9 Reasons Quantum Mechanics is Incomplete - Comments

[A. Neumaier]

(The reason I say "almost" is because measurement involves irreversible changes, and I'm not sure how to model irreversible changes using pure quantum mechanics.)

This needs interactions with the environment, which (for tractability) is usually taken to be a heat bath consisting of infinitely many harmonic oscillators with a continuous, unbounded frequency spectrum, and eliminating the bath degrees of freedom. The result is (after invoking the Markov approximation) a Lindblad-type equation for the reduced density operator. Pure states do not work because dissipation usually destroys pureness of the initial state.

If purity is to be preserved, one can use the Schrödinger equation with an optical potential, i.e., an imaginary contribution to the Hamiltonian. This is less accurate than Lindblad models but often adequate.

 

[PeterDonis]

Now let us consider an entangled state:
$$
|\uparrow\uparrow\rangle + |\downarrow\downarrow\rangle = \{1.1, 2.2, 3.3, 4.4\}
$$

This doesn't look right. It seems like $|\uparrow\uparrow\rangle$ should be $\{1.1, 1.2, 2.1, 2.2\}$, and similarly for $|\downarrow\downarrow\rangle$ with ontic states $3$ and $4$.

 

[DarMM]

This doesn't look right. It seems like $|\uparrow\uparrow\rangle$ should be $\{1.1, 1.2, 2.1, 2.2\}$, and similarly for $|\downarrow\downarrow\rangle$ with ontic states $3$ and $4$.

That is the correct form for $|\uparrow\uparrow\rangle$, but I gave the entangled state not $|\uparrow\uparrow\rangle$.

 

[PeterDonis]

That is the correct form for $|\uparrow\uparrow\rangle$, but I gave the entangled state not $|\uparrow\uparrow\rangle$.

The entangled state is just a superposition of $|\uparrow\uparrow\rangle$ and $|\downarrow\downarrow\rangle$, which just means you include all the ontic possibilities from both one-qubit states in the set of ontic possibilities for the two-qubit states. So I still don't see how ontic states like $1.2$ or $2.1$ (or $3.4$ or $4.3$) are excluded.

 

[DarMM]

The entangled state is just a superposition of $|\uparrow\uparrow\rangle$ and $|\downarrow\downarrow\rangle$, which just means you include all the ontic possibilities from both one-qubit states in the set of ontic possibilities for the two-qubit states.

No it doesn't as that would lead to a non-maximal knowledge state. The $+$ of superposition is a mapping between maximal knowledge (in standard terminology "pure") states. See for example the case given at the start of my post, we have:
$$
|\uparrow\rangle = \{1,2\} \\
|\downarrow\rangle = \{3,4\}
$$
However it is not the case that $|+\rangle = |\uparrow\rangle + |\downarrow\rangle$ is given by $\{1,2,3,4\}$ as that is a mixed state, instead it is given by $\{1,3\}$

 

[vanhees71]

Can you explain your notation? How can a ket be the same as a bunch of numbers in curly brackets?

 

[DarMM]

It's in my post #69 above. You have an ontic space with four elements or states $1,2,3,4$. Restricted epistemic states over such a space replicate many, though not all, features of quantum mechanics. That is they provide a local classical model that replicates what I listed in post #70. Spekkens paper I linked to goes into more detail.

 

[PeterDonis]

The $+$ of superposition is a mapping between maximal knowledge (in standard terminology "pure") states.

I understand that for the standard notation using kets; I'm just not seeing how that standard notation maps to the ontic states for more than one qubit. I'll look at the Spekkens paper.

 

[DarMM]

I understand that for the standard notation using kets; I'm just not seeing how that standard notation maps to the ontic states for more than one qubit. I'll look at the Spekkens paper.

It's very "formal" just to tell you not transparently obvious, i.e. it's just a bilinear map with the right properties.

