This forum gives conflicting info on the HUP

[DevilsAvocado]

All my posts above can be summarized as follows:
Whenever you measure something in QM, you really measure something else.

Essentially, that is because in QM you never really measure frequency. Instead, you really measure something else (the position of some macroscopic pointer) which turns out to be ENTANGLED with frequency.

Why do I get the feeling you are saying something very deep that's completely beyond my perception...

In QM we are not measuring reality but an abstract mathematical model that by some amazing 'coupling' turns out to be the most precise description of 'reality' we ever had – it simply works – but nobody really understands why.

* dizzy *

 

[Demystifier]

Okay thanks DM, think I understand now. Simultaneous "sharp" measurement of position and momentum is possible, but arbitrarily precise (including exact) values are not possible, right?

HUP does not prevent arbitrarily precise measurements.

But... what about 'the wall' of Planck time...?

Some physicists believe that time cannot be measured with a precision better than Planck time, but it really depends on which version of quantum gravity you believe in, and there is no any "standard" theory of quantum gravity. After all, we certainly can measure energy with a precision better that Planck energy, so the same could be valid for time as well.

But... if "the fast camera" is severely restricted by Mr. Planck, whom is a very close friend to Mr. Heisenberg... doesn't HUP always win in the end...?

My point is that Mr. Planck and Mr. Heisenberg are not really such close friends.

 

[Demystifier]

Why do I get the feeling you are saying something very deep that's completely beyond my perception...

In QM we are not measuring reality but an abstract mathematical model that by some amazing 'coupling' turns out to be the most precise description of 'reality' we ever had – it simply works – but nobody really understands why.

* dizzy *

To paraphrase Bohr, if you are not confused about QM, then you don't understand it.

But you can deconfuse yourself a lot with a Bohmian way of thinking. With Bohmian thinking, it is quite clear why all measurements reduce to observations of positions of some macroscopic pointers. See e.g.
http://lanl.arxiv.org/abs/1112.2034 [Int. J. Quantum Inf. 10 (2012) 1241016]
Sec. 2.

 

[DevilsAvocado]

Thanks DM, need some time to 'digest' ...

 

[Ken G]

All my posts above can be summarized as follows:
Whenever you measure something in QM, you really measure something else.

But I agree with what you said above, which is that one must be agnostic about QM when doing measurements. This means we first say what a measurement is, and then we use measurements to test theories. We can use indirect measurements if we put our trust in one theory as contingent on the testing of another, but any measurements that involve an element that requires a QM computation to be meaningful is not really a measurement that can be used to test QM.

Also, we can try doing two measurements at once, but we cannot be sure the measurements still mean anything, because the two apparatuses could function at cross purposes. For example, I can have a scale that measures my weight, and a tape measure that measures the circumference of my waist, but if I make both measurements at once then the weight measurement will be altered by the weight of the tape measure. We can assure that we use a light tape measure and get away with those simultaneous measurements, but that only means that we are taking pains to assure the results of both measurements are meaningful. That's not so easy for non-commuting measurements in quantum mechanics, because we describe measurements in quantum mechanics in terms of a superposition of eigenstates of operators, and how we use an apparatus to scramble the coherences within that superposition, but you cannot necessarily scramble the coherences within the eigenstate superpositions of two separate non-commuting operators. So the devices may read something sharp, but it might not qualify as a measurement, just as I would not be measuring my weight if I had a lead tape measure around my waist.

 

[atyy]

What you presented is one particular example of a measuring apparatus which does not allow simultaneous measurement of noncommuting observables. But nothing in your argument shows that this is a general conclusion.

Let me try to describe a variation of SG device which, in principle, would allow simultaneous measurement.

Suppose you want to measure the spin of particle A. For that purpose you first make the particle A interact with another particle B, such that the two particles become entangled. The entanglement may be such that, by measuring spin s_x of particle B, you also know spin s_x of particle A. Then you separate the entangled particles and route particle B in a standard SG device. In this way you measure s_x of particle A without ever routing particle A in a standard SG device.

Similarly, you can measure spin s_y of particle A by entangling it with a third particle C and routing particle C in another standard SG device.

Finally, you can combine these two modified SG devices, such that you can simultaneously measure s_x via particle B and s_y via particle C.

But can this work on an arbitrary state? When you entangle you may alter the state of the particle you are measuring, and when you measure the state of the probe, you will collapse the state of both the probe and the particle.

But I agree with what you said above, which is that one must be agnostic about QM when doing measurements.

But it is the theory that says what we measure, in particular, whether a measurement is "sharp" or not is defined by QM.

 

[Demystifier]

Ken G, I fully agree with you, and I like your analogy with weight-and-circumference measurement.

 

[Demystifier]

But can this work on an arbitrary state?

