During recent years, there is increasing interest in the ontological status and meaning of the wave function, and it seems that there is even a shift in research focus from the measurement problem to the problem of interpreting the wave function. This motivates us to organize an online workshop on the meaning of the wave function. This group aims to address the controversies surrounding the different viewpoints (Bayesian, epistemic, nomological, ontic, etc).
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Why protective measurement does not establish the reality of the wave function
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At Shan’s suggestion I have switched to a text-based talk, since they seem to be working better. The attached notes (no paper yet, sorry!) contain roughly what I intended to say. In short, the claim is that thinking carefully about the role of the protector in protective measurement, both operationally and in simple ontological toy models, undermindes arguments from protective measurement to the reality of the wave function.
I know that several participants have been thinking about the implications of protective measurement and so I welcome comments and questions.
Dear Matts,
I just got up. I am reading your interesting paper, and trying to understand your ideas. I will respond soon.
Shan
A very short note: I have read to 3.1. I noticed that 2.2 and 3.1 may be problematic. The measurement you discussed is not precisely PM.
Dear Matt,
As you know, as a consistent historian I stand a bit outside the controversy,
since I have what I consider a consistent quantum ontology into which the wave
function enters in a specific way. However, I would like to make the following
comment. Consider the situation you arrive at in Sec. 3 where Bob knows the
basis and makes a projective measurement of the state sent him by Alice.
Assuming Bob to be a competent experimentalist, when he gets a particular
measurement outcome i he is justified in concluding that before the measurement
took place the particle (or whatever) was in the ontological state |psi_i>. I
would not say this establishes the reality of the wave function, but it points
to a situation in which if you have a fully consistent interpretation of
quantum mechanics it is appropriate to treat this |psi> as a quantum property.
I add, to avoid confusion, that my ontology is based on Hilbert subspaces, not
hidden variables. Bob GriffithsHi Bob,
Thanks for your comment. Just to be clear about something I didn’t make explicit in the notes, I’m certainly not trying to argue that protective measurement is incompatible with the reality of the wave-function, and indeed in interpretations in which the wave-function is (always or sometimes) real it may well be a perfectly good method for determining it. In some interpretations, and I take your point to be that yours is one such, the measurement could even be the thing that ‘brings about’ the reality of the wave-function.
The claim is simply that protective measurement itself (outside of a specific interpretation) does not provide support for the notion that a correct understanding of quantum mechanics must require the wave-function to be real.
Yours,
MattHi Matt,
Thanks for your intriguing contribution. I just managed a cursory read this afternoon, and I will still have to take a closer look, and unfortunately now is not a good time (the family is calling to dinner). But I’ll get back to it later. And I look forward to seeing what the other participants will have to say.
Best,
MaxDear Matt,
I think your argument is valid. But I notice that, (I think Max will also agree with me), the measurement you discussed is not exactly the protective measurement, but a series of weak measurements (plus projective measurements) on one of a known set of orthogonal bases.
For an (adiabatic) protective measurements, the only information which is needed is that the system is in a nondegenerate eigenstate of its Hamiltonian (and a
lower bound of the energy gaps to other levels, which is enough to fix the strength and the adiabaticity of the measuring coupling), while the Hamiltonian and the state may be unknown. An example is a trapped atom, where the potential may not be known beforehand, but one does know that after a sufficiently long time the atom is to be found in the ground state.So, in my opinion, your analysis does not argue against the possible implications of PM for the ontological status of the wave function.
Shan
PS. There is an essential difference between a weak measurement and a protective measurement. The former, as you noted, can only provide an extremely inaccurate estimate of the expectation value of the measured observable, but the latter can obtain the expectation value of the measured observable with arbitrary accuracy.
To be clear, there are two versions of protective measurement discussed in the literature. The “Zeno” version in which weak measurements are intersperesed with projective measurements, as discussed in Matt’s notes, and the adiabatic version referred to by Shan. For the adiabatic version, we do have a model of how a Gaussian state, e.g. the ground state of a Harmonic oscillator, can be protectively measured without implying the reality of the wavefunction, but that is not discussed in the notes. Matt may wish to outline how this works.
