What Are the Key Controversies in Quantum Measurement Theory?

[A. Neumaier]

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 a|0\rangle + b|1\rangle into the state represented by the density matrix |a|^2 |0\rangle\langle 0| + |b|^2 |1\rangle\langle 1|.

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]

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.

 

[Hurkyl]

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]

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]

IOn the (control) system, the interaction truly is the non-unitary projection that turns the state represented by the ket a|0\rangle + b|1\rangle into the state represented by the density matrix |a|^2 |0\rangle\langle 0| + |b|^2 |1\rangle\langle 1|.

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]

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.

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]

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]

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))

 

[QuantumClue]

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 possesses memory. Memory does not make an event happen, no more than erasing the memory reverse the event.

 

[A. Neumaier]

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
\psi&#039;=M_i \psi/||M_i \psi_i||
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
\psi&#039;=M_{i_n}...M_{i_1} \psi/||M_{i_n}...M_{i_1}\psi_i||
(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]

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.[/size]



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.[/size]

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).