Meaning of the Wave Function

[Quantum Speculations]

The ontological status of the wave function in quantum mechanics is usually analyzed in the context of conventional impulsive measurements. These analyses are always based on some nontrivial assumptions, e.g. a preparation independence assumption is needed to prove the PBR theorem. In this talk, we will point out that the reality of the wave function can be argued without resorting to nontrivial assumptions by analyzing protective measurements, by which one can measure the expectation values of observables on a single quantum system. The existing objections to this argument will be answered. Moreover, we will also give a PBR-like argument for the reality of the wave function in terms of protective measurements.

Notes on the reality of the wave function

[Quantum Speculations]

A long-standing question in QF is whether the wave function relates only to an ensemble of identically prepared systems or directly to the state of a single system.

Recently, Pusey, Barrett and Rudolph demonstrated that under an appropriate assumption, the wave function is a representation of the physical state of a single quantum system (Pusey, Barrett and Rudolph 2012).

This poses a further interesting question, namely whether the reality of the wave function can be argued without resorting to nontrivial assumptions (cf. Lewis et al 2012; Colbeck and Renner 2012; Schlosshauer and Fine 2012, 2013; Leifer and Maroney 2013; etc).

In this presentation, I will argue that protective measurements, by which one can measure the expectation values of observables on a single quantum system (Aharonov and Vaidman 1993; Aharonov, Anandan and Vaidman 1993), already provide such an argument.

[Quantum Speculations]

Protective measurements

During a protective measurement, the measured state is protected by an appropriate procedure (e.g. via the quantum Zeno effect) so that it neither changes nor becomes entangled with the state of the measuring device appreciably.

In this way, such protective measurements can measure the expectation values of observables on a single quantum system, even if the system is initially not in an eigenstate of the measured observable.

Moreover, the whole wave function of the system can also be measured as expectation values of certain observables in principle.

[Quantum Speculations]

A PBR-like argument for psi-ontology in terms of protective measurements

For two arbitrary (protected) nonorthogonal states of a quantum system, select an observable whose expectation values in these two states are different. Then the overlap of the probability distributions of the results of protective measurements of the observable on these two states can be arbitrarily close to zero (e.g. when the measurement interval T approaches infinity for adiabatic measurements).

If there exists a non-zero probability p that these two (known) nonorthogonal states correspond to the same physical state λ, then when assuming the same λ yields the same probability distribution of measurement results as the PBR theorem assumes, the overlap of the probability distributions of the results of protective measurements of the above observable on these two states will be not smaller than p. Since p is a determinate number, this leads to a contradiction.

This argument only considers a single quantum system, and avoids the nontrivial assumptions used by the PBR theorem and improved theorems.

[Quantum Speculations]

Further clarifications of the PBR-like argument

1. The above protective measurements on the two protected nonorthogonal states are the same. (The protection can be regarded as one part of the state preparation procedure.)

2. Although a realistic (adiabatic) protective measurement with finite measurement time T can never be performed on a single quantum system with absolute certainty, the uncertainty can be made arbitrarily small when the measurement time T is arbitrarily long according to QM.

3. Protective measurements generally require the measured states be known. But knowing the wave function beforehand is not a weak point in my argument.

The wave function is only a mathematical object associated with the prepared physical system. Although the wave function is known, the physical state of the system is still unknown. Thus, strictly 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.

[Maximilian Schlosshauer]

Hi Shan,

Unfortunately your talk will be during a time slot when I will have my two young kids to look after. So I will not be able to chime in in real time. But I’ll be sure to check back later.

Also, I’m thinking that some of the points about the foundational implications of protective measurement we have talked about during Matt Pusey’s talk and my talk might be pertinent to today’s discussion.

Have fun!

Best,
Max

[Ken Wharton]

Hi Shan,

Apologies that I can’t participate at the scheduled time of your chat, but I have two questions that I can throw out now, and that we might discuss afterwards.

1) When you apply your revised ‘reality criterion’, you’re talking about predicting “with probability arbitrarily close to unity the value of a physical quantity”. But in this case those very “quantities” are themselves probabilities. Do you think this changes the strength of your conclusions here? (If one rewrote this reality criterion in terms of accurately predicting a (non-unity) probability distribution, do you think that would undermine the validity or usefulness of this criterion, vis-a-vis “reality”?)

