A Assumptions of the Bell theorem

[Demystifier]

The Born rule is fundamental. Ho often do you want to hear this answer?

Until you make it consistent with your other claims.

But measurement is emergent, am I right? Hence the Born rule must be valid even without measurement, is that correct? Then, in the absence of measurement, what determines the basis in the Born rule?

 

[Kolmo]

It's also true that there is no limit in the size of a system to show "quantum behavior" like interference/superposition and even entanglement

I always liked the last few sections of Daneri et al's classical 1962 paper "Quantum theory of measurement and ergodicity conditions" in this regard:
https://doi.org/10.1016/0029-5582(62)90528-X

They basically show that an experimental situation where one could display the relevant interference terms for coarse-grained observables of a macroscopic measuring device¹ necessarily breaks the molecular bonds allowing it to function as a macroscopic body, effectively reducing it to a heated soup of atoms.

It always seemed an easier to understand version of Bohr's talk about the dual nature of the apparatus as a measuring device or a quantum system, i.e. an experiment where you can see the interference terms in such a body explicitly means it doesn't induce irreversible processes.

¹By measuring some operator that doesn't commute with them. Gottfried's old text shows of course how difficult this is regardless since such operators are often "highly nonlocal" in his terminology. "Nonlocal" here meaning requiring simultaneous measurement of nearly each atom individually across the device, not "Nonlocal" as in faster than light.

 

[stevendaryl]

As for the nonlocality I don't see it. Following the evolution of the state it would predict either Alice's detector clicked or Bob's detector clicked, i.e. the possible histories are click here or click there. Just because the possible events are far apart doesn't to me indicate nonlocality.

If what is possible for Bob in the next second depends on what happens at Alice's detector 1 billion miles away, then that seems nonlocal to me.

 

[Demystifier]

If what is possible for Bob in the next second depends on what happens at Alice's detector 1000 miles away, then that seems nonlocal to me.

Then internet communication is nonlocal too.

 

[stevendaryl]

Then internet communication is nonlocal too.

Okay change 1000 miles to 1 billion miles.

 

[Kolmo]

If what is possible for Bob in the next second depends on what happens at Alice's detector 1 billion miles away, then that seems nonlocal to me.

I don't get it I have to say. There is a possibility for one of two events to happen, you wait and then you find out which happens. The fact that the two possible events are separated by billions of miles doesn't seem important to me. It's just that the particle might be detected here or there, it just seems probabilistic.

 

[stevendaryl]

I don't get it I have to say. There is a possibility for one of two events to happen, you wait and then you find out which happens. The fact that the two possible events are separated by billions of miles doesn't seem important to me. It's just that the particle might be detected here or there, it just seems probabilistic.

If a random event affects something far away, then that random has nonlocal effects. Sort of by definition.

The random event of Alice detecting a particle affects Bob's chance of detecting a particle.

 

[Kolmo]

If a random event affects something far away, then that random has nonlocal effects. Sort of by definition.

The random event of Alice detecting a particle affects Bob's chance of detecting a particle.

I would say given how you prepared the system there is a chance $p$ of Alice detecting a particle and a chance $1 - p$ of Bob detecting a particle. So the theory just predicts two mutually exclusive events with different probabilities.
If Alice detects a particle, then since the events are mutually exclusive we know Bob did not, i.e. the click only occurs in one place.

To ascribe this to nonlocality, in fact to say it is nonlocal "by definition" is very difficult to understand for me. I've never heard anybody describe mutually exclusive events being some form of nonlocality.

 

[stevendaryl]

To ascribe this to nonlocality, in fact to say it is nonlocal "by definition" is very difficult to understand for me. I've never heard anybody describe mutually exclusive events being some form of nonlocality.

I think some people work really hard at not understanding things.

Mutually exclusive events are only nonlocal if they are NONLOCAL events. If a particle nondeterministically turns left or right, that's local. If the particle nondeterministically decides to be here or a billion miles away, that's nonlocal.

I have made this example before. Suppose that there is a pair of coins, and by some strange law, the $nth$ time you flip one coin will always result in the opposite of the $nth$ time you flip the other coin. This works regardless of how far away the two coins are. But looking at one coin individually, it seems like it randomly produces heads or tails.