 

[vanhees71]

Well, it's highly misleading to label the toy model with the same symbols as the quantum (state) kets. The toy model obviously cannot reproduce all established quantum mechanical facts; for sure not the violation of Bell's inequality. I've a look at Spekken's paper later...

 

[PeterDonis]

it's just a bilinear map with the right properties.

There's a bilinear map for single qubits, yes; you give it explicitly in your post. But I don't see how the bilinear map for a two-qubit system is constructed.

 

[DarMM]

There's a bilinear map for single qubits, yes; you give it explicitly in your post. But I don't see how the bilinear map for a two-qubit system is constructed.

The details are given in Spekkens paper section III.A

You might prefer the exposition in https://arxiv.org/abs/1103.5037.

 

[DarMM]

Well, it's highly misleading to label the toy model with the same symbols as the quantum (state) kets. The toy model obviously cannot reproduce all established quantum mechanical facts; for sure not the violation of Bell's inequality. I've a look at Spekken's paper later...

Spekkens does it, I'm not going to deviate from his notation when quoting him. However in his paper it's obvious what he's doing.

And the model doesn't replicate the Bell Inequalities as I said, that's the point. It provides a local classical model of some features of QM in order to show that those features aren't specifically "quantum". Then the features it doesn't replicate are what is especially quantum. It's being used to pinpoint uniquely quantum features. See post #70

The reason I brought it up was because Wigner's friend shows up in this model rather simply.

 

[DarMM]

People might be interested to know that it's specifically because of models like Spekkens that people were led to trying the Frauchiger-Renner and Brukner Objectivity theorems.

(Brukner discussed here: https://www.physicsforums.com/threa...-basic-wigners-friend-type-experiment.968181/)

Initially there seems to be some contradiction between QM in the Wigner's friend scenario and the fact of definite outcomes for subsystems. However Spekkens toy model (and similar) shows that your not going to find such an issue because Wigner's friend can occur in purely local classical theories.

Thus the attempt to find a Wigner's friend that incorporates uniquely quantum features not replicated by these classical models like the CHSH inequalities or Hardy's paradox. Frauchiger-Renner is "Wigner + Hardy" and Brukner's objectivity theorem is "Wigner + CHSH".

However it seems neither do the job. There is no contradiction between a superobserver using a superposition and definite outcomes for subsystems, because the superposition refers to the statistics of superobservables of the friend's entire lab.

 

[Avodyne]

I'm trying to reconcile the following statements:

There is one reason that quantum mechanics is a complete physical theory.

If you look at @jambaugh 's post he's clearly taking an epistemic view of the wavefunction where collapse is just (generalized) Bayesian conditioning.

Secondly an epistemic account will by nature be taking the view that QM is incomplete as it views a central object in it as non-representational.

Question: is there a non-many-worlds interpretation of QM for which QM is "complete"?

 

[A. Neumaier]

Question: is there a non-many-worlds interpretation of QM for which QM is "complete"?

The thermal interpretation has this property.

 

[DarMM]

I'm trying to reconcile the following statements:

Basically @jambaugh is (I think) saying QM is complete because there is no further you can go, i.e. you cannot obtain the deeper explanation or any deeper theory will disagree with it. I'm saying epistemic views are not complete in the sense @stevendaryl gives of not providing the explanation for certain predictions.

Or to be brief I'm saying it's not complete and @jambaugh is saying it's as complete as it is going to get.

Question: is there a non-many-worlds interpretation of QM for which QM is "complete"?

Bohmian Mechanics. I would say there is no fully worked out complete interpretation for QFT.

 

[A. Neumaier]

Bohmian Mechanics. I would say there is no fully worked out complete interpretation for QFT.

Bohmian mechanics is a ''completion'' of ordinary QM by hidden variables. It is not QM itself but a different theory which contains QM as a subtheory.

 

[DarMM]

Bohmian mechanics is a ''completion'' of ordinary QM by hidden variables. It is not QM itself but a different theory which contains QM as a subtheory.