2) If your argument goes through, doesn’t it also establish the reality of multi-particle entangled states? You mentioned earlier that you share my goal of trying to find an ontology that would fit back in spacetime, and so I would think you wouldn’t also be arguing that these configuration-space-states are somehow ‘real’… If not, where does the argument break down for multi-particle states?

Best,

Ken

[Quantum Speculations]

Hi Max and Ken,

Thanks for your comments! As to Ken’s question #1, I think the quantities measured by PMs are not probabilities. They are expectation values of observables.

As to Ken’s second question: yes, the argument, if valid, also establishes the reality of multi-particle entangled states. But this does not imply ehese configuration-space-states are somehow ‘real’, such as according to Albert’s view. The state may describes ergodic motion of particles in real space. As you may know, this is the interpretation of the wave fucntion I have been studied (http://philsci-archive.pitt.edu/10659/).

Best,
Shan

[Quantum Speculations]

I should clarify one point for my argument. My argument, if valid, only says that the wave function describes certain state or property of a single quantum system, and it does not imply that the wave function itself is real, e.g. the wave function describes a continuous field a la Albert.

What physical state or property the wave function describes is a further question that needs to be studied. I do have an idea in this regard (http://philsci-archive.pitt.edu/10659/), which has been studied by me for 20 years. The idea is very similar to Bob’s picture of ergodic motion of particles.

According to this idea, the wave function of an N-body quantum system describes the state of random discontinuous motion of N particles, and in particular, the modulus squared of the wave function gives the probability density that the particles appear in every possible group of positions in space. At a deeper level, the wave function may represent the indeterministic dispositional property of the particles that determines their random discontinuous motion.

[Robert Griffiths]

Dear Shan,

A couple of comments. First on your reply #1087 to Ken. If PMs can measure excpectation values of observables, they can presumably also measure the probability distribution of an observable as well as its average. Thus if observable A has a spectral decomposition A = a_1 P_1 + _2 P_2 + a_3 P_3, where I = P_1 + P_2 + P_3 is the identity, then then same procedures which will allow you to determine will also allow you to determine <P_j> for j =1, 2, 3, and thus the probability distribution for A. At least I don’t see why this cannot be done.

Next on your #1090, that the wave function describes a certain state or property of a single system, I am a bit uncertain as to what you might have in mind. In hidden variable language there is the mysterious lambda, is that what you are thinking about? In orthodox QM there is no
lambda, but a wavefunction specifies a ray in the Hilbert space (which I would call a property)–is that what you’re thinking about? If the wave function is describing a “state” in the sense of a probability distribution, then this is a rather different beast. Would you care to comment?

Bob Griffiths

[Robert Griffiths]

Resend. I think I inserted some symbols which gave the computer indigestion; let me try again.

Dear Shan,
A couple of comments. First on your reply #1087 to Ken.
If PMs can measure expectation values of observables, they can presumably also measure the probability distribution of an observable as well as its average Avg(). Thus if observable A has a spectral decomposition A = a_1 P_1 + _2 P_2 + a_3 P_3,
where I = P_1 + P_2 + P_3 is the identity, then the same procedure which gives you Avg(A) will give you Avg(P_j) for each P_j, and thus the probability distribution for A. At least I don’t see why this cannot be done.

Next on your #1090, that the wave function describes
a certain state or property of a single system,
I am a bit uncertain as to what you might have in mind.
In hidden variable language there is the mysterious lambda, is that what you are thinking about? In orthodox QM there is no lambda, but a wavefunction specifies a ray in the Hilbert space (which I would call a property)–is that what you’re thinking about? If the wave function is describing a “state” in the sense of a probability distribution, then this is a rather different beast. Would you care to comment?

Let’s hope this comes through better. Bob Griffiths

[Matthew Leifer]

One big objection to your argument is that, if true, it would imply that classical probability distributions are also properties of individual systems. The reason is that we can do the whole protective measurement scheme for Gaussian states using only Gaussian operations. Quantum mechanics with only Gaussian states and operations is equivalent to classical Liouville mechanics with a restriction on the allowed probability distributions and operations and, under this correspondence, a Gaussian quantum state is equivalent to a Gaussian probability distribution on phase space. Therefore, your argument would imply that these Gaussian probability distributions must be ontic in classical mechanics.

