Undergrad Quantum theory - Nature Paper 18 Sept

[Demystifier]

Take polarized light, split it with PBS so that intensities in output beams are 1/3 and 2/3. In the beam with 2/3 intensity place a wave plate so that polarization is rotated by 45deg.
Will it do?

Yes, which helps me to give a better answer to the question by @stevendaryl (see my next post).

 

[Demystifier]

But we've already said what the state is, and all the participants know it. So how is it possible to acquire more information through observations?

This common knowledge is a knowledge of the state $|\psi(t_0)\rangle$ at some time $t_0$. But observation is performed at some later time $t_{\rm obs}>t_0$, so observation determines $|\psi(t_{\rm obs})\rangle$. The act of observation changes the state (that's called quantum contextuality), so $|\psi(t_{\rm obs})\rangle \neq |\psi(t_0)\rangle$. That's how observations acquire new information.

 

[eloheim]

A thought experiment only makes sense if a similar actual experiment is possible, at least in principle. So in an actual experiment, how would that presupposition be acquired?

I wish I could be more helpful than this, but did the specific experimental procedure in the Nature Communications paper, with its step-by-step (timing) recipe, do anything to address the problems you see with being able to actually perform an experiment like this in the real world? Here are a couple quotes that might be trying to answer such objections but I'm not really sure:

In the Gedankenexperiment proposed in this article, multiple
agents have access to different pieces of information, and draw
conclusions by reasoning about the information held by others. In
the general context of quantum theory, the rules for such nested
reasoning may be ambiguous, for the information held by one
agent can, from the viewpoint of another agent, be in a superposition
of different “classical” states. Crucially, however, in the
argument presented here, the agents’ conclusions are all restricted
to supposedly unproblematic “classical” cases. For example, agent
W only needs to derive a statement about agent F in the case
where, conditioned on his own information $\overline{w}$, the information z
held by F has a well-defined value (Table 3). Nevertheless, as we
have shown, the agents arrive at contradictory statements.

And:

Another noticeable difference to earlier no-go results is that the
argument presented here does not employ counterfactual reasoning.
That is, it does not refer to choices that could have been
made but have not actually been made. In fact, in the proposed
experiment, the agents never make any choices (also no delayed
ones, as e.g., in Wheeler’s “delayed choice” experiment[63]). Also,
none of the agents’ statements refers to values that are no longer
available at the time when the statement is made (cf. Table 3).

Anyone find this applicable to the objections at hand? Sorry, I can't decide. lol

 

[stevendaryl]

This common knowledge is a knowledge of the state $|\psi(t_0)\rangle$ at some time $t_0$. But observation is performed at some later time $t_{\rm obs}>t_0$, so observation determines $|\psi(t_{\rm obs})\rangle$. The act of observation changes the state (that's called quantum contextuality), so $|\psi(t_{\rm obs})\rangle \neq |\psi(t_0)\rangle$. That's how observations acquire new information.

Well, that's why I said that it's not a paradox for collapse models.

I do not exactly agree with the example of polarizing filters. I would say that a polarizing filter is not an observation. Seeing that a photon passed through a filter is an observation.

 

[stevendaryl]

Mathematical statement 3 does follow from 1 and 2 if different $\Pi$'s commute. And I think that they do commute if they represent different observers.

In the paper under discussion, there are the following observations:

1. $\overline{W}$ observes $\overline{ok}$
2. $F$ observes $+1/2$
3. $\overline{F}$ observes $\overline{t}$
4. $W$ observes $fail$

1&2 correspond to commuting projection operators, as do 2&3, as do 3&4. But 1&3 don't commute, and neither do 2&4.

 

[Demystifier]

Well, that's why I said that it's not a paradox for collapse models.

But even in the minimal interpretation without explicit collapse, there is some kind of "information update" postulate that effectively does the same thing. Something like that exists in all interpretations (in MWI it is split, in BM it is conditional wave function, in CH it is non-classical logic, ...), which is why all interpretations are consistent, which indeed is what the paper claims.

 

[stevendaryl]

In the paper under discussion, there are the following observations:

1. $\overline{W}$ observes $\overline{ok}$
2. $F$ observes $+1/2$
3. $\overline{F}$ observes $\overline{t}$
4. $W$ observes $fail$

1&2 correspond to commuting projection operators, as do 2&3, as do 3&4. But 1&3 don't commute, and neither do 2&4.

