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Has the measurement problem been resolved? Could someone explain the current state-of-the-art understanding of why deterministic evolution results in a random measurement outcome? Is there a model that explains where the randomness comes from?

I found many people talking about decoherence. But doesn't decoherence only claim that the reduced density matrix tends to be diagonal for large $t$? This explains nothing about randomness.

Let me clarify my question on randomness. It's known that quantum randomness is not due to a lack of information (which is the reason for classical randomness). I'm really asking for a natural way to introduce the Born rule.

For example, if one uses the MWI interpretation, this question would be: how do we rigorously define the probability measure? How do people solve the preferred basis problem? Can we define probability in a basis-free way (because Schmidt decomposition is not unique, defining probability on basis may not be good)? How do we define Born probability in a time-continuous way (usually, people just talk about two moments: before a measurement and after a measurement, and the world split into 2 branches, but a measurement, in fact, happens with time duration, and this split of branches should be continuous in time)? A great relevant question: Is the preferred basis problem solved?. I really would like to know more relevant works.

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  • $\begingroup$ Related: physics.stackexchange.com/questions/295527/… $\endgroup$
    – Ruffolo
    Commented Nov 18 at 11:51
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    $\begingroup$ The transition from determinism to perceived randomness is not the problematic one. Unknown degrees of freedom in the environment offer a sufficient source of unpredictability. The real problem is getting from a linear state evolution to a fundamentally non-linear one. $\endgroup$
    – Jazzmaniac
    Commented Nov 18 at 18:41
  • $\begingroup$ "It's known that quantum randomness is not due to a lack of information." Actually, that's not known. This is merely an assumption based on the fact that we have not been able to reduce the Born rule in such a way. But as I commented earlier, the challenge is not the randomness. If you are suggesting that Bell's inequality and its experimental verification imply such a statement, then this would be misguided. Bell only rules out that there is a local-realistic alternative to quantum theory. It does not rule out that there are unaccounted degrees of freedom. $\endgroup$
    – Jazzmaniac
    Commented Nov 19 at 12:47
  • $\begingroup$ Is there a (solvable) model that shows unknown external degree of freedom recovers prediction of Born rule? $\endgroup$
    – SSS
    Commented Nov 20 at 22:34
  • $\begingroup$ @SSS, there has be no such (published) derivation for a single measurement outcome, because that would require solving the linear vs. non-linear problem first. However, if you assume that the unknown degrees of freedom have a probability distribution with a certain degree of unitary invariance, then Gleason's theorem dictates that the only consistent probability measure on the system would be given by Born's rule. Make of that assumption what you want, but it's certainly not extremely far fetched. $\endgroup$
    – Jazzmaniac
    Commented Nov 21 at 7:47

3 Answers 3

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There is an algorithm that is used to make predictions from quantum theory. (1) You use equations of motion to predict the evolution of states and/or observables until you do a measurement. (2) Then you use the Born rule to predict the expectation values of measurable quantities represented by observables, POVMs etc. The measurement problem is about step (2).

The standard story one sees in textbooks about step 2 is that the state of a system is projected onto an eigenstate of a measured observable with a probability given by the Born rule: this alleged process is called collapse. This story is incompatible with the equations of motion used in step 1 and that is the measurement problem.

There are many proposed solutions to the measurement problem. Some physicists say you should ignore it and pretend there is no problem: advocates of the statistical and Copenhagen interpretations. This has the problem of treating measurement as a black box when in reality it is a physical process subject to some law of physics. If you're forbidden from looking in the box then you can't understand measurements and this prevents both experimental and theoretical progress if taken seriously. And if you're not going to take the theory seriously and use it why adopt it?

Some physicists say you should modify quantum theory to include collapse:

https://arxiv.org/abs/2310.14969

In this theory the Born rule is a postulate. Or you should modify quantum theory to include extra particles that are interacting with the wave function:

https://arxiv.org/abs/1906.10761

In pilot wave theory the Born rule comes from the statistical mechanics of a particle interacting with the wave function and there may be deviations from the Born rule under some circumstances.

