The minimal statistical interpretation is neither minimal nor statistical

[martinbn]

As an aside I think the minimal ensemble interpretation can be readily extended to successive measurements without having to worry about conceptualising new ensembles throughout. E.g. If we have a preparation $\rho$ and carry out a measurement of observables $\hat{A}=\sum a_i\Pi_{a_i}$ and $\hat{B}=\sum b_i\Pi_{b_i}$ at times $t_1$ and $t_2$ respectively, we can construct a compound observable $$\hat{C} = \sum_{i,j}a_ib_j\Pi_{a_i}(t_1)\Pi_{b_j}(t_2)$$The user can then compute probabilities and conditional probabilities like
\begin{eqnarray*}
p(a_i\land b_j) &=& \mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}\\
p(b_j | a_i) &=& \frac{\mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}}{ \mathrm{tr}\{ \rho \Pi_{a_i}(t_1) \} }\\
\end{eqnarray*}Without having to conceptualise any intermediate ensemble $\rho(t)$.

What about the evolution? Say in the Schrödinger picture, how does $\rho(t)$ evolve over time?

 

[atyy]

As an aside I think the minimal ensemble interpretation can be readily extended to successive measurements without having to worry about conceptualising new ensembles throughout. E.g. If we have a preparation $\rho$ and carry out a measurement of observables $\hat{A}=\sum a_i\Pi_{a_i}$ and $\hat{B}=\sum b_i\Pi_{b_i}$ at times $t_1$ and $t_2$ respectively, we can construct a compound observable $$\hat{C} = \sum_{i,j}a_ib_j\Pi_{a_i}(t_1)\Pi_{b_j}(t_2)$$The user can then compute probabilities and conditional probabilities like
\begin{eqnarray*}
p(a_i\land b_j) &=& \mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}\\
p(b_j | a_i) &=& \frac{\mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}}{ \mathrm{tr}\{ \rho \Pi_{a_i}(t_1) \} }\\
\end{eqnarray*}Without having to conceptualise any intermediate ensemble $\rho(t)$.

"... can be seen as a consequence of the wave packet reduction postulate of quantum mechanics, since we obtained it in this way. But it isalso possible to take it as a starting point, as a postulate initself: it then provides the probability of any sequence of measurements, in a perfectly unambiguous way, without resorting, either to the wave packet reduction, or even to the Schroedinger equation itself. The latter is actually contained inthe Heisenberg evolution of projection operators, but it remains true that a direct calculation of the evolution of|Ψ> is not really necessary. As for the wave packet reduction, it is also contained in a way in the trace operation of (37), but even less explicitly."
https://arxiv.org/abs/quant-ph/0209123

 

[Morbert]

What about the evolution? Say in the Schrödinger picture, how does $\rho(t)$ evolve over time?

I don't think a minimalist would be concerned with a question like that. The only thing they are interested in is starting with a preparation, dynamics, and possible outcomes $\mathcal{T} = (\rho, U, \{o_i\})$ and computing the probabilities $\mathcal{P} = \{p(o_i)\}$. Any computational technique used to get from $\mathcal{T}$ to $\mathcal{P}$ is just that: a computational technique.

 

[atyy]

I don't think a minimalist would be concerned with a question like that. The only thing they are interested in is starting with a preparation, dynamics, and possible outcomes $\mathcal{T} = (\rho, U, \{o_i\})$ and computing the probabilities $\mathcal{P} = \{p(o_i)\}$. Any computational technique used to get from $\mathcal{T}$ to $\mathcal{P}$ is just that: a computational technique.

Which is why I find @vanhees71's objection incomprehensible from the minimalist point of view. Why should one be bothered by state reduction? It's just a computational technique. That he is bothered by it suggests to me he is not a minimalist.

 

[martinbn]

I don't think a minimalist would be concerned with a question like that. The only thing they are interested in is starting with a preparation, dynamics, and possible outcomes $\mathcal{T} = (\rho, U, \{o_i\})$ and computing the probabilities $\mathcal{P} = \{p(o_i)\}$. Any computational technique used to get from $\mathcal{T}$ to $\mathcal{P}$ is just that: a computational technique.

But what do I put in the formulas for the probabilities? Either the state or the observables are subject to differential equations, the initial conditions are not enough.

 

[Lord Jestocost]

Just to stress that this is not about the ensemble interpretation of quantum mechanichs.