I think that most people would assume that there are two possibilities:

1. There is some yet-unknown internal mechanism determining the outcome. Maybe the coin is programmed so that the result is "heads" if the $n^{th}$ digit in the decimal expansion of $\pi$ is even, and the result is "tails" otherwise.
2. There is some nonlocal effect keeping the two results correlated.

I assume that you would say: There's nothing mysterious or nonlocal going on. It's just that there are two mutually exclusive events: One gets heads, or the other gets heads.

To my mind, such an attitude is deadly for physics. Once you stop noticing mysteries, you are no longer doing science.

 

[stevendaryl]

I think some people work really hard at not understanding things.

Mutually exclusive events are only nonlocal if they are NONLOCAL events. If a particle nondeterministically turns left or right, that's local. If the particle nondeterministically decides to be here or a billion miles away, that's nonlocal.

I have made this example before. Suppose that there is a pair of coins, and by some strange law, the $nth$ time you flip one coin will always result in the opposite of the $nth$ time you flip the other coin. This works regardless of how far away the two coins are. But looking at one coin individually, it seems like it randomly produces heads or tails.

I think that most people would assume that there are two possibilities:

1. There is some yet-unknown internal mechanism determining the outcome. Maybe the coin is programmed so that the result is "heads" if the $n^{th}$ digit in the decimal expansion of $\pi$ is even, and the result is "tails" otherwise.
2. There is some nonlocal effect keeping the two results correlated.

I assume that you would say: There's nothing mysterious or nonlocal going on. It's just that there are two mutually exclusive events: One gets heads, or the other gets heads.

To my mind, such an attitude is deadly for physics. Once you stop noticing mysteries, you are no longer doing science.

I would say that such a correlation is nonlocal by definition. It's a correlation between events that are far apart. What else should it be called, other than a nonlocal correlation?

 

[Kolmo]

I think some people work really hard at not understanding things.

I can assure you I'm not "working hard" here as it seems a trivial physical situation, but okay I guess I'm not one of these "deep conceptual" thinkers who notices these "mysteries". It just seems to me to be two mutually exclusive events which are spatially separated, not some sort of nonlocality and I have never really heard anybody consider such a basic "double slit"-type situation as nonlocal. The theory doesn't tell you which occurs because it is probabilistic, but I don't see that as nonlocality.

I think that most people would assume that there are two possibilities:

To my mind, such an attitude is deadly for physics. Once you stop noticing mysteries, you are no longer doing science.

That's a bit overblown I think. I do recognize the differences between quantum and classical probabilistic cases, I was the one originally arguing the theory was not Kolmogorovian after all, and that it's not a trivial one.

Your two possible choices have been investigated in several papers and are so highly constrained as to be effectively ruled out. If you consider these the only possible options and anybody who doesn't choose between them is "not doing science", fine I guess I don't do science or something.

 

[Nullstein]

I would say that such a correlation is nonlocal by definition. It's a correlation between events that are far apart. What else should it be called, other than a nonlocal correlation?

I think the word "non-local" should be reserved for causal relationships between spacelike separated regions. Causality is a stronger requirement than correlation. In the case of entanglement, I don't think we can infer such a causal connection quite yet, because we don't fundamentally understand quantum mechanics and the changes it might bring to our understanding of causality. In modern causality research, people generally acknowledge that our current, classical notion of causality might not be applicable to quantum entanglement.

 

[stevendaryl]

I think the word "non-local" should be reserved for causal relationships between spacelike separated regions.

But science never really tells about causes. It only tells us about correlations.

The closest that we come to giving a causal story is if we assume that certain variables are "freely chosen". Then we can say that the choice of those variables has a causal influence on anything that is correlated with them.

For example, in Newtonian physics, the initial positions and momenta are freely chosen.

Causality is a stronger requirement than correlation.

But correlations are what we directly observe. Of course, nonlocal correlations can often be explained in terms of local correlations, in the sense that we can deduce the nonlocal correlation from a sequence of local correlations. For example, we observe that Alice and Bob always wear the same color hat every day, no matter how far away they are. That's a nonlocal correlation, in my terminology. However, it can be explained using local correlations: Maybe in the past, Alice and Bob got together to decide what color hat they would wear each day for the rest of their lives.

