There is a simple question about quantum theory that has been increasingly bothering me. I keep hoping that my reading about interpretational issues will turn up a discussion of this point, but that hasn’t happened. I’m hoping someone expert in such issues can provide an answer and/or pointers to places where this question is discussed.
In the last posting I commented that I’m not sympathetic to recent attempts to “reconstruct” the foundations of quantum theory along some sort of probabilistic principles. To explain why, note that I wrote a long book about quantum mechanics, one that delved deeply into a range of topics at the fundamentals of the subject. Probability made no appearance at all, other than in comments at the beginning that it appeared when you had to come up with a “measurement theory” and relate elements of the quantum theory to expected measurement results. What happens when you make a “measurement” is clearly an extremely complex topic, involving large numbers of degrees of freedom, the phenomenon of decoherence and interaction with a very complicated environment, as well as the emergence of classical behavior in some particular limits of quantum mechanics. It has always seemed to me that the hard thing to understand is not quantum mechanics, but where classical mechanics comes from (in the sense of how it emerges from a “measurement”).
A central question of the interpretation of quantum mechanics is that of “where exactly does probability enter the theory?”. The simple question that has been bothering me is that of why one can’t just take as answer the same place as in the classical theory: in one’s lack of precise knowledge about the initial state. If you do a measurement by bringing in a “measuring apparatus”, and taking into account the environment, you don’t know exactly what your initial state is, so have to proceed probabilistically.
One event that made me think more seriously about this was watching Weinberg’s talk about QM at the SM at 50 conference. At the end of this talk Weinberg gets into a long discussion with ‘t Hooft about this issue, although I think ‘t Hooft is starting from some unconventional point of view about something underlying QM. Weinberg ends by saying that Tom Banks has made this argument to him, but that he thinks the problem is you need to independently assume the Born rule.
One difficulty here is that you need to precisely define what a “measurement” is, before you can think about “deriving” the Born rule for results of measurements, and I seem to have difficulty finding such a precise definition. What I wonder about is whether it is possible to argue that, given that your result is going to be probabilistic, and given some list of properties a “measurement” should satisfy, can you show that the Born rule is the only possibility?
So, my question for experts is whether they can point to good discussions of this topic. If this is a well-known possibility for “interpreting” QM, what is the name of this interpretation?
Update: I noticed that in 2011 Tom Banks wrote a detailed account of his views on the interpretation of quantum mechanics, posted at Sean Carroll’s blog, with an interesting discussion in the comment section. This makes somewhat clearer the views Weinberg was referring to. To clarify the question I’m asking, a better version might be: “is the source of probability in quantum mechanics the same as in classical mechanics: uncertainty in the initial state of the measurement apparatus + environment?”. I need to read Banks more carefully, together with his discussion with others, to understand if his answer to this would be “yes”, which I think is what Weinberg was saying.
Update: My naive questions here have attracted comments pointing to very interesting work I wasn’t aware of that is along the lines of what I’ve been looking for (a quantum model of what actually happens in a measurement that leads to the sort of classical outcomes expected, such that one could trace the role of probability to the characterization of the initial state and its decomposition into a system + apparatus). What I learned about was
- Work of Klaas Landsman, I liked a lot his survey article on the measurement problem
https://link.springer.com/chapter/10.1007/978-3-319-51777-3_11
where he describes the “flea on Schrodinger’s cat” speculative idea as an example of the “instability” approach to the measurement problem. Also interesting is his paper on “Bohrification”
https://arxiv.org/abs/1601.02794 - Work on a detailed analysis of a “Curie-Weiss” model of a measurement. The authors have a long, detailed expository paper
https://arxiv.org/abs/1107.2138
and more explanation of the relations to measurement theory here
https://arxiv.org/abs/1303.7257
and here
https://link.springer.com/chapter/10.1007%2F978-3-319-55420-4_9
There are also comments about this from Arnold Neumaier here
https://physicsoverflow.org/39105/curie-weiss-model-of-the-quantum-measurement-process
In these last references the implications for the measurement problem are discussed in great detail, but I’m still trying to absorb the subtleties of this story.
I’d be curious to hear what experts think of Landsman’s claim that there’s a possible distinct “instability” approach to the measurement problem that may be promising.
Update: From the comments, an explanation of the current state of my confusion about this.