These theories don't currently reproduce the predictions of relativistic quantum theories, i.e. - almost all tested predictions of quantum theory

https://arxiv.org/abs/2205.00568

Another option is to treat measurement devices using quantum theory. A measurement has to produce a record that can be copied and copying information out of a quantum system suppresses interference: decoherence

https://arxiv.org/abs/1911.06282

Since any system you see in everyday life has information copied out of it by its environment this explains the lack of interference:

https://arxiv.org/abs/1111.2187

It doesn't eliminate all but one of the possible states of the measurement result. But if those states don't interfere then they each evolve autonomously of one another and the world looks a bit like a collection of systems each of which looks a bit like the universe as described by classical physics on a suitably coarse level or description:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

This is often called the many worlds interpretation (MWI) but it is just a result of working out the implications quantum theory as we would for any other scientific theory. There are explanations of the Born rule in terms of decision theory and envariance in the MWI

https://arxiv.org/abs/0906.2718

https://arxiv.org/abs/quant-ph/0405161

https://arxiv.org/abs/quant-ph/9906015

https://arxiv.org/abs/1508.02048

The reason you can predict the probability but not the specific outcome is that there is no single outcome but the quantum state has properties that can be described by square amplitudes that act like probabilities under decoherence.

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  • $\begingroup$ Just to note, that expectation value predicted does not say anything about exact eigenstate observed, like expectation value of $3.5$ of dice rolling says nothing about next rolled number... We can surely validate fact that in the $100\times$ of rolling dice we get $3.5$ average result within some confidence level $\Delta$, but it says nothing about classical Physical system,- Why or How number $X$ has fallen on a try labeled with number $N$. QM probabilities just differs in that respect that it's proven that we can't build a better theory (which could eliminate the need of probabilities). $\endgroup$
    – Agnius Vasiliauskas
    Commented Nov 19 at 13:06
  • $\begingroup$ That is not the core of the measurement problem. In principle, there is no problem that there are two different evolution laws, applicable in different situations. The core of the measurement problem is in unclear details on when exactly and how exactly (in time) the projection process happens. $\endgroup$
    – Ján Lalinský
    Commented Nov 23 at 22:18
  • $\begingroup$ @JánLalinský What you are saying is false for reasons explained in the answer above. I should also point out that your answer is incompatible with measurement theory as it is actually used arxiv.org/abs/1604.05973 Collapse is both operationally and theoretically unviable $\endgroup$
    – alanf
    Commented Nov 24 at 8:34
  • $\begingroup$ @alanf I don't see how your answer explains why my comment is wrong; you take a particular view on what the measurement problem is, and I point out that is not the only view. Also, you wrote "there is no single outcome" -this contradicts observations. There is single outcome and single record in an experiment properly designed to produce such. Parallel reality of different exclusive outcomes in such experiment is, in physics, superfluous unfalsifiable speculation. Maybe it's fine as a weird worldview/image in mind, but it is not a scientific theory. $\endgroup$
    – Ján Lalinský
    Commented Nov 25 at 3:10
  • $\begingroup$ Check out Andrew's answer, and how he defines the measurement problem. $\endgroup$
    – Ján Lalinský
    Commented Nov 25 at 3:13
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One way to frame the discussion is that there are three principles that everyone likes (I am stealing this from philosopher Tim Maudlin, eg see the YouTube Video Tim Maudlin | Bell’s Theorem and Beyond: Nobody Understands Quantum Mechanics | The Cartesian Cafe at 2:30:30)

  1. Completeness of the wavefunction. The wavefunction contains all the information needed to describe the quantum state.
  2. Unitary evolution. Solving the Schrodinger equation (and possibly other deterministic equations) given some initial condition tells you what happens to the state of the system.
  3. Uniqueness of outcomes. Whatever we measure in an experiment is the unique state of the world.

However, the measurement process apparently forces us to give up one of these three principles.

In the Bohmian interpretation, we give up completeness of the wavefunction. There is a fully deterministic description of the state of the Universe in terms of "hidden variables," or particles that we are not able to directly observe, that carry extra information not in the wavefunction. When we perform a measurement, we see an apparently random outcome, but that is only because we do not know what these underlying particles are doing -- if we could measure them, we would see everything was actually deterministic. The wavefunction evolves according to the Schrodinger equation, and the particles interact with the wavefunction and with each other. However, a very strange aspect of these interactions -- especially given the causal structure of special relativity -- is that the particle interactions are non-local, meaning they influence each other faster than light can travel from one to the other. It isn't yet understood how to build a relativistic version of Bohmian mechanics, which might be unsurprising given the nonlocality.

In the Copenhagen (or, perhaps, "textbook") interpretation, and related ideas. we give up unitary evolution. This means that there are two evolution rules. One is that the wavefunction evolves according to the Schrodinger equation in a deterministic way. Given an initial state, we can solve the Schrodinger equation to see what is happening at a later time (or, go backward to see what happened at an earlier time). The second rule is measurement. During measurement, the wavefunction collapses to an eigenfunction of the observable that was measured. The measurement process is random. Defining exactly what constitutes a measurement is essentially what the "measurement problem" is.