In a many-worlds ontology, you can think of an observer's "world" to be the projection of the universal wavefunction using the projection operator corresponding to the observer's current state of knowledge. So there is the world in which $\overline{W}$ observes $\overline{ok}$. In that world, $F$ observes $+1/2$. So there are the following "worlds":

1. $W_1$: A world in which $\overline{W}$ observes $\overline{ok}$, and $F$ observes $+1/2$ and $\overline{F}$ and $W$ are in indefinite states (like Schrodinger's cat)
2. $W_2$: A world in which $F$ observes $+1/2$ and $\overline{F}$ observes $\overline{t}$ and $\overline{W}$ and $W$ are in indefinite states.
3. $W_3$: A world in which $\overline{F}$ observes $\overline{t}$ and $W$ observes $fail$ and $\overline{W}$ and $F$ are in indefinite states.

Even within one world, different observers thus disagree about which world they are. In world $W_1$, $F$ believes that he is in world $W_2$.

Quantum mechanics is most coherent when all observers agree about all macroscopic facts, but this example is specifically designed to violate that.

 

[stevendaryl]

But even in the minimal interpretation without explicit collapse, there is some kind of "information update" postulate that effectively does the same thing.

I don't see how updating can make any sense unless you believe either that (1) observations change the things that are being observed, or (2) there is extra information above and beyond the wave function. So collapse or hidden variables. What does updating mean in the minimal interpretation?

 

[Demystifier]

I don't see how updating can make any sense unless you believe either that (1) observations change the things that are being observed, or (2) there is extra information above and beyond the wave function. So collapse or hidden variables.

I agree.

What does updating mean in the minimal interpretation?

Minimal interpretation is agnostic on that. That's why it is called minimal. In fact, minimal interpretation is the same as shut-up-and-calculate. ... The only problem with minimal interpretation is that some of its adherents (we have one on this forum) do not just shut up but try to explain what it "really" means, and in such attempts easily fall into a contradiction. When you tell them that it's a contradiction, then they reply that it's philosophy without empirical consequences. And few hours later they forget all that and repeat the same contradiction again. But otherwise, when such guys do the "shut-up-and-calculate" stuff, they are great. (I think you know who I am talking about.)

 

[stevendaryl]

Minimal interpretation is agnostic on that. That's why it is called minimal. In fact, minimal interpretation is the same as shut-up-and-calculate. ...

I think that there is some slight disagreement about what the minimal interpretation means. There is a distinction between having a minimum ontology and having minimal assumptions. Minimal assumptions are in some sense maximalist about ontologies: many ontologies are possible, nothing is ruled out.

The minimal ontology might be something like this:

1. There is only one world.
2. Quantum mechanics works the same on any system (microscopic or macroscopic, measurement devices or not)
3. The probabilities of measurement results are given by the Born rule.

I actually don't think the minimal ontology is self-consistent. You can't have the Born rule being true in general if there is nothing special about measurements.

 

[Demystifier]

I fact, I think I have a new definition of the minimal interpretation of quantum mechanics. Like many other interpretations, it consists of
a) Computation rules for probabilities of measurement outcomes.
b) Explanation of what a) means.
However, other interpretations contain something weird in b). The minimal interpretation, by contrast, contains nothing weird in b). That's why the minimal interpretation is so great, unlike other interpretations it contains nothing weird in b). The only price paid for this absence of weirdness is that some of non-weird claims in b) are in logical contradiction with each other. But logical contradiction is only a philosophical problem, which does not affect the really important fact that the claims in a) are logically consistent and in agreement with experiments. So whenever someone points to a logical inconsistency in non-weird claims in b), you ignore b) entirely and concentrate on a). That's the essence of minimal interpretation.

 

[Demystifier]

The minimal ontology might be something like this:

1. There is only one world.
2. Quantum mechanics works the same on any system (microscopic or macroscopic, measurement devices or not)
3. The probabilities of measurement results are given by the Born rule.

I don't think that 2. and 3. should be called ontology.

You can't have the Born rule being true in general if there is nothing special about measurements.

I agree.

 

[PeterDonis]

In world $W_1$, $F$ believes that he is in world $W_2$.

Why?

 

[stevendaryl]

Why?

The universal wave function is $\sqrt{\frac{1}{3}} |\overline{h}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{+1}{2}\rangle$ If you project for $F$ measuring $+1/2$, you get (after adjusting the normalization):

$|\overline{t}\rangle |\frac{+1}{2}\rangle$

That's the relative state for the world corresponding to the second component having value +1/2. That's world $W_2$.

