that are differently prepared. This shows that the CNOT gate by itself is not a measurement device, but only the dissipative system that consists of the CNOT gate together with another gate that prepares the target line in a definite zero state. The latter requires already a definite outcome of a measurement, and hence must be itself a measurement device.

Indeed, in quantum information theory, one has specific measurement gates that perform a binary projective measurement and produce a definite outcome. These gates exist as real devices, and are necessary for any quantum information technology.

Thus while CNOT gates explain the working of decoherence in a very elegant and simple way, they - like decoherence itself - do not explain the working of measurement gates (or any other measurement devices).