In the case of entanglement, I don't think we can infer such a causal connection quite yet, because we don't fundamentally understand quantum mechanics and the changes it might bring to our understanding of causality. In modern causality research, people generally acknowledge that our current, classical notion of causality might not be applicable to quantum entanglement.

I guess it depends on what the default position is. I would say that a connection between two events is nonlocal unless it can be "implemented" using local correlations. The other default is that the connection is local unless it can be proved to involve nonlocal influences.

 

[PeterDonis]

There is no collapse

This claim is interpretation dependent, and the rules for this forum clearly state that you should not state claims made by particular interpretations as being established facts. Please take note.

 

[PeterDonis]

The Born rule is fundamental.

This claim is also interpretation dependent. See my previous post.

 

[RUTA]

I think the word "non-local" should be reserved for causal relationships between spacelike separated regions. Causality is a stronger requirement than correlation. In the case of entanglement, I don't think we can infer such a causal connection quite yet, because we don't fundamentally understand quantum mechanics and the changes it might bring to our understanding of causality. In modern causality research, people generally acknowledge that our current, classical notion of causality might not be applicable to quantum entanglement.

That's consistent with our position in this paper https://www.mdpi.com/1099-4300/23/1/114/htm

 

[Nullstein]

But science never really tells about causes. It only tells us about correlations.

I don't think that's true. There has been quite a revolution in causality research in the last few decades. People are able to algorithmically derive causal relationships from statistical data. The definite reference is "Causality" by Pearl. These algorithms work very well, but if applied to quantum theory, they break down.

But correlations are what we directly observe.

Right, but we need to be careful about terminology in order not to accidentally draw wrong conclusions. Correlations over spacelike intervals aren't necessarily mysterious. As you say, one could e.g. look for a common cause in the past, or more generally, one could look for chains of causally related events (as formalized in the causal Markov condition today). And another serious possibility is that our current understanding of causal explanations might require revision. At least, this attitude seems to be taken serious in the modern causality research community.

I guess it depends on what the default position is. I would say that a connection between two events is nonlocal unless it can be "implemented" using local correlations. The other default is that the connection is local unless it can be proved to involve nonlocal influences.

I don't think there should be a default position at all. If we can't decide it one way or the other, we should just admit that we don't know and keep researching it further.

 

[Nullstein]

That's consistent with our position in this paper https://www.mdpi.com/1099-4300/23/1/114/htm

That's very interesting, I will have a look at it!

 

[Kolmo]

"Causality" by Pearl

That's certainly the standard reference. I also liked "Causation, prediction, and search" 2nd Edition by Spirtes, Glymour and Scheines.

 

[vanhees71]

This claim is interpretation dependent, and the rules for this forum clearly state that you should not state claims made by particular interpretations as being established facts. Please take note.

I thought we have decided on the scientific part of the theory, according to which there is no collapse.

https://www.physicsforums.com/insights/the-7-basic-rules-of-quantum-mechanics/

 

[vanhees71]

This claim is also interpretation dependent. See my previous post.

I thought we have decided on the scientific part of the theory, according to which the Born rule is one of the postulates.

 

[PeterDonis]

I thought we have decided on the scientific part of the theory, according to which there is no collapse.

The 7 Basic Rules do not say "there is no collapse". They make no claim either way about whether collapse is a real physical process or not. They just tell how to use the projection postulate under the appropriate circumstances to calculate predictions.

 

[PeterDonis]

the Born rule is one of the postulates.

The Born Rule is one of the 7 Basic Rules, but, as in my previous post, those rules make no claim about what is "fundamental". They just tell you how to calculate predictions. The Born Rule is one of the rules to be applied when doing that.

 

[vanhees71]

Indeed, I'm uneasy about the projection postulate too, but that's the only point of the agreement, I'm a bit worried about since in my opinion the state of the system after a measurement cannot be postulated but depends on the specific measurement done on the system. Von Neumann filter measurements are pretty rare and feasible only for very simple systems. The Born postulate says less, and it's part of our "7 Basic Rules". It just tells you the probabilities when accurately measuring an observable when the (pure or mixed) state the system is prepared in is known.