The state of the world is described at a fixed time by a state vector, which evolves unitarily by the Schrodinger equation. No probability here.
If I pick a suitable operator, e.g. the momentum operator, then if the state is an eigenstate, the world has a well-defined momentum, the eigenvalue. If I couple the state to an experimental apparatus designed to measure momenta, it produces a macroscopic, classically describable, readout of this number. No probability here.
If I decide I want to know the position of my state, one thing the basic formalism of QM says is “a momentum eigenstate just doesn’t have a well-defined position, that’s a meaningless question. If you look carefully at how position and momentum work, if you know the momentum, you can’t know the position”. No probability here.
If I decide that, even though my state has no position, I want to couple it to an experimental apparatus designed to measure the position (i.e. one that gives the right answer for position eigenstates), then the Born rule tells me what will happen. In this case the “position” pointer is equally likely to give any value. Probability has appeared.
So, probability appeared when I introduced a macroscopic apparatus of a special sort: one with emergent classical behavior (the pointer) specially designed to behave in a certain way when presented with position eigenstates. This makes me tempted to say that probability has no fundamental role in quantum theory, it’s a subtle feature of the emergence of classical behavior from the more fundamental quantum behavior, that will appear in certain circumstances, governed by the Born rule. Everyone tells me the Born rule itself is easily explicable (it’s the only possibility) once you assume you will only get a probabilistic answer to your question (e.g. what is the position?)
A macroscopic experimental apparatus never has a known pure state. If I want to carefully analyze such a setup, I need to describe it by quantum statistical mechanics, using a mixed state. Balian and collaborators claim that if they do this for a specific realistic model of an experimental apparatus, they get as output not the problematic superposition of states of the measurement problem, but definite outcomes, with probabilities given by the Born rule. When I try and follow their argument, I get confused, realize I am confused by the whole concept: tracking a mixed quantum state as it evolves through the apparatus, until at some point one wants to talk about what is going on in classical terms. How do you match your classical language to the mixed quantum state? The whole thing makes me appreciate Bohr and the Copenhagen interpretation (in the form “better not to try and think about this”) a lot more…
Peter,
You seem to be hoping that the basic problem might be resolved by some deep analysis of a complicated many-body model capturing interactions between a system and a measuring device. But the critical facts are already on the table, and such an analysis doesn’t have any room to get around them–namely the facts:
1. An initial system state comprising a superposition of mutually orthogonal terms, unentangled with the measuring device state
2. An entangling (unitary) operation between system and device
3. A resulting superposition of FAPP orthogonal terms, each one with a different system state and corresponding device state
I am curious how you think there is any chance of this final superposition going away in favor of a single term? How could a more detailed analysis of a specific model possibly achieve that?
Have I misunderstood your goal here? If not, Lee Smolin’s remark should be clear: there is no way to magically inject the empirical concept of (measurement result) *probability* into this abstract Hilbert space set-up without adding something else. And there is also no way to get a unique measurement result out of that superposition without adding something else. So your solution is going to fall into one of three buckets:
1. Renounce the assumption that there is a unique result
2. Add something else
3. Flip the table over, scream incoherently, and run away
Or using their technical names:
1. Many Worlds
2. “Hidden Variables”
3. Copenhagen, postulated state reduction, QBism, or something similar
1. An initial system state comprising a superposition of mutually orthogonal terms, unentangled with the measuring device state
2. An entangling (unitary) operation between system and device
3. A resulting superposition of FAPP orthogonal terms, each one with a different system state and corresponding device state
*****
How can we be sure #3 follows from #2? If I understand Peter’s position correctly, the unitary evolution of “device”+”system” is horrendously complicated. The device is almost certainly not in an energy eigen-state at finite temperature, rendering its state is unknown. So the question becomes: can we contrive a simple enough “device” and show what happens after unitary evolution of the interacting system. For example system=1 Qubit and device = N Qubits where N is small enough that we can see what’s going on during unitary evolution? **Perhaps** one would find that all coefficients in the superposition except one, approach zero and N grows larger, AND that the one selected to grow with N depends on the initial state of the device (N Qubits).
Is this just wishful thinking?
I would think this is at least worth a shot before giving up and diving into MWI and pilot wave fantasies. Maybe one of the references demonstrates something along these lines.