Dear Shan,
Relative to your #1028, I am inclined to think that Matt’s formulation is pretty close to one of the things Vaidman put in his Compendium article, using repeated measurements and the Zeno effect to do the protection. It seems to me it isn’t all that different. And I might add that I thought the Bob-Charlie exchange being turned into a channel was an interesting way to look at the problem
Bob Griffiths
I think it is also probably possible to derive some sort of mathematical equivalence between the two types of protective measurement, i.e. to convert an adiabatic scheme into a Zeno scheme and vice versa. This may have been discussed in the literature but I am not sure. If so, it would probably be possible to adapt the Alice/Bob/Charlie story to the adiabatic case also.
Hi Shan,
Having returned to one of the original papers, I can see that you’re right that my “recap” of protective measurement does not quite agree with the original scheme, in which the “protecting” measurements are done during Bob’s measurement rather than only between them. Perhaps I heard about this scheme elsewhere and somehow I confused it with the original. My apologies for the confusion. Section 4 probably needs some revision in order to provide an analogy to the original scheme.
All is not lost, however. The argument in Section 3 only relies on the resources available to Bob (1 system prepared in the state + the protecting channel), which still match the initial scheme.
Futhermore, as the other Matt has already alluded to, a Section 4-style argument is actually easier and more complete in the case of the protection-by-time-evolution scheme. If the Hamiltonian is that of the simple harmonic oscillator, and the state is the ground state, then the entire protective measurement scheme can be carried out within Gaussian quantum mechanics, which admits a simple ψ-epistemic interpretation.
Yours,
MattDear Matt,
This may be a bit off the subject, but you said you were asking for
comments. I haven’t had time to work through the toy model in your
section 4, which you say is based on work by Spekkens and
collaborators, including Bartlett, Rudolph, and Spekkens,
“Reconstruction of Gaussian quantum mechanics from Liouville mechanics
with an epistemic restriction”, Phys. Rev. A 86, 012103. The comment
is that I published a paper in Phys. Rev. A 88, 042122 (2013) in which
I noted that what Bartlett et al. did using a toy model could be
complemented by a fully quantum-mechanical approach based on
histories. Whether something similar might be a useful complement to
your toy model in section 4 I cannot say, but I toss the idea into the
discussion for whatever it’s worth.Hi Matt,
Thank you very much for your clarification!
I think the measurement in Matt’s notes is not exactly the Zeno-type PM either. There is at least one difference as far as I can see. In the Zeno version, there are indeed frequent projective measurements to protect the measured state. But the measurement of the studied observable is usually a strong measurement. Moreover, the frequency of the projective measurements is required to be very large to ensure the accuracy of the obtained expectation value of the measured observable. These features do not appear in your scheme of measurements.
I will think more about your scheme.
Shan
Hi Matt (L),
Relative to #1033, I think it is an interesting idea, which has not been discussed in the literature. That will require to transform the time parameter in the adiabatic scheme into the frequency parameter in the Zeno scheme.
Shan
Thinking on the spot now, is there actually any difference between a small time-slice of a von-Neumann type strong measurement and a weak measurement? Weak measurements are normally obtained by reducing the interaction strength, but wouldn’t interacting for a shorter time amount to exactly the same thing?
If there is no difference, then my “recap” was just a different way of talking about the original scheme rather than a fundamentally different one.
Yours,
MattHi Matt (P),
Thanks for your kind reply!
My point is still that the resources provided by your protecting channel is not exactly those provide by a PM. An adiabatic PM can provide more resources (when the protection based on energy conservation is ensured, e.g. when a
lower bound of the energy gaps to other levels is known beforehand). These resources will allow the whole spatial wave function of the system to be measured points by points.Shan
Shan #1042,
We should be able to reach agreement at least on this narrow point: Thinking of the protection-by-measurement (aka Zeno) scheme, does the protection amount to repeated applications of the channel given by eq. (1) in my notes?
(How this compares to the protection in the Hamiltonian-based scheme is probably a question for another day. If it provides even more, that would only strengthen my argument, since I’m saying that the information Bob obtains from Charlie’s repeated application of (1) is already sufficient to undermine the argument to the reality of the wave-function.)