True, but most people include it in the notion of "complete" interpretations, however out of equilibrium it is more general than QM as you say. Strictly speaking it's not an interpretation, but then few of the interpretations of QM are when you look at them closely.

I'd say your Thermal Interpretation is the clearest realist and complete view of QFT I've seen.

 

[DarMM]

Thus Many-Worlds, fantasy or not, is a consequence of quantum mechanics. It really is. Unless you want to add hidden variables that say that only one of the possibilities is "real"

That depends on a very particular view of what a superposition is. How can Many-Worlds for example cope with the fact that any system confined to a finite volume is always in a mixed state?

 

[PeterDonis]

The details are given in Spekkens paper section III.A

This section doesn't say anything about two-qubit systems; the paper only starts discussing those in Section IV. The first time in that section that I see a state that looks like $| \uparrow \uparrow \rangle + | \downarrow \downarrow \rangle$ is in Section IV.C, p. 15, just before equation 78. There the state $[ (1\ \text{V}\ 2) \cdot (1\ \text{V}\ 2)] \ \text{V}\ [(3\ \text{V}\ 4) \cdot (3\ \text{V}\ 4)]$ is given, which looks to me like the equivalent in the paper's notation of $| \uparrow \uparrow \rangle + | \downarrow \downarrow \rangle$--but the paper says this state is a state of non-maximal knowledge, whereas you're saying the state you mean by $| \uparrow \uparrow \rangle + | \downarrow \downarrow \rangle$ is a state of maximal knowledge.

So either I'm misunderstanding the paper's notation and how it relates to standard ket notation, or you're using standard ket notation to mean something other than what it obviously seems to map to in the paper's notation.

 

[PeterDonis]

You might prefer the exposition in https://arxiv.org/abs/1103.5037.

This isn't really helping me because I'm not familiar with qubit stabilizer notation to begin with, so I would have to learn two unfamiliar things instead of one.

 

[DarMM]

This section doesn't say anything about two-qubit systems; the paper only starts discussing those in Section IV.

Sorry how the mapping is defined there can just be extended to two qubit systems to give the results I have. The same method of construction goes through. I can show how based on the single qubit case the two qubit case works if you wish.

The first time in that section that I see a state that looks like $| \uparrow \uparrow \rangle + | \downarrow \downarrow \rangle$ is in Section IV.C, p. 15, just before equation 78. There the state $[ (1\ \text{V}\ 2) \cdot (1\ \text{V}\ 2)] \ \text{V}\ [(3\ \text{V}\ 4) \cdot (3\ \text{V}\ 4)]$ is given, which looks to me like the equivalent in the paper's notation of $| \uparrow \uparrow \rangle + | \downarrow \downarrow \rangle$--but the paper says this state is a state of non-maximal knowledge

No that's the papers version of the mixed state:
$$\frac{1}{2}|\uparrow\uparrow\rangle \langle\uparrow\uparrow\rangle + \frac{1}{2}|\downarrow\downarrow\rangle \langle\downarrow\downarrow|$$
the state $| \uparrow \uparrow \rangle + | \downarrow \downarrow \rangle$ would be:
$$(1.1) \lor (2.2) \lor (3.3) \lor (4.4)$$
I gave this as:
$$\{1.1, 2.2, 3.3, 4.4\}$$
Which was actually a bit stupid of me as I should stick to Spekkens notation.

 

[PeterDonis]

I can show how based on the single qubit case the two qubit case works if you wish.

That would be helpful. In particular, I'm still not seeing how this

$$
| \uparrow \uparrow \rangle + | \downarrow \downarrow \rangle
$$

matches up to this

$$
(1.1) \lor (2.2) \lor (3.3) \lor (4.4)
$$

I understand fine what the latter means in terms of the ontic states of the model. I just don't understand how to get from the former to the latter.

the mixed state:
$$
\frac{1}{2}|\uparrow\uparrow\rangle \langle\uparrow\uparrow\rangle + \frac{1}{2}|\downarrow\downarrow\rangle \langle\downarrow\downarrow\rangle
$$