Yes.

When you entangle you may alter the state of the particle you are measuring, and when you measure the state of the probe, you will collapse the state of both the probe and the particle.

True, but this can also be said for a measurement of a single observable.

But it is the theory that say what we measure, in particular, whether a measurement is "sharp" or not is defined by QM.

I both agree and disagree with this. I agree probably for the same reason as you do, so let me only explain why I disagree.

If theory determines what we measure, that it can be argued that the same experiments measure different quantities, depending on which interpretation of QM you adopt. Hence, to avoid interpretation-dependence, it is wise to remain partially agnostic about the theory, and to define measurement in terms of concepts which do not depend much on the theory. One way to reduce the amount of assumed theory is to define all measurements in terms of classical concepts only. Of course, there are other ways too, so in reality it is important to define precisely what do you mean by a "measurement" when you claim that you have performed one.

 

[Fredrik]

That is actually quite easy. If you know
1) how A entangles with B when C is not present, and
2) how A entangles with C when B is not present,
then linearity alone is sufficient to determine how A entangles with both B and C when both are present.

I still don't understand what you have in mind. Are you referring to the linearity of the time evolution operator? We're not talking about having one time evolution operator act on two different states. We would have to deal with two different (and non-commuting) time evolution operators acting on one state. So linearity doesn't help.

I also don't see an obvious way to combine something like
$$\frac{1}{\sqrt 2}\big(|z+\rangle|z-\rangle|\sigma\rangle - |z-\rangle|z+\rangle|\sigma\rangle\big)$$ and
$$\frac{1}{\sqrt 2}\big(|x+\rangle|\rho\rangle|x-\rangle - |x-\rangle|\rho\rangle|x+\rangle\big)$$ into one state. Do you just add them and normalize? What about $|\sigma\rangle$ and $|\rho\rangle$? How do you choose them?

 

[atyy]

@Demystifier, I meant: can it work for accurate simultaneous joint measurements on arbitrary states? I agree this works for accurate measurements of a single observable. The entangling with a probe should be equivalent to the usual measurements which collapse the wave function. So if we measure position accurately it should collapse to a position eigenstate, and if we measure momentum it should collapse to a momentum eigenstate. If you try to measure both, it seems that you will measure something else.

 

[Ken G]

That's my issue as well, I think Demystifier's point is well taken that all measurements involve something that QM would regard as entanglement, so invoking EPR-type entanglement is not something completely unheard of. But we have to be careful about what we mean by a simultaneous measurement, and what we mean by simultaneous knowledge of two observables. I wonder if we can agree on the following things:

1) To be able to claim we have "simultaneous knowledge" that observables X and Y take on values x and y, to precision sigma(X) and sigma(Y), we are making the claim that if we next choose to do a very precise measurement of either X or Y, then we should get results within sigma(X) of x and within sigma(Y) of y. Is that not a reasonable meaning of simultaneous knowledge?

2) To be able to claim we have done a "simultaneous measurement" of X and Y, we must claim we are doing something that conveys simultaneous knowledge of X and Y.

 

[atyy]

That's my issue as well, I think Demystifier's point is well taken that all measurements involve something that QM would regard as entanglement, so invoking EPR-type entanglement is not something completely unheard of. But we have to be careful about what we mean by a simultaneous measurement, and what we mean by simultaneous knowledge of two observables. I wonder if we can agree on the following things:

1) To be able to claim we have "simultaneous knowledge" that observables X and Y take on values x and y, to precision sigma(X) and sigma(Y), we are making the claim that if we next choose to do a very precise measurement of either X or Y, then we should get results within sigma(X) of x and within sigma(Y) of y. Is that not a reasonable meaning of simultaneous knowledge?

2) To be able to claim we have done a "simultaneous measurement" of X and Y, we must claim we are doing something that conveys simultaneous knowledge of X and Y.

From my understanding, Ozawa's definition of an accurate measurement of A is simply that the distribution of measurement outcomes is the same as that when a projective measurement of A is performed on an ensemble of identically prepared particles in that state. So it depends on the textbook definition of accurate measurement that collapses the state into an eigenstate of the observable.

It seems that if we have some knowledge of the state, then we can tailor the measurement procedure for that state. However, these special procedures will not work on arbitrary states, so they cannot work for an an ensemble of particles in an unknown state. I think a special case is where one already knows the exact state, then a special procedure that accurately measures all observables simultaneously is to toss the state into the garbage, and just output measurement outcomes calculated using knowledge of the state and quantum mechanics.

 

[Demystifier]

We would have to deal with two different (and non-commuting) time evolution operators acting on one state. So linearity doesn't help.