In a "collapse" interpretation, $F$ measuring +1/2 would collapse the wave function of the world to collapse to the state $|\overline{t}\rangle |\frac{+1}{2}\rangle$. In a many-worlds interpretation, $F$ still thinks that the world is described by $|\overline{t}\rangle |\frac{+1}{2}\rangle$, but he allows for alternative possible worlds in which (for example) the other lab got result $\overline{h}$ rather than $\overline{t}$.

So when $F$ is in the state $|\frac{+1}{2}\rangle$ (which is the state of his having observed +1/2), his brain believes the world to be $W_2$.

But world $W_1$ is the state described by the wave function $|\overline{ok}\rangle |\frac{+1}{2}\rangle = \sqrt{\frac{1}{2}} |\overline{h}\rangle |\frac{+1}{2}\rangle - \sqrt{\frac{1}{2}} |\overline{t}\rangle |\frac{+1}{2}\rangle$. In this world, $F$ is in the state $|\frac{+1}{2}\rangle$. So we have already argued that $F$ believes that the world is $W_2$. "Believes" might not be the appropriate word, here, because it's not a matter of his wrong about that---there is no world that has the status of being "real", and so there is no right or wrong about what world we are in.

 

[PeterDonis]

when $F$ is in the state $|\frac{+1}{2}\rangle$ (which is the state of his having observed +1/2), his brain believes the world to be $W_2$.

That part I get, yes.

world $W_1$ is the state described by the wave function...

This part I don't get, because this wave function is not the projection of the universal wave function you give for $\bar{W}$ measuring ok.

 

[stevendaryl]

This part I don't get, because this wave function is not the projection of the universal wave function you give for $\bar{W}$ measuring ok.

Yes, it is. The definition of the state $|\overline{ok}\rangle$ is $\sqrt{\frac{1}{2}} |\overline{h}\rangle + \sqrt{\frac{1}{2}} |\overline{t}\rangle$. The state $|\overline{fail}\rangle$ is $\sqrt{\frac{1}{2}} |\overline{h}\rangle - \sqrt{\frac{1}{2}} |\overline{t}\rangle$.

[edit: I got the minus signs wrong. It should be:]
$|\overline{ok}\rangle = \sqrt{\frac{1}{2}} |\overline{h}\rangle - \sqrt{\frac{1}{2}} |\overline{t}\rangle$
$|\overline{fail}\rangle = \sqrt{\frac{1}{2}} |\overline{h}\rangle + \sqrt{\frac{1}{2}} |\overline{t}\rangle$

So in terms of the basis $|\overline{ok}\rangle$, $|\overline{fail}\rangle$, the state of the world is:

$\sqrt{\frac{2}{3}} |\overline{fail}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{6}} |\overline{fail}\rangle |\frac{+1}{2}\rangle - \sqrt{\frac{1}{6}} |\overline{ok}\rangle |\frac{+1}{2}\rangle$

So the projection onto the subspace where the first component is $\overline{ok}$ yields: $|\overline{ok}\rangle |\frac{+1}{2}\rangle$. That's $W_1$

 

[DarMM]

What is at stake here is the Born rule in the subsystem. Is probability of a subsystem given by $|\psi|^2$?

Thanks for the response. I assume you saw their Table 4 where they mention that HV theories like Bohmian Mechanics drop Consistency when applied to subsystems, but the Born rule when applied to the universe.

 

[PeterDonis]

in terms of the basis $|\overline{ok}\rangle$, $|\overline{fail}\rangle$, the state of the world is:

I get something different than you; here are the steps of my calculation:

$$
\sqrt{\frac{1}{3}} |\overline{h}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{1}{2}\rangle
$$

$$
\sqrt{\frac{2}{3}} |\overline{ok}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{1}{2}\rangle
$$

$$
\sqrt{\frac{2}{3}} |\overline{ok}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{6}} |\overline{ok}\rangle |\frac{1}{2}\rangle - \sqrt{\frac{1}{6}} |\overline{fail}\rangle |\frac{1}{2}\rangle
$$

The projection of this onto the $\overline{ok}$ subspace has components for both $+ 1/2$ and $- 1/2$ for $F$.

 

[stevendaryl]

I get something different than you; here are the steps of my calculation:

$$
\sqrt{\frac{1}{3}} |\overline{h}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{1}{2}\rangle
$$

$$
\sqrt{\frac{2}{3}} |\overline{ok}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{1}{2}\rangle
$$

I made a mistake (corrected now): The state $|\overline{ok}\rangle$ is defined with a minus sign: $|\overline{ok}\rangle = \sqrt{\frac{1}{2}} |\overline{h}\rangle - \sqrt{\frac{1}{2}} |\overline{t}\rangle$

 

[PeterDonis]

I made a mistake (corrected now): The state $|\overline{ok}\rangle$ is defined with a minus sign

Ah, ok.