I think you mis-formatted this; it looks to me like it should be

$$
\frac{1}{2}\vert\uparrow\uparrow\rangle \langle\uparrow\uparrow\vert + \frac{1}{2}\vert\downarrow\downarrow\rangle \langle\downarrow\downarrow\vert
$$

 

[DarMM]

That would be helpful. In particular, I'm still not seeing how this

$$
| \uparrow \uparrow \rangle + | \downarrow \downarrow \rangle
$$

matches up to this

$$
(1.1) \lor (2.2) \lor (3.3) \lor (4.4)
$$

I understand fine what the latter means in terms of the ontic states of the model. I just don't understand how to get from the former to the latter.

Perfect, I'll put it up tomorrow.

I think you mis-formatted this; it looks to me like it should be

$$
\frac{1}{2}\vert\uparrow\uparrow\rangle \langle\uparrow\uparrow\vert + \frac{1}{2}\vert\downarrow\downarrow\rangle \langle\downarrow\downarrow\vert
$$

Yes indeed, I was even staring there at my post trying to see what was wrong! Took me a while to see the little bracket. Corrected now.

 

[Avodyne]

IMO, many-worlds with the Vaidman interpretation of the Born rule is complete:

http://philsci-archive.pitt.edu/14590/

 

[DarMM]

IMO, many-worlds with the Vaidman interpretation of the Born rule is complete:

http://philsci-archive.pitt.edu/14590/

If I'm reading him right, he's saying that the measure of the subset of uncountably infinite worlds with outcome $A$ is given by the square of the coefficient of $|A\rangle$ in the expansion of the universal wavefunction in the (quasi-)classical worlds basis?

 

[Avodyne]

Yes. But the starting point is counting branches in simpler situations. See also Carroll & Sebens, who elaborate basically the same idea, with slightly different language:

https://arxiv.org/abs/1405.7907

 

[stevendaryl]

That depends on a very particular view of what a superposition is. How can Many-Worlds for example cope with the fact that any system confined to a finite volume is always in a mixed state?

What do you mean by that? What theorem is that?

 

[DarMM]

What do you mean by that? What theorem is that?

In QFT for a variety of subregions of spacetime you can prove the algebra of observables is a type $III$ C*-algebra factor and these have no pure states.

The meaning of this for Copenhagen-like views and the Thermal Interpretation is clear enough for me, but what it means for MWI is not clear.

 

[stevendaryl]

In QFT for a variety of subregions of spacetime you can prove the algebra of observables is a type $III$ C*-algebra factor and these have no pure states.

The meaning of this for Copenhagen-like views and the Thermal Interpretation is clear enough for me, but what it means for MWI is not clear.

I had not heard of this, although I found an article mentioning it here:

https://arxiv.org/pdf/1401.2652.pdf

I haven't grokked what it means yet. Maybe someone could make an Insights article about it?

 

[DarMM]

When I get the chance I'm hoping to write a Insights series on Algebraic QFT.

 

[A. Neumaier]

I had not heard of this, although I found an article mentioning it here:

https://arxiv.org/pdf/1401.2652.pdf

I haven't grokked what it means yet. Maybe someone could make an Insights article about it?

This was discussed two years ago here on PF.

 

[atyy]

In QFT for a variety of subregions of spacetime you can prove the algebra of observables is a type $III$ C*-algebra factor and these have no pure states.

The meaning of this for Copenhagen-like views and the Thermal Interpretation is clear enough for me, but what it means for MWI is not clear.

Are these subregions big enough to be the whole universe? I think it would affect MWI only if the universe itself could not be in a pure state.

 

[DarMM]

Are these subregions big enough to be the whole universe? I think it would affect MWI only if the universe itself could not be in a pure state.

It's possible that the global algebra of QED doesn't have pure states due to massless particles:
Buchholz, D., and Doplicher, S., Exotic infrared representations of interacting systems, Ann. Inst. H. Poincare, 40, 175-184, (1984).