 

[vanhees71]

The Born Rule is one of the 7 Basic Rules, but, as in my previous post, those rules make no claim about what is "fundamental". They just tell you how to calculate predictions. The Born Rule is one of the rules to be applied when doing that.

I don't know, what's fundamental, if not the "basic rules".

 

[PeterDonis]

Von Neumann filter measurements are pretty rare and feasible only for very simple systems. The Born postulate says less, and it's part of our "7 Basic Rules". It just tells you the probabilities when accurately measuring an observable when the (pure or mixed) state the system is prepared in is known.

Yes, that's intentional. There is not general agreement about how to handle more general or more complicated cases, so they aren't included in the 7 Basic Rules, which are only intended to capture the minimal set of things on which there is general agreement.

don't know, what's fundamental, if not the "basic rules".

"Fundamental" implies a claim about "how things really are". Our 7 Basic Rules make no such claim.

 

[Kolmo]

the state of the system after a measurement cannot be postulated but depends on the specific measurement done on the system

Do you mean in general you'd need to update using Kraus operators:
$\rho \rightarrow \rho^{'} = \frac{K_{i}\rho K^{\dagger}_{i}}{Tr(K_{i}\rho K^{\dagger}_{i})}$ with $\sum_{i}K_{i} K^{\dagger}_{i} = \mathbb{I}$
or did you mean something else?

 

[vanhees71]

No, we are just in the next debate about, what I am allowed to say though we are in the interpretations subsection of the quantum physics forum. Now we are again fighting about semantics, i.e., what I am allowed to call fundamental and what not.

My point of view is that one cannot generally say, in which state the system is after a measurement, because it depends on the specific apparatus used to measure, and of course the theory of measurement has evolved much further in the recent years, including the use of projection operator valued measurement formalisms etc. In your language, the Kraus operators are determined not by some postulate but by the apparatus used to do the measurement.

 

[Kolmo]

In your language, the Kraus operators are determined not by some postulate but by the apparatus used to do the measurement.

Indeed fully agree, it often requires a form of tomography to find the Kraus operators of the device and they cannot just be postulated. Indeed it is now a subfield of quantum information to find optimal operational algorithms to determine the Kraus operators in various settings.
Just checking I understood your meaning, thanks.

 

[PeterDonis]

Now we are again fighting about semantics, i.e., what I am allowed to call fundamental and what not.

You are allowed to say it is your opinion that something is fundamental, or that collapse is not a real physical process. You are not allowed to state those things as facts. They are your opinions. The guidelines for this forum (the interpretations forum) make that clear.

My point of view is...

All of this is fine from the standpoint of the guidelines for this forum.

 

[vanhees71]

I'm glad to hear that. I've the impression for discussing in the Quantum Forum you need to have more and more the abilities of a lawyer rather than a physicist .

 

[Demystifier]

I don't know, what's fundamental, if not the "basic rules".

Fundamental are those basic rules that cannot be violated even in principle. For instance, energy conservation and entropy increase are both basic rules, but, as far as we know, only the former is fundamental.

 

[Fra]

From my perspective I think of "fundamental" elements of a theory, are those that we exclude from requiring further motivation. Ie. the have axiomatic status and taken as unquestionable and not subject to beeing inferrable or "observable".

Things that can be verified by inferences within the theory, or from experiment as true or false, using the methods of the theory, are IMO then not "fundamental", they are inferences to which we assign degrees of belief.

But in the context of evolving theories, and lossy compression, it seems a plausible idea still that an evolving agent, can at some point, as part of lossy compression, "approximate" a confident historical inference, with an axiom.

This is why I think what is fundamental is also interpretation or theory dependent. I personally thing, it's not necessarily critial to assess what is fundamental and what is not, just as its' not important to assess what is true or false, if you have a framework where one can make evolutionary transformations between fundamental and non-fundamental things, as a form of recoding. So if we can observe someting, that ALWAYS holds, it is an inference, but instead of that and agent is to waste resources to maintain a confident opinion, it can instead may an internal remapping, to make a lossy compression it to an axiom, and this resource planning may be economical for the agent, and possibly be well motivated in such a model. We know from biology that evolution has a lot to do not only only with survival, but also about "economy" such as "regulatory economy" and other things. Because using resourcs at hand optimally, is a survival trait indirectly - this is what they information theoretic perspective to QM is IMO highly relevant! But it has many components.