Eric Dennis,
I’m not assuming the initial apparatus is in a pure state. Not that I know how to do this, but my question was based on treating the apparatus probabilistically (as opposed to the system state). What do you see wrong with
https://arxiv.org/abs/1107.2138
They treat the apparatus using quantum statistical mechanics, and claim:
“Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run. “
There is a nice interview with Roger Balian (of the Curie-Weiss model of quantum measurement) in which he and Francois David discuss their responses to Schlosshauer’s questions in Elegance and Enigma here: https://www.youtube.com/watch?v=-K_8W9lnSp8
Peter Woit
Here are my thoughts on measurement, informed by my work in quantum optics. (apologies for length, 35 years ago I was gonn make a unfied field theory! I got stuck in interference and how detectors work. And our atom-field interaction, the Jaynes Cummmings model is a non-relativistic SUSY SHO so….)
Measurement is an interaction between two subsystems, one subsystem in principle knowable at least in part, and one not. It need not be a classical and quantum cut but it often is. The moon is there because it scatters light from the sun, it is there when you don’t look at it.
But if I am observing part of a system, interacting with it, there has to be a cut. Defined by what the user can and cannot know in principle. And a cut means a partial trace of the density matrix and hence a mixed states.
Lets start with photodetection. When a single photon hits a PMT, it excites an electron to the continnum, and from there we get amplification and an avalanche to make a classical current. There is noise in this process called shot noise. Does it come from the source or is it a property of the detector. The “interpretation” depends on which operator ordering (normal, antinormal, symmetric, or any other you want) is used in the calculation. It can be either or some of both. All I know is a photon has been emitted and caused a click. I cannot ascribe a “path” or “cause”. And at what point did that current become classical??
With just emission from an excited state, spontaneous emission, that is “due” to two effects. Vacuum fluctuations and radiation reaction. Again, what percentage of those “causes” SpE depends on the operator ordering. And with commutators, this is different from putting things in normal ordering with no commutators as that is what you do for absorptive photodetection. If I use symmetric ordering, I get 1/2 RR and ½ VF. Symmetric ordering (i.e. adag*a=1/2(adag*a+a*adag+1) by definition leaves us with Hermitian operators. So in principle the two things could be separately measured, and there might be some physical reason to say ½ RR and ½ VF.
And not all measurements result into projection onto an eigenstate of a Hermitian operator, as in continuous weak measurement. Vaidman has talked about this somewhat correctly. You have a weak interaction and the “measurement” takes some non-zero time.
As an example, consider how we detect squeezed light. You have two photodetectors behind a beam splitter, and subtract the two photocurrents. With classical local oscillator(coherent state) at one input port of the beam splitter, the currents cancel. Even classical noise is cancelled. EE’s have used this for years, called coherent detection, or in QO terms “balanced homodyne detection”. Now just have one classical beam input, zero photocurrent (except for the shot noise level described above), and a quantum field on the other port of the BS. The two photodetectors click like crazy due to the strong classical field, and the presence of the quantum signal will unbalance the thing and give rise to a difference current. For strong local oscillator the output is proportional to E_{LO}*a_{theta}, you can measure the fluctuations in a_{theta}, any quadrature you want.
a_0 =1/2(a+adag), a_pi/2=i/2(a-adag)
You cannot measure THE field as that would mean you can have an amplitude operator (adag a) and a phase. The latter does not exist; if it did we would have a time operator.
So you change that phase and watch the noise go BELOW the shot noise limit, and then above. I think they are bolting squeezed light into LIGO as we speak.
You can have Quantum Non-Demolition experiments where you can measure x over and over again, and all the “detection” noise goes into p, or vice versa.
Stimulated emission is kind of classical, spontaneous quantum (for the field anyway), but you cannot tell the difference between the two in detection .A photon is a photon is an excitation of a field leading to a click in a detector). Glauber taught us (knowing QFT) that one had to look to correlations involving two detections. A two point Greens function if you will. Then you can tell a SP em source from a Stem source.
As an example of a “cut”, think about a black hole. Hawking radiation is a sum of two-mode squeezed states. There is a horizon, so you trace over the bits you can’t see. Then you get a thermal state. It’s like a Rindler observer seeing a thermal field in Unruh, and Minkowski can see both (or their effects)
There is no unique classical limit in QM. For the SHO, oh lets say large n is classical. Well it is except for the n zeros in the wave function. OK so then eigenstates of a are classical, coherent states. Then you have a wavepacket bouncing back and forth in the well. The first is appropriate if you look at the same time every look. If you look randomly you will see a coherent state. What is the classical limit?