With all due respect, there is no need for beating about the bush. As D. Home and M.A.B. Whitaker put it in “Ensemble interpretations of quantum mechanics. A modern perspective” (Physics Reports, Volume 210, Issue 4, January 1992, Pages 223-317):

“Ensemble interpretations of quantum theory contend that the wave function describes an ensemble of identically prepared systems. They are thus in contrast to ‘orthodox’ or ‘Copenhagen’ interpretations, in which the wave function provides as complete a description as is possible of an individual system.”
[Italics in original, LJ]

 

[martinbn]

With all due respect, there is no need for beating about the bush. As D. Home and M.A.B. Whitaker put it in “Ensemble interpretations of quantum mechanics. A modern perspective” (Physics Reports, Volume 210, Issue 4, January 1992, Pages 223-317):

“Ensemble interpretations of quantum theory contend that the wave function describes an ensemble of identically prepared systems. They are thus in contrast to ‘orthodox’ or ‘Copenhagen’ interpretations, in which the wave function provides as complete a description as is possible of an individual system.”
[Italics in original, LJ]

Yes, but your other quote was about something different. It was about whether one can interpret a superposition as lack of information just like one does in classical mechanics. That is incomaptible with QM and is not what the statistical interpretation says.

 

[AlexCaledin]

- but that "individual system" (unless it's the whole universe) might be merely a thinking tool! So that quote seems presenting a complementarity rather than contradiction...

 

[Demystifier]

If you don't find it ok, can you please tell me what you think is not ok.

ps: This is a discussion thread started by you. I would have thought that you'd be interested in the topic. So why do I have to pull one word answers from you. You are not a spy and this is not an interogation. If you are not interested fine. But it is hard for me to take siriously short "witty" responses with lots of emojies.

Sorry, but you are never satisfied with my answers, so it's hard to find a motivation for a serious answer.

 

[Morbert]

Yes, but your other quote was about something different. It was about whether one can interpret a superposition as lack of information just like one does in classical mechanics. That is incomaptible with QM and is not what the statistical interpretation says.

So e.g. say we want to carry out a measurement of $\hat{A}$ and $\hat{B}$ at times $t_1$ and $t_2$ respectively. This gives us $(\rho,U,\{\sigma_{ij}\})$ where $\rho$ is the preparation, $U$ is the dynamics, and $\{\sigma_{ij}\}$ are possible outcomes where $\sigma_{ij} = \Pi_{a_i}(t_1) \Pi_{b_j}(t_2) = U^\dagger(t_1)\Pi_{a_i}U(t_1)U^\dagger(t_2)\Pi_{b_j}U(t_2)$, then the probability $p(\sigma_{ij}) = \mathrm{tr}\{\rho\sigma_{ij}\}$

 

[Lord Jestocost]

Yes, but your other quote was about something different. It was about whether one can interpret a superposition as lack of information just like one does in classical mechanics. That is incomaptible with QM and is not what the statistical interpretation says. [bold by LJ]

In post #18, I pointed out that one should clearly define what the term “statistical interpretation” means in connection with quantum mechanics. Please, define what you mean by "statistical interpretation". As Andrew Whitaker puts it in his book “Einstein, Bohr and the Quantum Dilemma”:

“Thus all interpretations of quantum theory may be termed statistical if one is thinking of the results of experiments; indeed one may just say it is quantum theory that is statistical in that sense. However, if one thinks of the premeasurement situation, orthodox interpretations are probabilistic, while a Gibbs ensemble interpretation is statistical.”

 

[Lord Jestocost]

...the difference between a proper and improper mixed state?

An improper mixture cannot be given a classical ignorance interpretation.

 

[martinbn]

So e.g. say we want to carry out a measurement of $\hat{A}$ and $\hat{B}$ at times $t_1$ and $t_2$ respectively. This gives us $(\rho,U,\{\sigma_{ij}\})$ where $\rho$ is the preparation, $U$ is the dynamics, and $\{\sigma_{ij}\}$ are possible outcomes where $\sigma_{ij} = \Pi_{a_i}(t_1) \Pi_{b_j}(t_2) = U^\dagger(t_1)\Pi_{a_i}U(t_1)U^\dagger(t_2)\Pi_{b_j}U(t_2)$, then the probability $p(\sigma_{ij}) = \mathrm{tr}\{\rho\sigma_{ij}\}$

I see, now i understand.