It's how I think, but I am sure others with a more ontological theory on mind, finds this obscure. But even so, I do think they they need to contradict each other.

/Fredrik

 

[vanhees71]

I think it's a bit misleading to talk about "axioms". I'd rather call them "postulates". Physics is not a mathematical theory but an empirical science. I'd say the "fundamental laws" or "postulates" are those parts of a physical theory which we consider to be right by experience, i.e., all observations as far as we know are in accordance with them, and which cannot be derived within the theory from other laws.

Fundamental are those basic rules that cannot be violated even in principle. For instance, energy conservation and entropy increase are both basic rules, but, as far as we know, only the former is fundamental.

Well, in this strict sense neither is fundamental. Entropy increase is a statistical law, i.e., it holds true up to statistical fluctuations. Energy conservation holds only locally (within General relativity).

I think, what's considered as "fundamental" is (a) dependent on the theoretical context and (b) also sometimes a matter of opinion. Concerning (a): E.g., energy conservation is a fundamental law in Newtonian and special relativistic physics and follows from time-translation invariance of the underlying spacetime model(s). Whether you consider energy conservation or the symmetry as the fundamental law is a matter of taste. I'd say it's energy conservation, because it is the physics content and can (at least in principle) be confirmed or falsified by observation.

Concerning QT I'd say our "7 rules" are the fundamental laws except the projection postulate (as you see that's again an example, where it's also a matter of opinion, what you call fundamental), because the projection postulate is true in very rare special cases of ideal filter measurements, which are rather an exception than standard in our labs. That's not in contradiction with my opinion that the Born rule is a fundamental law within QT. Without it the entire formalism wouldn't have physical meaning, because the Born rule connects the abstract description of the quantum state to observable facts in terms of a probability for the outcome of (accurate) measurements. You can do a measurement, and almost always does, without preparing the measured system in an corresponding eigenstate dependent on the state the system was prepared in before the measurement: E.g., if a photon detector makes "click" it says that at its position there was a photon at the time it made "click". Afterwards the photon is not in a corresponding eigenstate of its position, because (a) it's simply absorbed by the apparatus and not there anymore as an "object" separated from the measurement apparatus and (b) a photon doesn't have a position observable to begin with, so that there cannot be position eigenstates either.

 

[Fra]

I think it's a bit misleading to talk about "axioms". I'd rather call them "postulates". Physics is not a mathematical theory but an empirical science. I'd say the "fundamental laws" or "postulates" are those parts of a physical theory which we consider to be right by experience, i.e., all observations as far as we know are in accordance with them, and which cannot be derived within the theory from other laws.

I am fine with either term and I agree with this paragraph for the purpose here.

/Fredrik

 

[Demystifier]

Without it the entire formalism wouldn't have physical meaning, because the Born rule connects the abstract description of the quantum state to observable facts in terms of a probability for the outcome of (accurate) measurements.

I understand that. But I don't see how to reconcile it with your other claims, namely:
(i) Measurement can only be understood in open systems.
(ii) Born rule is valid also in closed systems.
If Born rule makes sense only for measurements (the quote above) and if measurement requires an open system, then Born rule makes sense only for open systems, which contradicts (ii). Don't you see a logical contradiction and a need to revise at least one of the claims above?

 

[vanhees71]

There is no contradiction. You just haven't read my statement carefully enough.

I said the apparent "measurement problem" can be solved only considering open systems, making use of the fact that a measurement device must be macroscopic to produce an irreversibly stored measurement result. You don't need to solve this apparent problem to use quantum theory with its probabilistic meaning of the state to calculate the probabilities. As a theorist you don't need to bother about the measurement device constructed by the experimentalists, except you want to solve that apparent problem some philosophers still have with quantum theory. The probabilities you calculate with Born's rule are simply measured with devices constructed by experimentalists to measure the observables they want to measure on an ensemble of equally prepared systems and then do the statistics to compare the frequencies of the different outcomes with the predictted probabilities and estimate the statistical significance etc etc.