Regards,
MattDear Shan,
I am a bit bothered by your idea that measuring the wave function
piece by piece is evidence that it is “real”. Imagine a classical
particle whizzing around in some sort of confining potential, moving
so fast that it looks like a cloud. You send slow projectiles through
the cloud and get is density at different points. But you are just
measuring a probability distribution, and for those of us who don’t
think probability distributions are real, the fact that you can
measure it does not make it more real.Bob Griffiths
Hi Matt,
Relative to #1041, I now think there are no essential difference, and yours may be regarded as an interesting extension of the original Zeno PM.
My another point is that even if your alternative protocol does not establish the reality of the wave-function, this does not necessarily imply that the original PM cannot do that.
Moreover, your claim “But the only POVM that can perfectly distinguish a set of basis states is the projective measurement in that basis” seems not right for PMs where we know the measured wave function beforehand.
Also, I think an adiabatic PM can obtain more information than logN bits from Alice’s system. One may not know the Hamiltonian and the wave function of the system beforehand, but one can obtain the values of the wave function in the whole space by PMs.
No doubt, I will need to think more deeply about your interesting ideas. I will get back to you when I have any other comments. Thanks again!
Best,
ShanBob #1045,
You might be interested to know that your example is pretty much exactly what protective measurement amounts to when carried out within the Bartlett, Rudolph, Spekkens model.
Yours,
MattShan #1046,
The claims you refer to apply only when one considers the totality of Bob and Charlie’s actions as a measurement procedure on the system from Alice. (Imagine putting Bob and Charlie in a huge black box, that has an input for the quantum system and a classical output of Bob’s estimate of the state.)
Since an arbitrarily sequence of quantum operations that takes in a quantum system and gives out a classical outcome can always be represented as a POVM, the facts about POVMs that I used are applicable regardless of how elaborate the implementation (the inside of the black box) may be.
Regards,
MattHi Bob,
Thanks for your further comments! I fully agree with you. Yes, what PMs measures are just the density and flux density. The wave function is then constructed from these. My conclusion from PMs is that the wave function describes a property of a single quantum system. But which property it describes is a further problem. I have even proposed an interpretation of the wave function in terms of ergodic motion of particles, which is quite like your picture (see http://philsci-archive.pitt.edu/10659/ or relevant papers).
Best,
ShanHi Max,
why not say something about Matt’s ideas. Please join in! We may extend our discussions if you have time.
Shan
I’m also done for today, but I’ll keep on eye on this tomorrow in case there are further thoughts.
Thanks to everyone for the stimulating responses that I think will, in the true spirit of a workshop, lead to improvements in the paper whenever it finally appears!
Matt
Matt #1048:
I think there is an essential difference between we know nothing about the measured system and we know something about it.
If we know nothing about the measured system, then we cannot surely measure the wave function of a single system, and we cannot distinguish nonorthogonal states either, while if we know something about it, we may have enough information to do these by PMs.
I think this may influence your claim in 3.2.
Best,
ShanHi all,
Just back from dinner, sorry to be joining so late.
Since you’ve been discussing the two different implementations of a PM (adiabatic and Zeno), here’s a question for Shan. Do you think a Zeno-type PM is equally indicative of the reality of the wave function as an adiabatic PM? If yes, why? If not, what do you see as the differences?
(I think there are conceptual differences that may have interpretive consequences — even though one might be able to show a formal equivalence of the two schemes, as Matt L has suggested.)
Best,
MaxTo Shan #1028:
An example is a trapped atom, where the potential may not be known beforehand, but one does know that after a sufficiently long time the atom is to be found in the ground state.
At the risk of beating this horse to death: I would rephrase this to read “one does know that after a sufficiently long time the atom is to be found in the ground state with high probability.”
My point is that once again there is an irreducible indeterministic element. You can never know for sure if the system is actually in an eigenstate of the Hamiltonian (which is what you need for a PM to proceed). That is, of course, unless you have actively prepared it in such an eigenstate, but then you know the state from the outset and there’s no need for a protective measurement anyway. Or what am I missing here?
Best,
MaxHi Max,
Thanks for your comments! I just noticed them. I am reading Matt’s notes about the toy model of PM.
I think your above worry in #1056 is right, which is still related to the limit problem we discussed in your presentation. I think I have two solutions: one is the new criterion, and the other is my PBR-like argument.