 

[vanhees71]

But wouldn't you say that if you enlarge the system you would still get a closed system, and by partial tracing you can get the quantum state of the subsystem?

In principle yes, but in practice you can't describe a many-body system like a measurement device in all microscopic details (neither in quantum nor in classical physics). That's why you have to use effective theories to describe the relevant macroscopic degrees of freedom. That's in fact what's really the challenge of theoretical physics, i.e., to find the right description, answering the questions what are the "relevant degrees of freedom" and how to effectively describe their dynamics. That's where all the complex behavior of macroscopic systems comes from. As Anderson famously said "more is different"!

 

[vanhees71]

Which is why I find @vanhees71's objection incomprehensible from the minimalist point of view. Why should one be bothered by state reduction? It's just a computational technique. That he is bothered by it suggests to me he is not a minimalist.

I never ever have seen state reduction as a computational technique. To the contrary there's nothing calculated nor calculable at all. It's just assuming that the state instantaneously changes after a measurement is done, and the change is not described as a dynamical process at all. My objection is precisely that it is a vague ad-hoc rule with neither theoretical nor experimental foundation or necessity. Even worse, it contradicts the very fundamental construction of relativistic QFTs, where microcausality clearly excludes (by construction) such an instantaneous collapse. At best it's a FAPP hand-wavy rule to describe a preparation procedure in terms of a filtering process a la von Neumann (who called it a measurement rather than a preparation, which leads to further confusion).

I'm at least in the sense a minimalist that I don't understand, where you need this collapse or, closely related, a "Heisenberg cut" at all.

 

[atyy]

In principle yes, but in practice you can't describe a many-body system like a measurement device in all microscopic details (neither in quantum nor in classical physics). That's why you have to use effective theories to describe the relevant macroscopic degrees of freedom. That's in fact what's really the challenge of theoretical physics, i.e., to find the right description, answering the questions what are the "relevant degrees of freedom" and how to effectively describe their dynamics. That's where all the complex behavior of macroscopic systems comes from. As Anderson famously said "more is different"!

This is wrong. The partial trace does not work even in principle to derive the state reduction.

 

[Fra]

But wouldn't you say that if you enlarge the system you would still get a closed system, and by partial tracing you can get the quantum state of the subsystem?

One easily forgets the original justification when doing like this. The reduced state is then not closed, so the unitary axioms will not geneally hold is it may in general be an improper mixture. I would say that it is fallacious to think of a "reduced state" in the same way as an actual quantum state of a closed system, where unitary evolution holds. If you enlarge a system, you change the whole corroboration basis as well.

Edit: I think this touches again upon issues with QM itself, not just intepretations. IMO, the only distinguished cut for one observer is between itself and its environment. The artificial cuts where one observer considers a subsystem of the actual system, should imo not be confused with the original cut, the cut vs subsystem are just different abstractions.

/Fredrik

 

[Demystifier]

As an aside I think the minimal ensemble interpretation can be readily extended to successive measurements without having to worry about conceptualising new ensembles throughout. E.g. If we have a preparation $\rho$ and carry out a measurement of observables $\hat{A}=\sum a_i\Pi_{a_i}$ and $\hat{B}=\sum b_i\Pi_{b_i}$ at times $t_1$ and $t_2$ respectively, we can construct a compound observable $$\hat{C} = \sum_{i,j}a_ib_j\Pi_{a_i}(t_1)\Pi_{b_j}(t_2)$$The user can then compute probabilities and conditional probabilities like
\begin{eqnarray*}
p(a_i\land b_j) &=& \mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}\\
p(b_j | a_i) &=& \frac{\mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}}{ \mathrm{tr}\{ \rho \Pi_{a_i}(t_1) \} }\\
\end{eqnarray*}Without having to conceptualise any intermediate ensemble $\rho(t)$.

Yes, the part of consistent histories interpretation that I like the most is that it is based on a formula for probability of a history that can be applied in any interpretation.

 

[Fra]

I'm at least in the sense a minimalist that I don't understand, where you need this collapse or, closely related, a "Heisenberg cut" at all.

Do we not need have an observer? The that at least in principle defines the cut.