I think the issue with the argument is as Matt Pusey pointed out in his “talk”, but let me try to say the same thing in a different way. The issue is that the protection is repreparing the system in an independent copy of the initial state, so you effectively have an ensemble of independent copies which you are measuring. To make a classical analogy, suppose you have a classical continuous-time Markov chain and, for simplicity, suppose it has a unique fixed point and the system starts out in the fixed point. Suppose that the Markov dynamics is very rapidly mixing, ergodic even, so that it spends an amount of time at each sample space point proportional to the probability assigned in the fixed point distribution. Also, if the system departs from the fixed point for some reason, it will be driven back very rapidly. Now suppose we make some measurements on the system, with the background Markovian dynamics still in operation all the time. We can even suppose that our measurements disturb the system, so long as this disturbance operates on a much longer timescale than the relaxation to the fixed point in the background Markovian dynamics.

What will happen? We can measure some property of the system and perhaps disturb it a little, but the Markovian “protection” will then drive it back into the initial probability distribution. We will also lose correlations with the measured property because the background dynamics is rapidly mixing and so effectively reprepares the system in an independent copy of the initial probability distribution. We can then go ahead and measure again, and ultimately reconstruct the probability distribution by making a sequence of such measurements.

This is not exactly what happens in the Gaussian model of protective measurements, which works deterministically, but I think it is a useful analogue nonetheless. The point is that you have to pay attention to what additional resources the protection is giving you. If you can argue that it effectively gives you an ensemble of independent states then your argument collapses.

[Quantum Speculations]

Dear Bob,

Thanks for your very helpful comments!

Yes, I agree that PMs can also measures <P_j>. My point is that these measured quantities describe a single quantum system, and not just an usual probability distribution in QM. The latter interpretation depends on what happens during a conventional measurement.

As to the second comment, I would like give an example. In my interpretation of the wave function (as said above), the wave function of an N-body quantum system describes the state of random discontinuous motion of N particles (which is composed of density and flux density in configuration space), and in particular, the modulus squared of the wave function gives the probability density that the particles appear in every possible group of positions in space. At a deeper level, the wave function may represent the indeterministic dispositional property of the particles that determines their random discontinuous motion.

I hope these answers may be helpful.

Best,
Shan

[Quantum Speculations]

Dear Matt,

Thanks for your very helpful criticisms!

I am not familiar with the equivalence between quantum mechanics and classical Liouville mechanics. But I think this equivalence will not imply the physical interpretations of these two theories must be the same; otherwise will already has a classical int. of QM. (Moreover, I think your objection is not only for my argument, but also for the psi-ontic view.)

I will respond to your second objection below.

Shan

[Robert Griffiths]

Following up a bit on Matt Leifer’s comment in #1094. I myself find it very helpful to think of the sample space (or state space) of a Markov chain of length n, in which there are, let us say, just 6 possibilities at each time step, as the same thing as the sample space for a collection of n dice all rolled at once, except that you have to imagine that the way in which nearby dice fall is correlated (to imitate the temporal correlations of a Markov chain). Thus while the protective measurement is carried out on just one system, it looks formally like a set of measurements carried out on several systems. This to my mind raises the question of whether there is all that much of a difference between a wave function describing a SINGLE system and describing an ensemble. Bob G

[Quantum Speculations]

I agree with most of your analyses in your ergodic motion example.

In my interpretation of the wave function in terms of random motion of particles, what PM measures is indeed a probability distribution, such as the position probability density of a particle. But this probability density describes the single particle, not an ensemble. This probability should be better understood as an indeterministic dispositional property of the particle that determines its random discontinuous motion. This is consistent with the conclusion of my PM argument.

[Quantum Speculations]

Matt #1094:

In my picture of random motion of particles, it seems that we may say that classical probability distributions are also properties of individual systems. The word “classical” is in the sense of an ergodic, random motion of individual particles.

Shan

[Quantum Speculations]

Hi Bob,

Thanks for your further comments!

I think the point is that people usually believe that a wave function ONLY describes an ensemble, but if the psi-ontic view is right, a wave function also describes a SINGLE system. This is new. Certainly, a wave function can still describe an ensemble.

Shan

[Matthew Pusey]

Hi Shan,

As the other Matt has already mentioned, the existence of psi-epistemic models of protective measurement makes your argument difficult to swallow. But let me focus here on two more specific questions here:

1) Why couldn’t somebody also run your argument using the tomography-of-protector then projective measurement-of-system scheme?

2) Are you saying the same protection can apply to two non-orthogonal states? Can you give an example of how that would work?