Yes, I still think a Zeno-type PM is equally indicative of the reality of the wave function as an adiabatic PM. I have argued that knowing the wave function beforehand does not influence my argument. The main reason, as you may know, is as follows:
The wave function is only a mathematical object associated with the prepared physical system, and we need to determine whether it refers to the physical state of the system or to the state of an ensemble of identically prepared systems. In this sense, although the wave function is known, the physical state of the system is still unknown. Thus, precisely speaking, what the protective measurements measure is not the known wave function, but the unknown physical state, which turns out to be represented by the wave function.
Best,
ShanHi Shan,
Thank you for your reply, that’s helpful.
I will need to take a closer look at Matt’s notes myself. Tomorrow!
Best,
MaxHi Matt,
After I have re-read your notes, I still think that your scheme is different from the PM scheme in either form (Zeno or adiabatic). Your scheme is equivalent to weak measurements of an ensemble of identically prepared systems, as Bob noted. So, it might be not surprising that one cannot argue for the reality of the wave function based on the scheme.
In your scheme, one get many inaccurate expectation values, from which the right one is derived.
In a PM, one directly get the right expectation value with arbitrary accuracy.
I would like to know your further response.
Best,
ShanThanks, Matt, for that very clear piece! I don’t think I have much to add or contribute, although I will say that your arguments helped me reframe how I think about protective measurements, and as it happens it’s quite similar to how I’ve recently been framing weak measurements, with one key difference. Have you looked at weak measurements in either of the contexts explore here?
Shan #1053,
What we do or don’t know has no bearing on which POVMs exist. Of course it may affect our choice of POVM – if we already know what basis the system was prepared in, we can measure it in that basis and determine the correct state. I don’t think anybody would argue that this establishes the reality of the wave-function. The claim is that, because of the limits on which POVMs exist, when you put Bob and Charlie into a black box their actions must amount to exactly the same measurement on the initial system (a projective measurement in the protected basis).
Yours,
MattShan #1070,
We definitely need to think more about which schemes are or are not equivalent to the original ones. (There is also a question of what equivalence means exactly – for example if one scheme requires classical post-processing of the data whilst another does the same processing “as it goes along”, does that necessarily mean they are not equivalent in the sense that matters?)
But I still don’t see how this affects are operational argument, which is independent of the details of Bob’s strategy – we only need to to be right about what Charlie (the protector) does.
Regards,
MattKen #1074,
I’m afraid I don’t quite get your question. Which contexts are you talking about?
Cheers,
MattI meant the contexts from section 3 and section 4, but I see there’s quite a lot of work done on the operational aspects of weak measurements already, so how about we focus on section 4. Is there an analog that tells us something about general weak measurements in the Spekkens toy model, for instance? ( I’m particularly interested in understanding the imaginary part of the weak value, if you have any insights on that front.)
Ken #1081,
The best toy model for thinking about weak measurements is the Gaussian theory, because then you already have continuous variables to act as your pointer, the pointer can be prepared in a Gaussian state, and the “von-Neumann measurement” interaction is present in the theory.
A nice example for imaginary weak values is to prepare the system in a Gaussian centred at the origin, do a weak measurement of position, and then post-select on momentum p. The weak value goes like i p. If you look at this in the toy theory, there’s a natural explanation for why that value manifests itself in the way it does.
Regards,
MattThanks, Matt! If there’s anything that’s been written up on weak values in the Gaussian theory, please give me a pointer to it. I don’t recall anything like that in the original paper, but it’s been awhile since I’ve looked at it..
Matt #1079:
“But I still don’t see how this affects are operational argument, which is independent of the details of Bob’s strategy – we only need to to be right about what Charlie (the protector) does.”
Yes, this is the crux; what Charlie (the protector) does is not a PM. For example, in your scheme, one get many inaccurate expectation values, from which the right one is derived. But in a PM, one directly get the right expectation value with arbitrary accuracy.
SO, I think your argument is indeed valid, but your conclusion is not.
Best,
ShanShan #1089,
Getting inaccurate expectation values is something that may or may not happen to Bob. Charlie just sits there doing projective measurements in a fixed basis over and over again, right?
Regards,
Matt
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