But this gets blurred when on moves the cut around independelty from the original observer. To me that is conceptually inconsistent. Its not the observer that "chooses" the cut by considering an artificial "subsystem", the observer is a manifestation of the cut, no choices left! This is how i see it. But there are still problems with this view, but which i attribute to the theory itself.

/Fredrik

 

[Lord Jestocost]

I'm at least in the sense a minimalist that I don't understand, where you need this collapse or, closely related, a "Heisenberg cut" at all.

In case you would be a minimalist, everything should boil down - in mathematical language - simply to the purely quantum-mechanical von Neumann measurement chain. As the quantum mechanical time evolution (Schrödinger equation) is valid for all physical systems, that would be the end of the story. There is no natural location within this chain where quantum mechanical potentialities emerge into classical actualities (our empirical reality). The cut is a purely epistemological move (introduced by hand) to connect the mathematical formalism of QM to the empirical reality.

 

[vanhees71]

This is wrong. The partial trace does not work even in principle to derive the state reduction.

I don't want to derive state reduction, because I don't need it to make predictions with quantum mechanics. I don't think that there is something like state reduction at all in nature. It's all working with local interactions and thus there cannot be any state reduction.

 

[vanhees71]

In case you would be a minimalist, everything should boil down - in mathematical language - simply to the purely quantum-mechanical von Neumann measurement chain. As the quantum mechanical time evolution (Schrödinger equation) is valid for all physical systems, that would be the end of the story. There is no natural location within this chain where quantum mechanical potentialities emerge into classical actualities (our empirical reality). The cut is a purely epistemological move (introduced by hand) to connect the mathematical formalism of QM to the empirical reality.

All I'm saying is that there's no cut nor a collapse of the state. Measurement devices behave classically because at the level of precision I need to describe them it's sufficient to describe macroscopic coarse-grained observables (e.g., the pointer position of a galvanometer).

 

[atyy]

I don't want to derive state reduction, because I don't need it to make predictions with quantum mechanics. I don't think that there is something like state reduction at all in nature. It's all working with local interactions and thus there cannot be any state reduction.

But if you have only unitary evolution, you have no measurements, no measurement outcomes, and no physics?

 

[Fra]

It's all working with local interactions and thus there cannot be any state reduction.

Conceptually, there is not necessarily a contradiction between these views, if you think that the collapse itself is localized to the agents internal structure.

(Ie agents collapsing internal state, and the locality of agent-agent interactions are not a contradiction. On the contrary does the agents collapsing state offer a way to explain agent-agent interactions. Only problem is that quantum theory as it stands is too heavy to be put into a lightweight agent, so we need more than just an interpretation to make sense of this )

/Fredrik

 

[Demystifier]

But if you have only unitary evolution, you have no measurements, no measurement outcomes, and no physics?

Many worlds?

 

[martinbn]

Many worlds?

What about BM? There is no collapse in BM, right?

 

[Demystifier]

It's all working with local interactions

But those local interactions are deterministic, and yet quantum events are random. So if you believe in fundamental randomness (which you do), then consistency requires the existence of something which is not described by those local interactions. So if you cared about consistency (which you don't, because for you consistency is just an aspect of philosophy), you would conclude that it cannot be true that all is working with those local interactions.

 

[Demystifier]

What about BM? There is no collapse in BM, right?

Yes, but BM is not "unitary only" in the sense in which many worlds are. In BM there are additional variables for which unitarity does not apply.

 

[Demystifier]

I don't want to derive state reduction, because I don't need it to make predictions with quantum mechanics. I don't think that there is something like state reduction at all in nature. It's all working with local interactions and thus there cannot be any state reduction.

I think I finally get the essence of the so-called "minimal" interpretation. It's called minimal not because it's minimal but because it's maximal. More precisely, it is a maximal denial interpretation. Every interpretation denies something, but this interpretation denies more than any other interpretation. It denies almost any interpretative idea that can be denied without directly contradicting experimental facts. When denial is maximal, then the set of non-denied claims is minimal, which is why it's called "minimal". But this interpretation is not defined by the set of accepted claims, for if it was, it would remain agnostic or silent on other claims. This interpretation is defined by what it denies (not by what it accepts), and by this criterion this is the maximal interpretation.

 

[martinbn]

Yes, but BM is not "unitary only" in the sense in which many worlds are. In BM there are additional variables for which unitarity does not apply.

Yes, but the wave function evolves only unitarily. So according to @atyy BM is not physics.