And for an electron dropping down to a lower level, that is indeed an interaction. The ATOM and the vacuum field know that it happened, and in principle you could detect it. No looking at the meter, or rather not HAVING a meter is not natures fault
It seems funny that QM pretty much tells us you can just know transition amplitudes and probabilities between some initial and final states. Yet we persist in looking for the watchmaker in there somewhere. I guess I am a Bayesianist, but not those that have elevated it to consciousness.
So what is a measurement? An irreversible interaction (at least on the lifetime of the universe. And to give “causes” you have to make a split of the users choice. And its not unique (as density matrices are not)
Any this is in my somewhat informed opinion, I am an experimentalist trapped inside a theorists body, so this suits me. Density matrix is fundamental and represents what we are able to know about a system at any time. You want “cause?” Invent one and see if it matches the voltmeter.
Hi Peter,
Regarding your preferred approach: to model quantum measurement as a dynamical process and its outcome probabilities as explainable by quantum statistical mechanics (without assuming any underlying ontic states beyond quantum theory), you would still have to assume the initial states of measured system or the apparatus (or both) to be describable by quantum states.
Then one can always ask why is it being described by quantum to begin with? A quantum state is essentially a list of outcome probabilities. So this approach seems to assume a list of probabilities from the start, and therefore begs the question of where the quantum probabilities come from.
Daniel Tung,
I don’t see why I have to think of a quantum state as “a list of outcome probabilities”. It’s a fundamental, complete, mathematically very deep description of the the state of the world at a given time. If I want to do a “measurement”, I couple it to a macroscopic quantum object which I have to define by a density matrix due to its macroscopic nature. This evolves in time during measurement as a density matrix. If I believe Balian and collaborators, they give an example of a solution of the “outcome” problem, showing why the final result can be thought of as a single outcome (the cat is dead or alive, not a superposition). I’m not a positivist, so don’t see why I have to think of the quantum state just in terms of the frequencies of these outcomes of very special setups coupled to the state I care about.
I’m surprised that this work hasn’t gotten a lot more attention. I’ll add to the posting a bit more information about it, would love to hear from experts why or why not this isn’t “a solution to the measurement problem.”.
Peter,
I have not read that mammoth paper (1107.2138) but it looks like a goose chase. The opaque terminology is inauspicious. Based on some selective skimming, this “hierarchic property” is related to how, e.g., the probability of a number chosen from {1, 2, 3, …, 100} being prime is independent of how you order the subset of primes from that set. Great, so some density matrices have a property isomorphic to a property of probability distributions. Of course we already knew that density matrices have many such isomorphic properties. I don’t see anything earth-shattering about adding this particular property to that list.
More fundamentally, the structure of their argument doesn’t show any evidence of getting to where you want to go. The apparatus starting in a mixed state, rather than a pure state, means simply this: for an experiment run many times, within each run we don’t know *which* pure state the apparatus starts in. But, rest assured, in each run, the apparatus starts in *some* particular pure state.
And now we are back to the basic problem I named above. In *each run*, unitary evolution demands that the final system+apparatus state still contains all the possible system eigenstates (entangled with corresponding apparatus/pointer states), all superposed. Even if in some model they find that density matrices describing *multiple runs* share some extra mathematical properties with classical probability distributions, they are still totally failing to show (i) how a particular eigenstate can possibly be singled out in each run, or (ii) how one could possibly *derive* a probability statement about that undescribed singling-out process. Hilbert spaces in, Hilbert spaces out. No real-event probabilities are in sight.
A smart person tells you he can turn a snail into a giraffe. How? He says he follows it around for 12 miles and administers a very special sequence of taps to its shell along the way. Which is more likely? That he actually can do it. Or that, at some point close to mile 12, he will declare success, and when you look down and say “Still seems like a snail to me,” he will reveal that by “giraffe” he meant a snail underneath a tree.