 

[zonde]

I do not exactly agree with the example of polarizing filters. I would say that a polarizing filter is not an observation. Seeing that a photon passed through a filter is an observation.

Of course my example does not represent the experiment fairly. But that's because the thought experiment is problematic. It seems to me that this thought experiment violates no-cloning theorem. Observer $\overline{F}$ is measuring $|\overline{h}\rangle$ or $|\overline{t}\rangle$, then depending on result he sends to $F$ either $|\frac{-1}{2}\rangle$ or $|fail\rangle$. But phase relationship between observer $\overline{F}$ memory states $|\overline{h}\rangle$ and $|\overline{t}\rangle$ is still there and observer $F$ gets copy of that phase relationship with the system he gets. So there are two copies of arbitrary state. That's the source of the contradiction I would say.
If the paper fairly represents reasoning of MWI then it is a problem of MWI. But I'm not sure about that. When the worlds split is it required that each system in separate world remains in coherent superposition with it's complementary copy in the other world? The world as a whole can be in coherent superposition with the other world, but the subsystems of worlds are then only entangled.

 

[vanhees71]

Reading the marvelous blog by Aaronson posted in #44, I come to the conclusion that once more the authors (and the referees of the paper except one) got lost in philosophy. It's a bit sad that Aaronson doesn't explain its states. If I understand it right he has a 2D Hilbert space (spin 1/2) with one basis he labels with $|0 \rangle$ and $1 \rangle$. Taking the usual spin-$z$ eigenstates, these are the eigenvectors of the operators $2 \hat{s}_z +1$ with eigenvalues $1$ and $0$. Then, I figured out that the other basis is that of the spin operator in $\pm 45^{\circ}$ direction, i.e.,
$$|\pm \rangle=\frac{1}{\sqrt{2}} (|1 \rangle \pm |0 \rangle),$$
and the Hadamard transform is
$$\hat{H}=|+ \rangle \langle 1|+ |- \rangle \langle 0 |.$$

Then, there's indeed obviously no more "weirdness" than the usual "weirdness" and thus, within the minimal interpretation, no weirdness at all. To boil down the argument from an experimental point of view, what's wrong in the author's argument is the assumption that the various "labs are in isololation" and then nevertheless assuming the various agents measuring each other and the labs still are in isolation. The apparent contradiction is thus a classical example for the old rule "ex falso quodlibet".

So the assumption of isolated labs which are nevertheless measured is contradicting QT right in the assumptions and one of the reasons for much "quantum weirdness", namely that it is impossible to measure something microscopic without disturbing it, because one needs at least another microscopic thing of the same order of magnitude as the measured one. The most obvious example are charges of elementary particles. It makes no sense to talk about the Coulomb field of a single electron, say, since an electric field is defined by the action of this field on a test charge, which is so small that it doesn't affect the measured field significantly. That's of course impossible if the measured field is that of an electron since one needs at least another charged test particle. To keep the classical notion of the Coulomb field you'd need a charge much smaller than the electron's charge which doesn't exist in Nature (given that quarks with there 1/3 and 2/3 elementary charges are confined and thus not available, besides the fact that even this charge is not "negligibly" small against the electron charge).

 

[zonde]

To boil down the argument from an experimental point of view, what's wrong in the author's argument is the assumption that the various "labs are in isololation" and then nevertheless assuming the various agents measuring each other and the labs still are in isolation.

I don't see that this is assumed. Observer $\overline{F}$ is isolated up to the point when he is measured by $\overline{W}$ (except that he sends a system to $F$ before that). After $\overline{W}$ measured $\overline{F}$, there is no need for $\overline{F}$ to be isolated, he just has to keep his memories. And $\overline{F}$ is not measuring $\overline{W}$, he just knows what the setup is supposed to be and uses this knowledge in his reasoning.
It's similar for $F$ and $W$ pair.

 

[zonde]

The apparent contradiction is thus a classical example for the old rule "ex falso quodlibet".

Didn't knew that contradictory statement have such a disastrous consequences for logical reasoning: Principle of explosion.

 

[Demystifier]

it is impossible to measure something microscopic without disturbing it, because one needs at least another microscopic thing of the same order of magnitude as the measured one. The most obvious example are charges of elementary particles. It makes no sense to talk about the Coulomb field of a single electron, say, since an electric field is defined by the action of this field on a test charge, which is so small that it doesn't affect the measured field significantly. That's of course impossible if the measured field is that of an electron since one needs at least another charged test particle.