Also in classical physics I do not need to understand, e.g., the very complicated details of a digital volt meter measuring some voltage across a resistor in some circuit in order to compare it to the value predicted by standard circuit theory. I just have to trust the manufacturer that he had built a device that measures the voltage.

 

[Fra]

Fundamental are those basic rules that cannot be violated even in principle.

This is how I think of this. For me it helps to think of two forms of probabilities, these two versions are also what I take to illustrated the difference between correlation and causation in agent view. The first type is just correlation, the second type is what is "causes" actions. The feedback loop between the two would be more complex though.

1) Correlations - or distributional patterns in the agents or physicists raw observations.

For example the final results from many many trials in particle experiments, where you get an observed "probability" corresponding to the whole ensemble (never individual events). Without an explicit history of measurment records, this "probability" can not be established. This is what one measures in a real experiments. Inferring observed distributions, does not require born rule. In an abstract agent one could envision and event counter, and automatically builds up a frequentist distribution. It requires not "inferences", just observation and counting distinguishable events.

2) Expectations of the agent - that influences the agents instant action.

This is not based on measurements, and this is where the QM inferemnce comes in. It's based on the agents internal inference (expectation) of the future trial, based on it's own encoded (possibly lossy) information. (Or the "initial state" or "preparation") This kind of "probability" would ideally influence the choices made by the agent, in case it was interacting more than a normal "measurement device does". This is that requires the use of the born rule. As it the prescribed rule for constructing the p-value from the abstract reprensentation of the information (quantum state). Thus I imagine this born rule to be value also for closed systems (ie given that we take quantum theory for what it is), because it describes how the agents "aligns" even in between measurements.

Other than this, we expect that if the theory is GOOD, then the two probabilities (ie it's numbers) will agree!
Ie. the theories "expectated" probabiltiy, should match the observed one.

=> For this we need real measurements (and thus "open" up the system), yet the born rule does make sense for closed systems as it prescribes the inference on the agent side - not on the quantum side.

But I think it's helpful, to understand this, if one sees that they are really different, and don't HAVE to agree. Because the physical process of inference from current state, and the observational recordes are two different ones). It's when they do not agree, that we have the input for a learning loop? So I don't think one should think that a theory whose expectations fails to be equal to the actual future, is useless or wrong. I think there is a value to see how it is instead part of the game, and the theory is alive.

What makes the above indeed possible seems like hopeless soup, is that we do not yet know how, observational history "forges" the agents internal states, that implies the expectations. Ie how the "internal state" revision works. It's clearly not a simlpy bayesian update, as several non-commuting observables must be combinied into ONE state, and ontop of this makes this with the capacity the agent has at hand. But this is exactly what i personally take to be one of the key quests for the foundations. Smolin talked about "principle of precedence" which is just indicative of the idea, but I find this to be a largely crazy and immature field, but still extremely interesting and impossible to get off your mind.

/Fredrik

 

[Demystifier]

I said the apparent "measurement problem" can be solved only considering open systems, making use of the fact that a measurement device must be macroscopic to produce an irreversibly stored measurement result.

Perhaps a part of our mutual misunderstanding is that we don't mean the same by the "measurement problem". How would you say concisely what the apparent "measurement problem" is? I'm not asking for the solution, just briefly state the problem!

Also in classical physics I do not need to understand, e.g., the very complicated details of a digital volt meter measuring some voltage across a resistor in some circuit in order to compare it to the value predicted by standard circuit theory. I just have to trust the manufacturer that he had built a device that measures the voltage.

In comparison of classical and quantum measurements, my problem is this. The general principles of classical theoretical physics can be formulated without even mentioning measurement, while those of quantum physics cannot. The culprit is the Born rule. For instance, if I say that the probability is $p={\rm Tr}\rho\pi$, the projector $\pi$ is defined by the measurement. I don't see how to specify $\pi$ without specifying measurement (except in the consistent histories interpretation).

So I will ask you again, in your opinion, is the Born rule valid in the absence of measurement? More specifically, if we have one spin-1/2 particle isolated from the environment, does it make sense to speak of probability of its spin in the z-direction?