You are right and I have to withdraw the statement that linearity is enough. In reality, you need to do the following. If the first measurement is achieved with the Hamiltonian H_1 and the second measurement with the Hamiltonian H_2, then the combined measurement is described by the evolution with the Hamiltonian H_12=H_1+H_2. The corresponding evolution operator U_12 is well defined, but different from both U_1 U_2 and U_2 U_1.

 

[Demystifier]

So if we measure position accurately it should collapse to a position eigenstate, and if we measure momentum it should collapse to a momentum eigenstate. If you try to measure both, it seems that you will measure something else.

You are right. As I already said, if you try to measure both it will collapse to a coherent state. That's why the repeated measurement will not give the same value of either position and momentum.

But if you insist that the only meaningful measurement is the one in which repeated measurement gives the same value, then what about a photon measurement which typically destroys the photon? Does it mean that photon measurements are not meaningful?

 

[Demystifier]

I wonder if we can agree on the following things:

1) To be able to claim we have "simultaneous knowledge" that observables X and Y take on values x and y, to precision sigma(X) and sigma(Y), we are making the claim that if we next choose to do a very precise measurement of either X or Y, then we should get results within sigma(X) of x and within sigma(Y) of y. Is that not a reasonable meaning of simultaneous knowledge?

2) To be able to claim we have done a "simultaneous measurement" of X and Y, we must claim we are doing something that conveys simultaneous knowledge of X and Y.

I cannot agree on 1) because, as I explained in the post above, it would imply that we cannot have a reasonable knowledge about photons.

Concerning 2), you tacitly assume that X or Y is real when we measure it. But the example of Bohmian mechanics teachs us that spins may never be real, even when we "measure" them.

This, indeed, is why Bell insisted that we should not talk about measurement (which, as we see, is a misleading concept) but about experiment.

 

[atyy]

You are right. As I already said, if you try to measure both it will collapse to a coherent state. That's why the repeated measurement will not give the same value of either position and momentum.

But if you insist that the only meaningful measurement is the one in which repeated measurement gives the same value, then what about a photon measurement which typically destroys the photon? Does it mean that photon measurements are not meaningful?

Yes, that's why repeated measurement is not the only accurate measurement, which opens the door to the acccurate measurements on a specific state that Ozawa (and others) talk about.

How's this:

An accurate measurement of A on any state is a procedure that gives measurement outcomes according to the Born rule for any state. If the state survives, it will be in an eigenstate of A with probability given by the Born rule.

There are other procedures that give accurate measurements of A on specific states in the sense that the distribution of outcomes is the same as that given by the Born rule. These procedures require some knowledge of the state, and don't work for any state. In these cases, if the state survives, it need not be in an eigenstate of A with probability given by the Born rule. For example, if the state is known, a state-dependent measurement is to toss the state into the garbage or leave it unchanged and simply write down measurement outcomes according to the Born rule.

 

[Ken G]

From my understanding, Ozawa's definition of an accurate measurement of A is simply that the distribution of measurement outcomes is the same as that when a projective measurement of A is performed on an ensemble of identically prepared particles in that state. So it depends on the textbook definition of accurate measurement that collapses the state into an eigenstate of the observable.

True, he uses measurement in the sense of a predictable outcome, not in the sense of conveying knowledge of an observable. We may have to recognize different types of measurement-- getting away from our classical prejudices that any measurement that represents a testable outcome also conveys knowledge of the observable!

It seems that if we have some knowledge of the state, then we can tailor the measurement procedure for that state. However, these special procedures will not work on arbitrary states, so they cannot work for an an ensemble of particles in an unknown state.

That's an important issue as well, but is a separate feature in the landscape. So it seems we really have three levels of measurement here:
1)general measurements: they convey knowledge of the observable on any state, without any prior knowledge,
2)specific measurements: they convey knowledge of the observable, but only if you already know something about the state, and
3)non-repeatable measurements: they do not convey knowledge of the observable in the sense defined above, but they do register a result on a pointer. Destructive measurements are of this type, but so are non-destructive measurements involving EPR entanglements that are broken by the measurement.

It seems to me that not only are Ozawa's theorems not about simultaneous measurements of type (1), they are not even of type (2)-- they invoke type (3)! Since the measurements are non-repeatable, they make it impossible to test if the two measurements are interfering with each other.

As as example, consider a device that sends out two photons in opposite directions with the same known wave packet, and in a momentum-conserving way, so the photons are entangled. We do a precise p measurement on one, and a precise x measurement on the other. The wave packet determines the distributions we get, so we have a precise measurement in Ozawa's sense. Do we have simultaneous knowledge of x and p of both photons, in the sense I defined above? No, we have a form of the HUP. Yet doesn't the entanglement say that we are doing a simultaneous x and p measurement for both photons? In the sense Ozawa means, apparently yes, but not in the sense of conveying knowledge of both those observables on both photons, even given our knowledge of the entangled state.