Cheers,
Matt

[Quantum Speculations]

Hi Bob and Matt,

Thanks again for your very helpful discussions! I think both of you may be sympathy with my interpretation of the wave function in terms of random discontinuous motion of particles (see http://philsci-archive.pitt.edu/10659/), as you already presented similar examples.

I will catch a bus now. Your later comments and criticisms are very welcome!

Best,
Shan

[Robert Griffiths]

Dear Shan,

It seems to me that your argument is somehow getting weaker as time progresses. It is not the reality of the wavefunction that you want to argue for, but rather that the wavefunction applies to a single system, and if you buy Matt’s analogy, the wavefunction tells us something about statistical properties of the motion of the system. But at the end of the day we have probabilities, and they would seem, to me, to belong to the epistemological side of things. Is that where you finally arrive?

Bob G

[Maximilian Schlosshauer]

Hi Matt L.,

I’m hope I’m not joining this conversation too late. Regarding your comment in #1094:

The issue is that the protection is repreparing the system in an independent copy of the initial state, so you effectively have an ensemble of independent copies which you are measuring.

This way of describing the protection seems specific to the Zeno-type protective measurement. If the protective measurement is adiabatic (i.e., no intermediate projections back onto the initial state), then I don’t see the ensemble analogy you are drawing; at least it’s not obvious. Is your argument specific to the Zeno version?

Best,
Max

[editor]

Hi Max,

Thanks for your comments! I think you will agree that Matt’s scheme is not a PM. Right?

Shan

[editor]

For example, in his scheme, one get many inaccurate expectation values, from which the right one can only be constructed (which is very like the standard tomography based on projective measurements). But in a PM, one directly get the right expectation value with arbitrary accuracy.

[editor]

Hi Bob,

Thanks for your further comments!

I agree with you. It is just the meanings of words that differ between us. The “reality of the wavefunction” is usually understood as “the wavefunction applies to a single system” (e.g. in the PBR theorem).

As you have seen from my previous posts, my interpretation in terms of RDM of particles indeed says that “the wavefunction tells us something about statistical properties of the motion of a single system”. But this does not mean they belong to the epistemological side of things. the psi-epistemic view usually denotes that the wavefunction does not apply to a single system.

Best,
Shan

[Quantum Speculations]

Matt #1101:

Thanks for your further comments!

2) Are you saying the same protection can apply to two non-orthogonal states? Can you give an example of how that would work?

No, we need two different protections for two non-orthogonal states. The protection is taken as part of the state preparation procedure.

1) Why couldn’t somebody also run your argument using the tomography-of-protector then projective measurement-of-system scheme?

This needs a more careful analysis. My answer is negative. One reason is that scuh a scheme is not a usual measurement, like projective meaurement or PM. The result of your scheme is re-consctructed from many inaccurate WM results. In this case, I don’t know whether it is meaningful to say “the same physical state $\lambda$ yields the same probability distribution of measurement results.” Moreover, even if one can run my argument using your scheme, it seems that this does not refute the argument either. I will say more about your scheme later.

Best,
Shan

[Quantum Speculations]

Dear Matt,

I must say that your scheme let me think more deeply about PM. I am very grateful to you for this.

I think we should not make assertions quickly; we need a more careful analysis and more effective discussions, especially because we have different opinions from the beginning.

Here is my current thoughts:

1. Your scheme is not the same as adiabatic PM (as Max also noted). I think you should agree with this. It is unclear whether you can find a psi-epistemic model of adiabatic PM. So, it is fair to say that there is still no firm basis to reject the possible implications of such PMs.

2. Whether your scheme is equivalent to or similar to Zeno-type PM is a very interesting question, which certainly needs a careful analysis. I think they are different. My previous analysis is:

Your scheme is equivalent to many weak measurements of an ensemble of identically prepared systems. In your scheme, one get many inaccurate expectation values, from which the right expectation value is re-constructed.

By contrast, in a Zeno-type PM, although there are also frequent projective measurements (as a protection), one directly get the right expectation value with arbitrary accuracy in a single measurement.

[Quantum Speculations]

Here is my further idea (gotten on the bus to my office), which shows that there are more differences:

(1) The first one concerns the weak measurements (with readouts). These weak measurements result in strong wavefunction collapse. Here the strongness means that the probability to collapse most of the outcome states is large, although the initial state and the final state are almost the same. It is this feature makes your scheme be more like or even equivalent to projective measurements (as you have proved in your notes).