Eric Dennis,
Thanks, although I still would like to see a better explanation of exactly what these authors are claiming, along with what exactly goes wrong. A short version of their claims that others may find gives insight is
http://iop.uva.nl/binaries/content/assets/subsites/institute-of-physics/quantum-measurement.pdf?1489502855021
Longer version
https://arxiv.org/abs/1303.7257
At the level of sloganeering, Balian in an interview claims that the situation is analogous to that of classical stat mech and thermodynamics, where one consistently has reversible microphysics, irreversible macrophysics.
For a short critique, see
http://people.bss.phy.cam.ac.uk/~mjd1014/readings.html
One problem for me is that subtleties about what probability is seem to be coming up here, ones beyond my ability to see my way through.
Peter,
Those other links are helpful. Note how in the new paper you cite, they are shifting their claim from that of the previous paper, which says: “Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process.”
In the newer paper (1303.7257) it’s not “standard quantum statistical mechanics alone” anymore. Now it’s that augmented by some mild-mannered “interpretative principles.” The last one of these “principles” (i.e., separate postulates that must be made besides the Schrodinger equation) is as follows (p. 17, excuse my re-formatting):
“Interpretative principle 5. Consider a set of macroscopic orthogonal projectors Π_i, a state D~ associated at a given time with an ensemble E and the states Dˆ(k)_sub associated with its sub-ensembles E^(k)_sub. If the projectors have in these states the commutative behaviour expressed by Eqs. (12a-b), their q-expectation values q^(k)_i can be interpreted as physical probabilities for exclusive events, i. e., as relative frequencies.”
In the abstract of the paper, this move is described as follows:
“The latter property supports the introduction of a last interpretative principle, needed to switch from the statistical ensembles and sub-ensembles described by quantum theory to individual experimental events. It amounts to identify some formal “q-probabilities” with ordinary frequencies, but only those which refer to the final indications of the pointer.”
The “q-probabilities” are apparently just the squared-norms of terms in some decomposition of a Hilbert space vector. So we finally get actual probabilities only by “identifying” them with Hilbert space norms–by hand.
I conclude that this whole enterprise has moved itself squarely into my original bucket #3 containing Copenhagen, postulated state reduction, and other similar gambits. This was the bucket I had identified with the interpretative principle of “flip the table over, scream incoherently, and run away.”
This from Balian, et. al. on resolving Bell, GHZ type experiments:
“As a quantum measurement is a joint property of S{ystem} and A{pparatus}, we are not allowed to interpret simultaneously as real properties of the initial state of S the results of experiments obtained with different apparatuses (here with different directions of the detectors). This deep property of quantum measurements is in line with the absence, for quantum states, of a sample space as in ordinary probability theory [46, 47, 48, 49, 50].”
IMO, this might be the thing to look at to see if it all makes sense.
Anonyrat,
That just sounds to me like a standard fact going back to Copenhagen. What looks new and confusing here are claims about what “probability” means in the context of getting a supposed single outcome.
I’ve not commented on this, as all my previous posts on similar subjects have been rejected. But I must. I and my former boss have though about this a lot, especially because of experiments we did long ago.
The crux is that you cannot start with a single pure eigenstate. You must include all of the “apparatus” … including the measurement apparatus and the “test particle generation apparatus”. And the original state must be a very complicated state vector. This is because both apparatus must be above T = 0. In any real experiment this is true because both are macroscopic. “Near” zero won’t do because macroscopic things have low energy phonon states. You can’t deexcite them. You can of course choose an energy for the apparatus … but since you are limited in how long you can take to establish that, the uncertainty principle makes a spread in the energy of “true eigenstates” necessary. Because of this the “quantum recurrence time” of the apparatus is incredibly long.
And these complicated composite states are described by a state vector, which changes with time (Schrodinger picture). There are “observables” on that state vector that are macroscopic. Because of the huge number of eigenstates that are excited, these observables follow classical paths. You tell what the outcome is by looking at them (i.e. did a bunch of electrons flow to just one pixel on an LCD?). This can be seen on a computer solving the Scrodinger equation. And, just like in classical mechanics, a tiny change in a wave function a certain point is space and time can, and does, grow with time into a macroscopic change. This can and has been computer simulated. This is probabilistic, based on the state when, say, an electron wave packet hits the surface of a CCD from space. But remember that that wave packet was generated by an apparatus with a finite temperature. Its not a uniform plane wave.
The size of the “apparatus” necessary is surprisingly small, but macroscopic … at least a few hundred atoms. Its not yet exactly simulatable on a computer but is getting closer.