 

[Ken G]

I cannot agree on 1) because, as I explained in the post above, it would imply that we cannot have a reasonable knowledge about photons.

We can have reasonable knowledge of photons, but we have to recognize different types of knowledge. Usually we only use measurements to see if our predictions were right, so we don't care if the photon is destroyed or not. But we are only claiming "my theory worked", we are not claiming "I have knowledge that the photon is in state X." The latter is a very different kind of claim, and so we might have measurements that can support a prediction, yet not convey knowledge of the state of the photon. A state is a kind of preparation, so knowledge of a state must be knowledge about a preparation, not knowledge of a destruction.

Concerning 2), you tacitly assume that X or Y is real when we measure it. But the example of Bohmian mechanics teachs us that spins may never be real, even when we "measure" them.

To me, the only thing that is real is the knowledge, not the thing-in-itself.

This, indeed, is why Bell insisted that we should not talk about measurement (which, as we see, is a misleading concept) but about experiment.

Perhaps we can talk about both-- as long as we are clear what we mean.

 

[Ken G]

I think I've had an insight that will help with this. The problem seems to be that if we have two particles, we are wondering if there are really 4 things to precisely know there (two x and two p), or just 2 things to know there (an x or p from one, and an x or p from the other)? The HUP suggests the latter, but EPR entanglement might be used to make it seem like there actually are 4 things there we can know. This relates to the issue of what kind of experiment can impart "simultaneous knowledge" of x and p for both particles, if they are entangled.

I would argue that entanglement does not work to extend what we can simultaneously know about the state of two particles, for the simple reason that entanglement means that there are still only 2 things to know about those two particles. As when the particles are unentangled, we can choose between 4 precise measurements to do, but we can only do 2 measurements that can be reproduced immediately, so our ongoing knowledge of the system is restricted to 2 things. When the particles are entangled, this continues to be true-- so there are still only two things about that system we can obtain "simultaneous knowledge" of.

In other words, if we have two entangled particles that conserve total p=0, and we do a p measurement on one, we can predict precisely the p of the other. And if we instead do an x measurement on the other, we can say we know x and p of that particle, but actually what we measured was an x and p of the entangled system, so we still only know two things about a system that gives us only 2 things to know. If we want to say we know 4 things, then we have to treat the system as no longer entangled, which is true because we broke the entanglement, but now we cannot use the other p to get the p of a particle, we'd have to do a new p measurement on that no-longer-entangled particle. Bottom line: if one interprets one form of the HUP as saying that two particles only contain two pieces of precise x,p information that we could be in a position to predict, this continues to be true whether the two particles are entangled or not.

 

[atyy]

I think I've had an insight that will help with this. The problem seems to be that if we have two particles, we are wondering if there are really 4 things to precisely know there (two x and two p), or just 2 things to know there (an x or p from one, and an x or p from the other)? The HUP suggests the latter, but EPR entanglement might be used to make it seem like there actually are 4 things there we can know. This relates to the issue of what kind of experiment can impart "simultaneous knowledge" of x and p for both particles, if they are entangled.

I would argue that entanglement does not work to extend what we can simultaneously know about the state of two particles, for the simple reason that entanglement means that there are still only 2 things to know about those two particles. As when the particles are unentangled, we can choose between 4 precise measurements to do, but we can only do 2 measurements that can be reproduced immediately, so our ongoing knowledge of the system is restricted to 2 things. When the particles are entangled, this continues to be true-- so there are still only two things about that system we can obtain "simultaneous knowledge" of.

In other words, if we have two entangled particles that conserve total p=0, and we do a p measurement on one, we can predict precisely the p of the other. And if we instead do an x measurement on the other, we can say we know x and p of that particle, but actually what we measured was an x and p of the entangled system, so we still only know two things about a system that gives us only 2 things to know. If we want to say we know 4 things, then we have to treat the system as no longer entangled, which is true because we broke the entanglement, but now we cannot use the other p to get the p of a particle, we'd have to do a new p measurement on that no-longer-entangled particle. Bottom line: if one interprets one form of the HUP as saying that two particles only contain two pieces of precise x,p information that we could be in a position to predict, this continues to be true whether the two particles are entangled or not.

The way I've been thinking about measurements above doesn't have anything to do with knowledge or information. It's just a procedure to produce a distribution of outcomes. That's why I said, well if you know the state you can measure all observables simultaneously, but obviously you get no information.

But if one wants to use measurements to get information, then there is this notion of "no information without distrubance". I googled it, and this paper seems interesting: http://arxiv.org/abs/quant-ph/9512023v1 (10.1103/PhysRevA.53.2038).