I think it is also this feature that makes your scheme different from Zeno-type PM. In a Zeno-type PM, the wavefunction collapse is very weak in the sense that the probability to collapse the wrong outcome states is very small, and can be made arbitrarily small in principle.

In other words, wavefunction collapse has no role in obtaining the right expectation value in an (ideal) Zeno-type PM. But, as you also noted in the end of your notes, wavefunction collapse has a significant role in obtaining the weak measurements results, from which the right expectation value is re-constructed. This also leads to the second difference.

(2) It is that your scheme relies on the Born rule (so does the PBR theorem? I am not sure). This is not unexpected as it relies on wavefunction collapse. Imagine that the Born rule assumes a different form, e.g. not |psi|^2, then it seems possible that the right expectation value cannot be re-constructed from the weak measurements results.

By contrast, a Zeno-type PM does not replies on the Born rule. (It may only require that very small probability amplitude has small probability collapse outcome.) The right expectation value is obtained independently of the Born rule in a Zeno-type PM. In fact, the expectation value is generated from the continuous Schroeinger evolution. (It is also this avoidance of wavefunction collapse that makes PM probably have direct implications on the reality of the wave function)

It is these differences that makes one doubt the equivalence of your scheme and Zeno-type PM.

I would like to know your further thoughts.

Best,
Shan

[Matthew Pusey]

Max #1109,

I can tell you how I currently see this issue. As we’ve mentioned, the adiabatic scheme is present in the Gaussian toy theory. Recall that the ontology of that toy theory is just that of classical particle mechanics. This basically turns out to move the particle around in such a way that, for any observable allowed in the theory, the amount of time the particle has a specific value is proportional to the probability of it being measured to have that value. Since the protective measurement scheme operates slowly, it ends up seeing the time average. So the role of the protection is again to provide access to the entire probability distributions, but instead of providing repeated random samples as in the Zeno case it provides a deterministic dynamics such that if we sample at a random time we see the distribution.

As it happens, the basic idea of this explanation was anticipated in one of the original papers. Two responses are provided (p4625), here’s my paraphrase of them and thoughts on them:

1. It is difficult to reconcile this idea with the existence of nodes in the wavefunction – does the particle need to go infinity fast past these? This is an interesting point that I’d like to think about more. It doesn’t arise in the Gaussian theory because none of the wavefunctions have nodes. But we already now from things like Bell’s theorem that the toy theory ontology cannot be extended to all of quantum mechanics, I don’t think this undermines the argument that phenomena present in the toy theory are compatible with the ψ-epistemic view.
2. This is not how things work in Bohmian mechanics. That seems irrelevant since Bohmian mechanics is ψ-ontic.

Regards,
Matt

[Matthew Pusey]

Shan #1115-#1117,

I’ve agreed that more work is required to clarify exactly what Bob’s strategy is in the original scheme and whether or not this is equivalent to my “recap”. Thanks for your additional ideas on this matter.

But as I have tried to make clear, the operational argument does not depend in any way on what the protective measurement scheme tells Bob to do. The argument is based purely on what the scheme requires of Alice and Charlie. You have not disputed my characterization of what Alice and Charlie do, and so as far as I can see the operational argument stands. I think we are starting to go round in circles so perhaps we should agree to disagree for now. I’ll be happy to send you a draft of our paper whenever we finally write it.

Yours,
Matt

[Quantum Speculations]

Hi Matt,

Thanks for this! I look forward to reading your final paper.

Yes, I agree that “the operational argument does not depend in any way on what the protective measurement scheme tells Bob to do.”

But in your note, you say: “But since protective measurement uses exactly the same resources, how can it lead to a radically different conclusion about the role of the wave-function?”

My point is that what Bob has done in your scheme is not a PM, and protective measurement does not use exactly the same resources. So, if this is true, it will influence your operational argument.

Best,
Shan

PS. I noticed another feature of your scheme last night (when I am near sleeping). It seems that your scheme cannot measure the expectation value of an observable with arbitrary accuracy, and the state of the ensemble will be largely mixed in the end. Suppose the first weak measurement leads to small probability 1/n to collapse to other states, then after n steps the probability to stay in the initial state is (1-1/n)^n, which is not close to one when n approaches infinity. I hope this point may be useful for your further analysis.

[Author]

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