Many people have problems understanding because they try to use a descriptivive “system” and “apparatus” that are impossibly small. They say “well, we’ll use a “classical bath” “weakly coupled” to that small system/apparatus … you can’t do that, in time weak couplings add up. You can’t use perturbation theory, just like you can’t use perturbation theory in bound state QCD problems. The apparatus is bound states.
This is an attempt at a wordy explanation of what I believe
https://arxiv.org/abs/1107.2138 says.
There is an article by S. Weinberg in 2016 about measurement, Lindblad equation and the Born rule.
S. Weinberg, What Happens in a Measurement?
Phys Rev A93 (2016) 032124 – arXiv:1603.06008
followed up by Lindblad Decoherence in Atomic Clocks
Phys Rev A94 (2016) 042117 – arXiv:1610.02537
See also https://backreaction.blogspot.com/2017/02/testing-quantum-foundations-with-atomic.html
Mateus Araújo, and Peter,
I appreciate the contribution by Landsman and Reuvers to the understanding of the Born rule. Yet, they do not mention that the instability in their asymmetric double well scenario was described, in a more idealized “Einstein’s boxes” version, by J. Gea-Banacloche, “Splitting the wave function of a particle in a box”, Am. J. Phys. 70 (3), March 2002, p. 307-312. On the other hand, neither does J. Gea-Banacloche seem to know the work of Jona-Lasinio et al. (1981) cited by Landsman and Reuvers. A survey on “Einstein’s Boxes” can be found in T. Norsen, “Einstein’s Boxes”, arXiv:quant-ph/0404016v2, 8 Feb 2005 (Dated: February 1, 2008), published Am. J. Phys. 73, 164-176 (2005); and S. J. van Enk extends the subject to “Splitting the wavefunction of two particles in two boxes”, arXiv:0806.4932v2 [physics.ed-ph] 5 Aug 2008. I guess that Einstein would have been pleased about the instabilities described in these articles. –
One more thing: akhmeteli wrote on September 9, 2018 at 10:32 pm: “[The] projection postulate and the Born rule do not reflect the fact that any measurement requires some time to perform.” I agree: This requires that the Born rule must have a sibling on the time axis, where the actual “occurance” of the measurement is undetermined in standard QM, as it is in space. This of course plays an important role e. g. in radioactive decay, but is not covered by text book derivations of the (statistical) decay process.
It looks like I’ve missed the party – which for “interpretation of quantum mechanics” parties is usually fine – but since it’s interesting to see Peter getting interested in this, here are my quick thoughts.
For Hilbert space quantum mechanics I would just take probability as a primitive concept. For any two unit vectors u and v we have “the probability that the system is in state u given that it is in state v”. It’s odd that this can be nonzero when u is not equal to v, but that’s life. Then various choice of mathematically natural axioms will force this probability to be the square of the absolute value of the inner product of u and v.
(You can give this setup a more epistemological spin by saying “the probability that the system is found to be in state u if we check this when we know that it’s in state v”. However, most days I consider “is found” and “we know” to be distracting fluff. We could add such verbiage to the laws of classical mechanics as well. It doesn’t hurt much, but it doesn’t help much either.)
In the way I’m putting it, all the interpretive difficulty is packed into the word “probability”. By leaving probability as an undefined primitive, I’m leaving it as a separate task to say what “probability” actually means in experimental contexts. This is a very interesting quagmire, with various kinds of “Bayesians” and “frequentists” fighting it out. A lot of the fight over quantum mechanics, I claim, is secretly a fight over what “probability” actually means.
This seems like a completely different question. But it’s better in that I can give my preferred answer more rapidly: no.
John,
Thanks for writing. Now that the party has died down, I’m still confused, but can at least express a bit more clearly what is bothering me. I’ll also add this to the bottom of the posting itself.
The state of the world is described at a fixed time by a state vector, which evolves unitarily by the Schrodinger equation. No probability here.
If I pick a suitable operator, e.g. the momentum operator, then if the state is an eigenstate, the world has a well-defined momentum, the eigenvalue. If I couple the state to an experimental apparatus designed to measure momenta, it produces a macroscopic, classically describable, readout of this number. No probability here.
If I decide I want to know the position of my state, one thing the basic formalism of QM says is “a momentum eigenstate just doesn’t have a well-defined position, that’s a meaningless question. If you look carefully at how position and momentum work, if you know the momentum, you can’t know the position”. No probability here.