 

[Fredrik]

http://arxiv.org/abs/0911.1147

And a claimed proof of "Theorem 14. In any Hilbert space with dimension more than 3, there are nowhere commuting observables that are simultaneously measurable in a state that is not an eigenstate of either observable." is provided on p10.

There's one detail in theorem 14 (and elsewhere) that stands out. The theorem you're quoting isn't saying that a simultaneous measurement of two non-commuting observables is possible, period. It's saying that there's a state such that if the particle is in that state, then a simultaneous measurement is possible.

I still haven't read enough to see the significance of this. The author has a very specific definition of terms like simultaneous measurement. I would have to study those definitions to know what impact his theorems have on my conjecture that "simultaneous measurements are possible if and only if the measuring devices can exist in the same place without interfering with each other".

 

[atyy]

There's one detail in theorem 14 (and elsewhere) that stands out. The theorem you're quoting isn't saying that a simultaneous measurement of two non-commuting observables is possible, period. It's saying that there's a state such that if the particle is in that state, then a simultaneous measurement is possible.

I still haven't read enough to see the significance of this. The author has a very specific definition of terms like simultaneous measurement. I would have to study those definitions to know what impact his theorems have on my conjecture that "simultaneous measurements are possible if and only if the measuring devices can exist in the same place without interfering with each other".

I think quite general relations for simultaneous measurements are given in http://arxiv.org/abs/1312.1857. Some of those results are shown graphically in Fig. 1 of http://arxiv.org/abs/1304.2071, which also has a useful appendix A1 that uses the basic idea you mentioned that if you attempt to measure jointly you really measure something else. Branciard's papers follow work by Arthurs and Kelly, and by Ozawa http://arxiv.org/abs/quant-ph/0310070. This paper by Ozawa shows in section III that an accurate simultaneous measurement of two observables is possible only if the two observables commute.

There are somewhat different relations if one uses different definitions of error http://arxiv.org/abs/1306.1565. This paper is about sequential measurements, but it's related since the Ozawa relation for sequential measurements http://arxiv.org/abs/quant-ph/0207121 is pretty similar for to the above-mentioned Ozawa relation for simultaneous measurements.

Apparently there is some controversy over whose definitions of error are better http://physicsworld.com/cws/article...y-reigns-over-heisenbergs-measurement-analogy.

If the probes in sequential measurements are correlated, it seems that one can do better than Ozawa's relation for sequential measurements http://arxiv.org/abs/1212.2815. There's a comment just after Eq 21: "Notice, that in the EPR thought-experiment [37] the measurements can be carried out simultaneously, violating the joint uncertainty principle. This is a consequence of the probes and the measured system being initially correlated, contrary to what is assumed herein ...".

 

[Demystifier]

So it seems we really have three levels of measurement here:
1)general measurements: they convey knowledge of the observable on any state, without any prior knowledge,
2)specific measurements: they convey knowledge of the observable, but only if you already know something about the state, and
3)non-repeatable measurements: they do not convey knowledge of the observable in the sense defined above, but they do register a result on a pointer. Destructive measurements are of this type, but so are non-destructive measurements involving EPR entanglements that are broken by the measurement.

Well summarized!

 

[Demystifier]

How's this:

An accurate measurement of A on any state is a procedure that gives measurement outcomes according to the Born rule for any state. If the state survives, it will be in an eigenstate of A with probability given by the Born rule.

The notion of state survival is not well defined. For example, even when the photon is destroyed, you can say that the state survives because you still have some state of quantum electrodynamics. A state with zero number of photons is still a state.

If we accept only the first requirement above (that outcomes need to be given by the Born rule), then "my" simultaneous measurement of non-commuting observables is "accurate".

 

[Fredrik]

If the state survives, it will be in an eigenstate of A with probability given by the Born rule.

The article by Ozawa that you linked to in #24 says this:

In fact, it is widely accepted nowadays that any observable can be measured correctly without leaving the object in an eigenstate of the measured observable; for instance, a projection $E$ can be correctly measured in a state $\psi$ with the outcome being 1 leaving the object in the state $M\psi/\|M\psi\|$, where the operator $M$ depends on the apparatus and satisfies $E=M^\dagger M$ (see, for example, a widely accepted textbook by Nielsen and Chuang [10]).
​

I'm going to have to read up on this, because I have no idea what he's talking about.

I see that you're also saying this:

There are other procedures that give accurate measurements of A on specific states in the sense that the distribution of outcomes is the same as that given by the Born rule. These procedures require some knowledge of the state, and don't work for any state. In these cases, if the state survives, it need not be in an eigenstate of A with probability given by the Born rule.

Maybe he's just talking about that. (He doesn't say if his $\psi$ is arbitrary). I will think about it.

