The minimal statistical interpretation is neither minimal nor statistical

[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.

 

[vanhees71]

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

Why? A measurement device shows values for the measured quantity. Quantum mechanically the system is entangled with the measurement device such that the "macroscopic pointer state" of the measurement device is in one-to-one correlation with the value of the measured observable. It's clear that the state of the measured system (provided it can be separated from the measurement device after the measurement at all, which is, e.g., usually not the case for a photon which is absorbed due to the measurement by a photodetector) is given by tracing out the measurement device. Due to the entanglement necessary to have a precise measurment of the measured observable this state is necessarily a proper mixed state. You can call this "collapse", but it's not the naive collapse given handwavingly in a few words in textbooks but one that is completely consistent with the local nature of interactions and the "unitary" quantum dynamics of the composite system.

 

[Fra]

Yes, but the wave function evolves only unitarily.

But only for closed proper systems.

Not sure what the pure bohmist take is on that, but does bohmians think of the hidden variables as closed or if they allow for secret storage of info?

/Fredrik

 

[vanhees71]

I think the Bohmians claim that particles move in the trajectories they calculate. The trouble I have with this ontic interpretation of the Bohmian trajectories is that they cannot be observed. At least I'm not aware of any experiment which unambigously demonstrates Bohmian trajectories (leaving out the problem with BM in the relativistic context; I think it's in principle a sound and solid framework for non-relativistic QM but that's not convincingly the case in application to relativistic QFT).

 

[atyy]

Why? A measurement device shows values for the measured quantity. Quantum mechanically the system is entangled with the measurement device such that the "macroscopic pointer state" of the measurement device is in one-to-one correlation with the value of the measured observable. It's clear that the state of the measured system (provided it can be separated from the measurement device after the measurement at all, which is, e.g., usually not the case for a photon which is absorbed due to the measurement by a photodetector) is given by tracing out the measurement device. Due to the entanglement necessary to have a precise measurment of the measured observable this state is necessarily a proper mixed state. You can call this "collapse", but it's not the naive collapse given handwavingly in a few words in textbooks but one that is completely consistent with the local nature of interactions and the "unitary" quantum dynamics of the composite system.

But in this formalism, sequential measurements are not possible. For a sequence of measurements, you substitute one simultaneous measurement of everything after all the measurement devices have interacted with the system being measured. If you never have sequential measurements, you don't need state reduction.

 

[Lord Jestocost]

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).

What are you actually trying to tell the forum readers? Whether you think that measurement devices behave classically is completely irrelevant. Do you seriously think that your "measurement devices“ (when entangled with the quantum entity which is measured on) break physically the linear nature of quantum theory (a superpostion cannot be replaced by the logical alternative 'either … or').
Maybe, you don’t really understand what the ‘macro-objectification problem’ means in the framework of quantum theory. I therefore deeply recommend to read Giancarlo Ghirardi’s entry “Collapse Theories“ on the “Stanford Encyclopedia of Philosophy”. https://plato.stanford.edu/entries/qm-collapse/

 

[vanhees71]

If I can isolate the measured system again after the measurement, I have a new state given by the partial trace. Why shouldn't I be able to describe another measurement when the measured system is known to be prepared in this state?

 

[atyy]

If I can isolate the measured system again after the measurement, I have a new state given by the partial trace. Why shouldn't I be able to describe another measurement when the measured system is known to be prepared in this state?

If you only have a unitarily evolving wave function, the only rule you have is the Born rule, which gives the probabilities of outcomes at one times. So let's say you measure observable A with possible outcomes {ai} at t1, and observable B with possoble outcomes {bi} at t2. You do not have any rule to give you the joint probability P(ai, bj). So you are not able to predict all the probabilities for sequential measurement.

Instead, what you must do in this formalism, to avoid sequential measurement is to have all the measuring devices interact in sequence, then separate out and maintain their pointer states. After the sequence of unitary interactions, you make a simultaneous measurement of all measuring devices. The probabilities will be the same as what you get with sequential measurement and state reduction, but here one does not acknowledge the reality of sequential measurement outcomes.

 

[vanhees71]

I don't have only unitary time evolution but also the partial tracing. The time evolution of the statistical operator for a part of a closed system (which by definition is an open system) is not unitary.

Of course, if you can the entire sequence exactly then you don't need anything else than unitary time evolution and Born's rule, but I don't know any macroscopic system, where this is feasible.

 

[Morbert]

If you only have a unitarily evolving wave function, the only rule you have is the Born rule, which gives the probabilities of outcomes at one times. So let's say you measure observable A with possible outcomes {ai} at t1, and observable B with possoble outcomes {bi} at t2. You do not have any rule to give you the joint probability P(ai, bj). So you are not able to predict all the probabilities for sequential measurement.

You could just extend the state space to include both times: $\mathcal{H} = \mathcal{H}_{t_1} \otimes \mathcal{H}_{t_2}$

We could then define suitable projector $\Pi_{a_i\land b_j} =\Pi_{b_j}^{t_2} \Pi_{a_i}^{t_1}$

Applying the Born rule then tells us the probability of the outcome $a_i \land b_j$ is given by $\mathrm{tr}(\Pi_{a_i\land b_j}\rho)$

 

[atyy]

I don't have only unitary time evolution but also the partial tracing. The time evolution of the statistical operator for a part of a closed system (which by definition is an open system) is not unitary.

But the partial trace will not allow you do derive P(ai,bj). The partial trace derives the state reduction for non-selective measurements (where the results of the first measurement are discarded), but it does not derive the state reduction for selective measurements, which is needed for P(ai,bj).

 

[vanhees71]

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.

QFT is not relativistic. So I don't understand your first sentence. What do you think does not work with local interactions? Do you have a falsification of local relativistic QFTs?

 

[Demystifier]

QFT is not relativistic. So I don't understand your first sentence. What do you think does not work with local interactions? Do you have a falsification of local relativistic QFTs?

I'm afraid I don't understand your first sentence.

Anyway, interactions are deterministic because if you know the state at one time, then Hamiltonian evolution determines the state at any other time.

 

[vanhees71]

Argh. I meant of course "QFT is not deterministic".

Of course the QT state evolution is a deterministic law. That doesn't imply that the theory is deterministic. This is really what we are discussing about the whole time.

 

[atyy]

You could just extend the state space to include both times: $\mathcal{H} = \mathcal{H}_{t_1} \otimes \mathcal{H}_{t_2}$

We could then define suitable projector $\Pi_{a_i\land b_j} =\Pi_{b_j}^{t_2} \Pi_{a_i}^{t_1}$

Applying the Born rule then tells us the probability of the outcome $a_i \land b_j$ is given by $\mathrm{tr}(\Pi_{a_i\land b_j}\rho)$

Yes, but when we write this in the Schroedinger picture, we will find non-unitary evolution. The point is that state reduction is part of the standard operational view of QM, and one cannot have unitary evolution alone (without many worlds or Bohmian mechanics).

 

[vanhees71]

Why can one not have unitary evolution alone? If that were the case we'd need another dynamical law complementing it. I don't know of any such law nor of any necessity for it.