 

[Kolmo]

In comparison of classical and quantum measurements, my problem is this. The general principles of classical theoretical physics can be formulated without even mentioning measurement

This doesn't change too much of what you are saying but I'd say more it's can't be formulated without effectively Boolean systems.

If we have a state $\rho$ then the most general time evolution is a quantum operation or CPTP map and all such maps are sums of Kraus operators:
$\rho \rightarrow \mathcal{T}(\rho) = \sum_{k} E_{k}\rho E^{\dagger}_{k}$
which was funnily first proven for unknown reasons by Stinespring while working in the navy¹.

However when interacting with DOFs described by a Boolean algebra (sometimes called "classical" but this terminology is a bit ambiguous) then it is proven that all interactions with a Boolean system induce an evolution of the Kraus sum type above. This was most fully proven in papers by VP Belavkin, but a readable version is in Greg Kuperberg's old notes (https://www.math.ucdavis.edu/~greg/intro-2005.pdf).

The only question then is can one derive these effectively Boolean systems from within quantum theory itself.

This is closely related to conditionalization. For if a sequence of measurements $E_{i}$ is done for defining the conditional:

$$P(E_{i}|E_{1}\ldots E_{i-1}E_{i+1}\ldots E_{n}) = \frac{Tr(E_{1}\ldots E_{n}\rho)}{\sum_{k}Tr(E_{1}\ldots E^{k}_{i}\ldots E_{n}\rho)}$$

Where the $k$ sum is over some POVM containing $E_i$. The ambiguity of which POVM prevents a definition of conditional probability, thus only with a Boolean system which has no basis ambiguity can such conditional probabilities be defined (more explanation here: https://arxiv.org/abs/1310.1484).

¹ W. Forrest Stinespring, Positive functions on C ∗ - algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216

 

[vanhees71]

Perhaps a part of our mutual misunderstanding is that we don't mean the same by the "measurement problem". How would you say concisely what the apparent "measurement problem" is? I'm not asking for the solution, just briefly state the problem!

The measurement problem is the question, how it comes that an ideal measurement device leads to well-defined single outcomes while the quantum state only provides probabilities for this outcome.

For me the answer is obvious: Because the measurement device is constructed to do so.

Together with the projection postulate, which I don't consider a fundamental postulate but the description of a very special case of preparation procedures, which I'd always call "von Neumann filter measurement" to be clear, it's often considered a problem that there should be a dynamical description of the transition from the state $|\psi \rangle \langle \psi|$ with $|\psi \rangle=\sum_n \psi_n |u_n \rangle$ to $|u_{n_0} \rangle \langle u_{n_0}$ when a (here for simplicity considered non-degenerate) observable is measured and the outcome is the eigenvalue of the $n_0$-th eigenstate.

The answer to this, in my opinion, is that a projection can of course not be described by a unitary time evolution, and indeed a filter meausurement (or rather a filter preparation) needs a "filter" and thus you have to use the description in terms of an open quantum system to understand the said transition.

 

[vanhees71]

In comparison of classical and quantum measurements, my problem is this. The general principles of classical theoretical physics can be formulated without even mentioning measurement, while those of quantum physics cannot. The culprit is the Born rule. For instance, if I say that the probability is $p={\rm Tr}\rho\pi$, the projector $\pi$ is defined by the measurement. I don't see how to specify $\pi$ without specifying measurement (except in the consistent histories interpretation).

So I will ask you again, in your opinion, is the Born rule valid in the absence of measurement? More specifically, if we have one spin-1/2 particle isolated from the environment, does it make sense to speak of probability of its spin in the z-direction?

All of theoretical physics makes only sense with operationally defined observables, i.e., implicitly we always use "measurement protocols" do define even classical observables like position, momentum, etc. In other words theoretical physics is a mathematical description of observables, defined by "measurement protocols". Otherwise you would merely have some axiomatic system in the sense of math without any relation to what's observed in nature. It seems only much more familiar given our everyday experience to deal with classical physics than with quantum physics.

I don't know, what you mean by the validity of Born's rule "in absence of measurement". Born's rule is about the probabilities for the outcome of (precise) measurements of an observable. So I can't make sense of Born's rule "in absence of measurement".