 

[Fredrik]

The notion of state survival is not well defined. For example, even when the photon is destroyed, you can say that the state survives because you still have some state of quantum electrodynamics. A state with zero number of photons is still a state.

Good point, but I would choose to call state survival "theory-dependent" (as in "it depends on whether you're using a theory with a fixed number of particles"), rather than "ill-defined".

If we accept only the first requirement above (that outcomes need to be given by the Born rule), then "my" simultaneous measurement of non-commuting observables is "accurate".

Then how do you measure $S_z$ and $S_x$ (in the quantum theory of a single spin-1/2 particle with a magnetic moment) simultaneously? You can't put two Stern-Gerlach devices in the same place, because (if they each have a detector screen) the first one you put there physically prevents you from putting another one there. I suppose that a Stern-Gerlach device would still be considered a Stern-Gerlach device if we replace the detector screen with two small detectors at the appropriate locations. Then you could actually put two in devices in the same place. Now you have four detectors but they will never signal detection because the particles will miss them all. So two measuring devices in the same place equals no measuring device at all. You could also try to combine two Stern-Gerlach devices into one, by using one detector screen and the magnets from both, but this combination device would (presumably, because I haven't done the math) be a measuring device that's suitable for a measurement of $(S_z+S_x)/\sqrt 2$. You could interpret the position of the dot as a simultaneous measurement of $S_z$ and $S_x$, but then the results wouldn't be consistent with the Born rule (imagine doing a measurement on a particle prepared in an eigenstate of $S_x$).

Pretend that you know nothing about quantum theory, and just use two sophisticated gadgets for which you were told that they measure position and momentum. You don't even need to know how the gadgets work. All you need to know is how to use them, by pressing appropriate buttons. When you do that, the displays on the gadgets show some digital numbers which, you are told, are the measured position and momentum.

So when you turn on both gadgets at the same time, what do you expect to see? Do you expect that only one of the gadgets will display a number? If so, then which one?

No, you should not expect such a thing. There is no doubt that both gadgets will show some numbers. As long as you think like an experimentalist without any theoretical prejudices, there is nothing more natural than to interpret these two numbers as simultaneous measurement of position and momentum. That is all.

My comments about Stern-Gerlach devices apply to this as well. I described a scenario in which the gadgets wouldn't show any numbers at all, so I would say that there's plenty of doubt. I also described a scenario where "numbers" are shown, but it's not at all natural to interpret them as results of simultaneous measurements of non-commuting observables.

 

[Fredrik]

I think a few general comments are in order. A theory of physics can't be defined by mathematics alone. We also need correspondence rules, i.e. statements that tell us how to interpret the mathematics as predictions about results of experiments. The correspondence rules must describe the devices that we're supposed to use to test the predictions. (It is possible to do this without using the theory, but this isn't the topic of this post, so I'll put the explanation in a footnote*).

If we try to put two measuring devices that are represented by non-commuting self-adjoint operators A and B at the same place, we're typically going to find one of the following:

1. It's physically impossible to put them at the same place.
2. The result is a single device that isn't described by any correspondence rules.
3. The result is a single device that according to the correspondence rules is represented by a self-adjoint operator C, different from both A and B.

I haven't thought this through to the point where I can say that these are the only options, but at the moment, it seems to me that they are. Option 2 has a couple of sub-options: If you add a new correspondence rule that describes the device and tells you to use it for simultaneous measurements of A and B, will this make the theory worse?

Consider two compatible observables instead, like position components. You don't make a simultaneous measurement of position components by designing a device that only measures one component, and then using several such devices in the same place. I don't think it's even possible to measure only one component. The closest you can get is probably to measure one component more accurately than the others. If this is what we mean by only measuring one component, I don't think two measuring devices with different orientation can exist in the same place.

What you actually do to measure position is to use a particle detector. If it signals detection, then all components of position have been measured simultaneously. This makes me suspect that we need to think about simultaneous measurements of incompatible observables in the same way. It's not about putting two devices in the same place. It's about the possibility to build a new device that can be said to be measuring both observables simultaneously, in the sense that if we add a new correspondence rule to the theory, one that specifies the two observables and the situation in which the device is able to measure the observables simultaneously, we're not making the theory worse.

Ozawa seems to have found a theorem that tells us that if A and B are non-commuting observables, there's a state $\psi$ such that if the system is prepared in that state, then we can "simultaneously measure" A and B. (See post #24, by atyy). But Ozawa's definition of "simultaneously measurable in state $\psi$" is purely mathematical, as it must be, since the term was meant to be used in a theorem. This means that we can't be sure that it's possible to build a measuring device that does the measurement just because two incompatible observables are "simultaneously measurable in state $\psi$" in the sense of Ozawa's definition.