In the many worlds interpretation, we give up on uniqueness of outcomes. The nice thing about the many worlds interpretation is that it does not require any new elements -- no extra particles we don't observe (as in the Bohmian interpretation), no second random evolution rule (as in the Copenhagen interpretation). All we need is the wavefunction and the Schrodinger equation. The cost of that is that the wavefunction describes an infinite number of possible worlds we could be living in, and so without some principle to define which world is the "right" one, somehow all of those worlds must exist, and there is nothing about the world we actually experience that is "special," except for the small detail that it is the only one that we experience existing! Besides the philosophical quagmire of making sense of an infinite number of parallel realities, a major technical problem with the many worlds interpretation is that it is not agreed how to (or if it is possible to) derive the Born rule that assigns probabilities to different measurement outcomes given a wavefunction -- this is not just some minor detail, because those probabilities are the main thing we actually observe from quantum mechanics!

There are other interpretations, but I like that breakdown in showing that (as far as we know) any approach you take forces you to give up some principle that you like from classical mechanics.

Decoherence is important for understanding how the classical world emerges from quantum mechanics, although it does not explain how you get a random theory from a deterministic one. Without decoherence, you could build a clever interferometer that forced two parts of the wavefunction to interfere at a later time. So a natural question is, why don't we experience these kinds of interference effects in our everyday lives? With decoherence, any interference effects involving big, hot objects would be exponentially suppressed, and therefore practically impossible to measure. You are completely right that this does not explain how we go from deterministic to non-deterministic evolution, but it at least explains why interference effects are hard to observe in the classical world.

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Has the measurement problem been resolved?

Not in a widely accepted way. There are many ways to solve it, even dismiss it, but every such solution leaves most other people who understand the problem unimpressed.

Could someone explain the current state-of-the-art understanding of why deterministic evolution results in a random measurement outcome?

That is not the measurement problem. As @Jazzmaniac pointed out, apparently random outcomes can be consistent with deterministic evolution, if the deterministic evolution equation involves variables of unknown value (e.g. stochastic external field, or details of the state).

In orthodox quantum theory, results of measurements are not due to deterministic evolution. Measurement and projection of quantum state is evolution process of another kind (process of 1st kind), not describable by the Schrodinger equation (process of 2nd kind). Many experiments show the process of 1st kind is very much needed to account for observations, the most prominent is probably the experiment determining which way the atom went in the Stern-Gerlach experiment. Thus experimental facts show inadequacy of using just the Schroedinger equation for describing all evolution. Despite of how nice and attractive this equation is, it is not sufficient as the only evolution law.

One way to state the measurement problem is: assuming there are two incompatible rules for how quantum state evolves in time (this by itself need not be a problem, although some people don't like it and consider it part of the measurement problem), what is the criterion/rule to determine, for any given situation, whether the first or the second rule applies?

In practice one uses the process of 1st kind (projection) for that time for which one knows it started to be necessary to have $\psi$ be the projected state in order to be compatible with known facts, such as which way the atom went in the SG experiment, or in the double-slit experiment. However, we do not know why and how exactly (detailed account in time) the quantum state gets projected. It's a modification of calculation based on new information, and when and how we get this information varies. It is hard to bolt down the projection process in time to a real physical process in time, and if this is done in some way (e.g. the GRW theory), it is not described by the Schroedinger equation.

Is there a model that explains where the randomness comes from?

In classical physics, we think results of experiments aren't really random, they only look random, because we can't predict them, and we think this prediction would be possible in principle, if we had more data and better computers(sometimes people think about weather this way).

In orthodox quantum theory, results of experiments are believed to be more random than that, in that they are believed to not be determined by any set of variables in the past state. However, in some situations (entangled state of the super-system), results of measurements on distant systems are believed to be perfectly correlated, so there are some constraints on the randomness of results in quantum theory, it is not absolute.

There are ideas about more unknown variables that should be added to quantum state, determining the results, and thus randomness being only apparent - so-called hidden variables. In particular, there is the Bohm theory, in which the additional variables are positions of the particles. In double slit experiment, dots on screen only look random, but actually they are not, as they are due to actual particles that were there, continuously moving there from some distant positions they had before. Apparent randomness of the detected dots is explained as due to unknown past values of those past particle positions.

I found many people talking about decoherence. But doesn't decoherence only claim that the reduced density matrix tends to be diagonal for large $t$? This explains nothing about randomness.

I agree, decoherence suppresses effects of superpositions on density matrix, but it remains a probabilistic description. It does not tell us anything about which single result will be obtained in an experiment on single system, e.g. where a single atom in the double slit experiment will land on the screen).

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