It's an off-topic, but how then the charge of a single electron is measured?

 

[stevendaryl]

Of course my example does not represent the experiment fairly. But that's because the thought experiment is problematic. It seems to me that this thought experiment violates no-cloning theorem. Observer $\overline{F}$ is measuring $|\overline{h}\rangle$ or $|\overline{t}\rangle$, then depending on result he sends to $F$ either $|\frac{-1}{2}\rangle$ or $|fail\rangle$. But phase relationship between observer $\overline{F}$ memory states $|\overline{h}\rangle$ and $|\overline{t}\rangle$ is still there and observer $F$ gets copy of that phase relationship with the system he gets. So there are two copies of arbitrary state. That's the source of the contradiction I would say.

No, I don't think there is any cloning of states going on. What I would think state cloning would amount to, in a two-component system, is something like this:

$(\alpha |\overline{h}\rangle + \beta |\overline{t}\rangle) |0\rangle \Rightarrow (\alpha |\overline{h}\rangle + \beta |\overline{t}\rangle) (\alpha |\frac{+1}{2}\rangle + \beta |\frac{-1}{2} \rangle)$ (where $|0\rangle$ is some initial state of the $F$ system).

That would violate linearity. But there is nothing like that going on in this example.

I think that the point about phase relationships is relevant here, though. If $|\overline{h}\rangle$ and $|\overline{t}\rangle$ are macroscopically different states, then the phase relationship between them is probably unmeasurable. But that's not no-cloning (which applies to microscopic systems, as well).

If the paper fairly represents reasoning of MWI then it is a problem of MWI. But I'm not sure about that. When the worlds split is it required that each system in separate world remains in coherent superposition with it's complementary copy in the other world? The world as a whole can be in coherent superposition with the other world, but the subsystems of worlds are then only entangled.

The only problem that it presents for MWI is with the intuitive idea that in a particular world, all macroscopic facts are determinate (Schrodinger's live cat and his dead cat are in different worlds). But there is no guarantee of that. A world may contain macroscopic superpositions. Maybe that exotic possibility detracts from the appeal of MWI, but it doesn't (to me) seem to contradict MWI.

Actually, if you analyze this thought-experiment through MWI, I think you will get something more like the many-minds theory. Each observer is in his own possible world.

 

[stevendaryl]

So the assumption of isolated labs which are nevertheless measured is contradicting QT right in the assumptions and one of the reasons for much "quantum weirdness", namely that it is impossible to measure something microscopic without disturbing it, because one needs at least another microscopic thing of the same order of magnitude as the measured one.

It depends on what you mean by "measure". If you have two microscopic variables that are entangled, then you can learn something about one variable by measuring the other.

 

[vanhees71]

It's an off-topic, but how then the charge of a single electron is measured?

You do scattering experiments and deduce the coupling constant from the measured cross section. That's how all 20+x constants of the Standard Model + neutrino masses and mixing parameters have to be measured.

 

[vanhees71]

It depends on what you mean by "measure". If you have two microscopic variables that are entangled, then you can learn something about one variable by measuring the other.

Sure, but then you have to adapt your state according to the information gained. If the issue is really as simple as reviewed in the blog by Aaronson, there's nothing new with this gedanken experiment but good (or bad depending on your philosophical prejudices) old contextuality of QT.

 

[zonde]

No, I don't think there is any cloning of states going on. What I would think state cloning would amount to, in a two-component system, is something like this:

$(\alpha |\overline{h}\rangle + \beta |\overline{t}\rangle) |0\rangle \Rightarrow (\alpha |\overline{h}\rangle + \beta |\overline{t}\rangle) (\alpha |\frac{+1}{2}\rangle + \beta |\frac{-1}{2} \rangle)$ (where $|0\rangle$ is some initial state of the $F$ system).

That would violate linearity. But there is nothing like that going on in this example.

I think that the point about phase relationships is relevant here, though. If $|\overline{h}\rangle$ and $|\overline{t}\rangle$ are macroscopically different states, then the phase relationship between them is probably unmeasurable. But that's not no-cloning (which applies to microscopic systems, as well).

Ok, if there is only one copy of the state than the experiment is no more mysterious than three polarizers experiment. When $\overline{W}$ measures the state and gets $|\overline{ok}\rangle$ the state of the system he gets is updated from $|fail\rangle$ to $|\frac{+1}{2}$ and then $W$ measuring this system obviously can get $|ok\rangle$ half of the time.
I'm not sure however what happens with $\overline{F}$ and $F$ observers and their memories.