I'm inclined to say that if it's not possible to build the device, then Ozawa's definition of "simultaneously measurable" is inappropriate and misleading. But if is possible to build the device (and make a new correspondence rule about it without making the theory worse), I would say that his definition is perfectly appropriate.

I don't think you can prove that the device can always be built, so my impression is that what Ozawa's theorem is really telling us is just this: Every time we figure out a way to build a device that can be used to do a (state-dependent) simultaneous measurement of two incompatible observables, add a new correspondence rule for this device, and verify (with experiments) that this hasn't made the theory worse, it will strengthen our opinion that his definition of "simultaneously measurable in state $\psi$" is appropriate. And every time we can't think of a way to do it, it will weaken this opinion. If we find a compelling argument against the possibility that the device can be built, then we will reject his definition.*) I have thought of a procedure that can at least in principle be used to ensure that the correspondence rules can be described without using the theory that they're a part of. You need to build a hierarchy of theories. In the level-0 theories, measuring devices are so simple that no theory is needed to describe how to build them. For example an hourglass for time measurements and a rope with knots for length measurements. For each positive integer n, the measuring devices in a level-n theory are described by assembly instructions that can be understood and carried out by someone who understands level n-1 theories and has access to level n-1 measuring devices.

 

[Demystifier]

Then how do you measure $S_z$ and $S_x$ (in the quantum theory of a single spin-1/2 particle with a magnetic moment) simultaneously?

As I explained in #29, I need at least 3 particles, the two of which can be thought of as "micro-detectors". So if you insist that I must use only one particle, then I cannot do that.

 

[Demystifier]

I would have to study those definitions to know what impact his theorems have on my conjecture that "simultaneous measurements are possible if and only if the measuring devices can exist in the same place without interfering with each other".

Why do you insist that simultaneous measurements are made in the same place?

In my three-particle method of simultaneous measurements in #29, the three particles are at the same place at the time of their mutual interaction, but at different places at the time of detection. Do you clasify this as being at the same place?

 

[atyy]

The notion of state survival is not well defined. For example, even when the photon is destroyed, you can say that the state survives because you still have some state of quantum electrodynamics. A state with zero number of photons is still a state.

If we accept only the first requirement above (that outcomes need to be given by the Born rule), then "my" simultaneous measurement of non-commuting observables is "accurate".

Yes, for the collapse rule more generally I should use that for a POVM, so that I can treat the collapse to zero photons.

But I'm still skeptical that you can measure accurately for arbitrary state.

For simultaneous measurements, why doesn't the impossibility proof given in section III of http://arxiv.org/abs/quant-ph/0310070 apply?

For sequential measurements, why don't the inequalities in http://arxiv.org/abs/1304.2071 (Eq 12) or http://arxiv.org/abs/1211.4169 or http://arxiv.org/abs/1306.1565 apply?

It seems that at best, one can make an accurate sequential measurement for some states.

 

[Fredrik]

Why do you insist that simultaneous measurements are made in the same place?

In my three-particle method of simultaneous measurements in #29, the three particles are at the same place at the time of their mutual interaction, but at different places at the time of detection. Do you clasify this as being at the same place?

I don't think this three-particle trick works, at least not in this case. I haven't been able to find a state vector with the appropriate entanglement. When I tried to write one down, I failed in ways that made me suspect that no such state vector exists. It seems to me that we need a sum of terms of the form $|x\pm,x\mp,\sigma\rangle$ and $|y\pm,\rho,y\mp\rangle$. But no matter what we choose $\sigma$ to be, a measurement of $S_y$ on particle C that yields +1/2 will not tell us that particle A is in state y-.

This may however be a problem that's specific to systems with 2-dimensional Hilbert spaces. I haven't tried to work out any other case. If the three-particle trick works in in other cases, then I agree that it's not true in general that simultaneous measurements are done at the same place.

 

[atyy]

The article by Ozawa that you linked to in #24 says this:

In fact, it is widely accepted nowadays that any observable can be measured correctly without leaving the object in an eigenstate of the measured observable; for instance, a projection $E$ can be correctly measured in a state $\psi$ with the outcome being 1 leaving the object in the state $M\psi/\|M\psi\|$, where the operator $M$ depends on the apparatus and satisfies $E=M^\dagger M$ (see, for example, a widely accepted textbook by Nielsen and Chuang [10]).
​

I'm going to have to read up on this, because I have no idea what he's talking about.

These are given in http://arxiv.org/abs/1110.6815 (p9, statements [II.1]-[II.5]). These axioms can deal with the case that Demystifier mentions about zero photons (p13, R2).

Maybe he's just talking about that. (He doesn't say if his $\psi$ is arbitrary). I will think about it.