So if an electron is prepared in a spin state $|\psi \rangle \langle \psi$ with $\psi=|\hat{z},+1/2 \rangle$ the probabilities given by Born's rule to measure the spin component in some direction $\hat{n}$, $P(\hat{n},\sigma)=|\langle \hat{n},\sigma|\psi \rangle|^2$ are the probabilities to find the value $\sigma \in \{-1/2,1/2\}$ when accurately measuring the spin component in $\hat{n}$ direction.

For me, there's no other meaning in the quantum formalism than providing such probabilities for the outcome of measurements.

 

[Kolmo]

For me the answer is obvious: Because the measurement device is constructed to do so.

This is why I think it is worth reading the 1962 Daneri et al paper"Quantum theory of measurement and ergodicity conditions" where it is shown that a macroscopic body of the right constituency gives rise to irreversible storage. The most detailed modern treatment by Allahverdyan et al. is just further detail upon this, but various thermodynamic processes give rise to superselection¹ for the measurement result.

¹ This isn't to say a macroscopic body cannot display quantum effects when probed in enough detail. Also note "superselection" in modern measurement theory has a broader definition than found in older texts.

 

[WernerQH]

For me, there's no other meaning in the quantum formalism than providing such probabilities for the outcome of measurements.

For me, Q(F)T is a machinery for calculating correlation functions. This is most easily done for "closed" systems, but of course only for open systems is there a point of contact with the real world.

The "measurement problem" is not about measurements, but what it is that is "correlated". It is certainly a rather misleading idea that an electron always has a definite direction of spin.

 

[Kolmo]

The "measurement problem" is not about measurements, but what it is that is "correlated"

Measurement results is the usual answer, at least in areas I work in.

 

[WernerQH]

Measurement results is the usual answer, at least in areas I work in.

But physicists can't agree on what constitutes a "measurement".

 

[vanhees71]

Physicists know very well what constitutes a measurement. It's only too philosophy inclined theoreticians that don't agree ;-). SCNR.

 

[Kolmo]

But physicists can't agree on what constitutes a "measurement".

I've never really seen this in actual research honestly. A measurement is usually defined as a thermodynamically irreversible process leading to a stored result, normally due to the aggregate properties of a macroscopic body.
I know people have problems with this definition, but I don't see it very often in practice. Probably because the arguments are formally equivalent to arguing about does water really boil in Stat Mech because you only get sharp phase transitions in the infinite volume limit. Most wouldn't be interested in the "boiling water problem" either.

 

[Fra]

I don't know, what you mean by the validity of Born's rule "in absence of measurement". Born's rule is about the probabilities for the outcome of (precise) measurements of an observable. So I can't make sense of Born's rule "in absence of measurement".
...
For me, there's no other meaning in the quantum formalism than providing such probabilities for the outcome of measurements.

This is where I know we have different views. And I also find your position somehow incomplete, but perhaps(?) for a different reason than Demystifier, I can not tell for sure.

As I see it, the born rule defines the agents expectations of a "potential" future measurement. And if you (like me) are into the agent picture, this reflects the actual state of the agent, even WITHOUT measurments.

I am sure this would have observable consequences in terms of causality, as the agents actions are tuned as per the born rule, BEFORE the measurement takes place, and BEFORE the information update.

But I see our perspective too, as the agent view is a specific interpretation, mixed up with idea of howto modiy QM, so i agree it does not belong to the minimal QM. But this is why the minimal itnerpretation seems to be as "fine" in a FAPP sense, yet conceptually incomplete. This is why i find the minimal theory deeply insatisfactory.

Mvh
/Fredrik

 

[WernerQH]

Physicists know very well what constitutes a measurement. It's only too philosophy inclined theoreticians that don't agree ;-). SCNR.

Of course we have intuitive and stereotyped ideas about measurement ("Stern Gerlach apparatus"). But as the discussions here amply demonstrate, different people's ideas on this topic do not harmonize. Axiomatization of QM was fueled by the hope of making a vague term like "measurement" precise by embedding it in a rigid set of axioms. But is it really useful to introduce such a term as an irreducible primitive concept of a microscopic theory? John Bell railed against this.

A neutral hydrogen atom in interstellar space is the best example of a closed quantum system. Do we really need to introduce a radio astronomer and his dish to discuss the 21 cm radiation?