What Are the Key Controversies in Quantum Measurement Theory?

[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).