If I decide that, even though my state has no position, I want to couple it to an experimental apparatus designed to measure the position (i.e. one that gives the right answer for position eigenstates), then the Born rule tells me what will happen. In this case the “position” pointer is equally likely to give any value. Probability has appeared.
So, probability appeared when I introduced a macroscopic apparatus of a special sort: one with emergent classical behavior (the pointer) specially designed to behave in a certain way when presented with position eigenstates. This makes me tempted to say that probability has no fundamental role in quantum theory, it’s a subtle feature of the emergence of classical behavior from the more fundamental quantum behavior, that will appear in certain circumstances, governed by the Born rule. Everyone tells me the Born rule itself is easily explicable (it’s the only possibility) once you assume you will only get a probabilistic answer to your question (e.g. what is the position?)
A macroscopic experimental apparatus never has a known pure state. If I want to carefully analyze such a setup, I need to describe it by quantum statistical mechanics, using a mixed state. Balian and collaborators claim that if they do this for a specific realistic model of an experimental apparatus, they get as output not the problematic superposition of states of the measurement problem, but definite outcomes, with probabilities given by the Born rule. When I try and follow their argument, I get confused, realize I am confused by the whole concept: tracking a mixed quantum state as it evolves through the apparatus, until at some point one wants to talk about what is going on in classical terms. How do you match your classical language to the mixed quantum state? The whole thing makes me appreciate Bohr and the Copenhagen interpretation (in the form “better not to try and think about this”) a lot more…
Dear Peter,
Long-time reader, first time commenter (I think).
I agree with everything you write in your latest, third update. My problem with the Balian et al. paper is the same: once you start using density matrices which are not pure, so mixed, you have lost the possibility to describe ordinary unitary time evolution, and you have introduced a probabilistic feature in your theory. In decoherence language, if you neglect the off-diagonal parts of the density matrix in the pointer basis, you have gone from an exact to a probabilistic model.
I think Landsman is on the right track, but I also think most people in foundational QM are looking at this from a too microscopic and/or mathematical point of view, trying to analytically derive neat and exact results for messy many-body systems. Condensed-matter physicists know this is a foolish undertaking.
If you believe like Landsman and myself and perhaps you that everything obeys unitary time evolution, you should put your money where your mouth is and start a project to conduct large-scale computer quantum simulations in which you describe measurement apparatuses as quantum many-body systems like Balian et al., but *in pure states*, couple it to your test particle and see how the combined system evolves for a large number of initial states, for both test particle and apparatus (the last one with a slight perturbation from pure symmetric in the pointer basis). If Born’s rule is robust, it should emerge while each run is definite and unitarily evolving.
The problem will be to find the correct Hamiltonian to describe the apparatus: it needs to be in an almost-symmetric, metastable state, unstable to the tiny interaction with the test particle, while the stable states are those in the pointer basis. But any experimentalist can tell you this is a very real problem in nature.
Personally I think this will solve the measurement problem to a large extent, but it will not, immediately, solve non-locality. It everything happens by unitary time evolution, like measurement of one half of a Bell-photon pair, then information including the change of the wavefunction due to interaction with the detector can only travel at the speed of light. I also think the usual interpretation of particles as wavepackets (two photons as two wavepackets within one wavefunction?) may be insufficient to this end.
Finally let me mention this completely ignored and/or unknown paper with does something like Landsman in a QFT model:
https://arxiv.org/abs/quant-ph/0505077
And one unpublished paper which does something like the simulation I mentioned:
https://arxiv.org/abs/0809.1575
AB,
Thanks very much for the comments and references. The project you suggest sounds worth doing, but needs to be done by people with much different expertise than mine.
One thing I’ve always been curious about is whether some part of the quantum computing story intersects with this problem, which might make it much more interesting to people.
About non-locality, I confess I’ve never thought much about that, beyond the vague thought that “the apparatus is non-local, so what’s surprising about non-locality”? But I really don’t want to start a discussion of this here and educate myself right now, best to leave for another time.
Peter Woit wrote:
And perhaps no physics here, either, unless we say how the state of the world is described by that vector: that is, how we can use the vector to make predictions of experimental results.