Ozawa seems to have found a theorem that tells us that if A and B are non-commuting observables, there's a state $\psi$ such that if the system is prepared in that state, then we can "simultaneously measure" A and B. (See post #24, by atyy). But Ozawa's definition of "simultaneously measurable in state $\psi$" is purely mathematical, as it must be, since the term was meant to be used in a theorem. This means that we can't be sure that it's possible to build a measuring device that does the measurement just because two incompatible observables are "simultaneously measurable in state $\psi$" in the sense of Ozawa's definition.

I'm inclined to say that if it's not possible to build the device, then Ozawa's definition of "simultaneously measurable" is inappropriate and misleading. But if is possible to build the device (and make a new correspondence rule about it without making the theory worse), I would say that his definition is perfectly appropriate.

I don't think you can prove that the device can always be built, so my impression is that what Ozawa's theorem is really telling us is just this: Every time we figure out a way to build a device that can be used to do a (state-dependent) simultaneous measurement of two incompatible observables, add a new correspondence rule for this device, and verify (with experiments) that this hasn't made the theory worse, it will strengthen our opinion that his definition of "simultaneously measurable in state $\psi$" is appropriate. And every time we can't think of a way to do it, it will weaken this opinion. If we find a compelling argument against the possibility that the device can be built, then we will reject his definition.

My understanding is that these statements about the possibility of simultaneous or sequential measurements being possible only apply for very special states.

For example, sequential measurement of non-commuting observables A and B is possible if the state is an eigenstate of A, because measuring A leaves the state undisturbed, so that B can be measured accurately on the same state. But this procedure is very bad in general, since measuring A will cause the state to collapse to a completely different state, so that the subsequent "accurate" B measurement will be very inaccurate because it is carried out on the wrong state. In general, a more accurate procedure for all states is one that is not perfectly accurate for any state, but slightly inaccurate for all states.

The case of simultaneous accurate measurement that Ozawa discusses is basically an EPR-type argument that if the particles are appropriately entangled, a simultaneous measurement of A on particle 1 and B on particle 2 can give you an accurate measurement of B on particle 1. As he says, this is again a special case, and the procedure will fail for any other state.

 

[Demystifier]

I don't think this three-particle trick works, at least not in this case. I haven't been able to find a state vector with the appropriate entanglement. When I tried to write one down, I failed in ways that made me suspect that no such state vector exists. It seems to me that we need a sum of terms of the form $|x\pm,x\mp,\sigma\rangle$ and $|y\pm,\rho,y\mp\rangle$. But no matter what we choose $\sigma$ to be, a measurement of $S_y$ on particle C that yields +1/2 will not tell us that particle A is in state y-.

I understand your point, and mathematically you are right. However, there is a conceptual subtlety, which is best conveyed as a counter question: If you measure only one observable, say S_y, then what kind of measurement DOES tell you that the particle A is in state y?

If you try to answer it, you will see that your answer may be questioned by a different interpretation of QM. That's why, in a kind of a minimal interpretation, it may not be useful to think of a particle being in a state y. Instead, it may be more useful to concentrate only on the classical states of macroscopic pointers. It is only in this MACROSCOPIC CLASSICAL language that my three-particle trick makes sense.

 

[Demystifier]

But I'm still skeptical that you can measure accurately for arbitrary state.

For simultaneous measurements, why doesn't the impossibility proof given in section III of http://arxiv.org/abs/quant-ph/0310070 apply?

For sequential measurements, why don't the inequalities in http://arxiv.org/abs/1304.2071 (Eq 12) or http://arxiv.org/abs/1211.4169 or http://arxiv.org/abs/1306.1565 apply?

Different conclusions concerning whether something can or cannot be measured accurately depend on different DEFINITIONS of "accurate measurement" one adopts. In classical mechanics it is quite clear what an accurate measurement is, but in QM it is not. Essentially, that's because in QM it is not clear what is the reality or ontology the accurate measurement is supposed to be about. In other words, whether something can or cannot be measured accurately depends on the interpretation of QM.

For example, in one interpretation of QM, measurements (either accurate or inaccurate) do not exist at all. That is, experiments do not "measure" reality, but create it.

Of course, nobody is obligued to accept such an interpretation, but then one needs to carefully explain what interpretation one does adopt, and according to it, what exactly one means by a "measurement".

 

[atyy]

Different conclusions concerning whether something can or cannot be measured accurately depend on different DEFINITIONS of "accurate measurement" one adopts. In classical mechanics it is quite clear what an accurate measurement is, but in QM it is not. Essentially, that's because in QM it is not clear what is the reality or ontology the accurate measurement is supposed to be about. In other words, whether something can or cannot be measured accurately depends on the interpretation of QM.

Yes, what definition are you using? If I understand correctly, these papers define an accurate measurement of A as one that produces the same distribution of outcomes as a projective measurement of A.