Every theory of physics has this issue: it consists of precise mathematics surrounded by a vaguer and much more complicated cloud of “interpretation”: instructions on how to connect this mathematics to things we can do and see in a laboratory.
Those of us who like precision prefer to focus on the mathematics. We’d be happy if this mathematics were, ultimately, “all there really is”. Maybe it can somehow provide its own interpretation. But it’s hard to see how. To address this rigorously, we’d need to say very precisely what we mean by an interpretation, and prove theorems saying that such-and-such a theory can admit only the following interpretations. But I haven’t seen anyone do a good job of this yet.
It’s particularly hard to see how to get probabilities out of some mathematics unless we put them in by hand – that is, decree that certain quantities in the mathematics stand for probabilities. The reason is that it’s very hard to say what probabilities actually are, in a non-circular way.
It sounds like you’re trying to put the probabilities in by hand as follows. When an experimental apparatus meets a microscopic system, the apparatus is in a mixed state, and we decree that mixture has a probabilistic interpretation. We then evolve the joint system unitarily, see what happens, and try to derive probabilities of results.
In other words, we try to bring in the probabilities through a probabilistic description of the measuring apparatus.
I don’t see how this will work. When I take an electron with spin pointing up along the z axis, and measure its spin along the x axis, I should get “up” with probability 50%. I don’t see how this probability is going to emerge from the probabilities in the mixed state representing the measuring apparatus. Indeed, a good measuring apparatus is usually considered to be one where our lack of knowledge about its state doesn’t affect its functioning!
So I’m willing to entertain this line of thought, but I’m not optimistic.
I prefer to bite the bullet and say, right from the start, that a unit vector in a Hilbert space describes probabilities. The meaning of a unit vector v in Hilbert space is that given another unit vector u, the probability that “if the system is in state v, then it’s in state u” is the square of the absolute value of the inner product of u and v.
Of course this leads to a host of further puzzles, but at least it’s clear where I’m putting in the concept of probability.
(By the way, since a pure state of a joint system restricts to mixed states on each of its parts, if you’re willing to decree that mixed states describe probabilities, you can reason backwards and get the probability interpretation of pure states as a corollary, given suitable side-assumptions. Wojciech Zurek gave a nice explanation of this in the workshop “Statistical Mechanics, Information Processing and Biology” at the Santa Fe Institute, and he probably has a paper on it.)
John,
I guess it just doesn’t seem to me obviously either necessary or a good idea to try and directly interpret the fundamental quantum state in terms of something very complicated and very different, the experimentally accessible probabilistic classical emergent behavior of a pointer. When you say
“When I take an electron with spin pointing up along the z axis, and measure its spin along the x axis, I should get “up” with probability 50%.”
you’re assuming a very simple and direct relationship between the microscopic state and the macroscopic emergent classical behavior. But that seems to me exactly where the mystery of measurement problem is, and any attempt to think through how to correctly model this makes clear it’s a very hard problem. I’m having trouble finding convincing simple-minded (i.e., “it’s just some pure state, evolving according to the Schrodinger equation”) arguments about how a measurement apparatus is supposed to behave.
On a related note, I’m also not convinced by
“a good measuring apparatus is usually considered to be one where our lack of knowledge about its state doesn’t affect its functioning! ”
As I was trying to make clear, the problem occurs exactly when we’re talking about observable quantities that have no meaning until we couple our system to the apparatus. It is the apparatus itself which, coupled to the system, is giving (in a complicated way involving emergent classical behavior) meaning to the observable quantity. I don’t see at all why our uncertain knowledge of the state of the apparatus is not going to be relevant.
https://arxiv.org/abs/1805.08583
My summary: If your experiment is described in terms of a separate preparation and measurement step, and all you can do in the end is count events, i.e. you _start_ from statistics, you have to end up with the formalism of quantum mechanics without needing any weird postulates.
I appreciate John’s comment that the notion of probability is a source of controversy even outside of any quantum concerns.
My question to the experts is, does Consistent Histories (Griffiths, Gell-Mann, Hartle) provide a useful perspective on Peter’s dilemma? As an amateur in the subject of interpreting QM, it has always appealed to me.
David Metzler,
I’m no expert, but from what I’ve seen, Consistent Histories just ignores the problem that is bothering me of understanding exactly how classical behavior emerges from quantum. As far as I can tell, it’s just one of many ways of papering over that problem.