# Can interpretation-dependent facts be derived from the axioms?

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Nullstein
In the thread https://www.physicsforums.com/threa...correlations-or-action-at-a-distance.1049354/, @PeterDonis claims that a certain mathematical derivation from the basic axioms of QM is an interpretation-dependent proposition. I'm referring to post #54 here and this is how I understand that post.

In particular, let's suppose we are on a tensor product Hilbert space ##\mathcal H=\mathcal H_A\otimes \mathcal H_B## and are given a density matrix ##\rho## on that space. We then perform a measurement on system B. It is a theorem in QM that the experimentally measurable statistics of the full system after the measurement, if no conditional probabilities are taken, is given by
$$\rho^\prime = \sum_i (\mathbb 1\otimes P_i) \rho (\mathbb 1\otimes P_i) \text{.}$$
The ##P_i## are the projection operators at system B. The statistics of the full system after measurement is described by this state and this can be verified in the lab. Moreover, this is the only state that correctly describes the statistics in the lab. It is the unique possible choice.

Now if we are interested in the physics of subsystem A, we can evaluate the partial trace over ##\mathcal H_B##. It is yet another theorem of QM, which follows from the interpretation-independent axioms and math that
$$\rho_A = \mathrm{tr}_{\mathcal H_B}(\rho) = \mathrm{tr}_{\mathcal H_B}(\rho^\prime) \text{.}$$
So the state of ##\rho_A## is not changed by the measurement at system B. Again, the state ##\rho_A## accurately describes the statistics of system A, both prior and post measurement, as can be verified in the lab and again, it is the unique choice that does so.

Since ##\rho_A## doesn't change upon measurement, as proved from the interpretation-independent axioms of QM, it follows that ##\rho_A## is entangled after the measurement if and only if it was entangled prior to the measurement. Here, we apply the mathematical definition of entanglement: A state is entangled if it is not a product state.

Since all of these propositions follow from the mathematical axioms of QM that every interpretation must agree on, and moreover, these predictions can be verified in the lab, is seems to me that all these statements must be interpretation-independent. If some particular interpretation denies these results, then it contradicts the predictions of QM and can be experimentally excluded by a simple lab experiment. Can there be any other possibility? How can a statement that is testable in the lab and derived from interpretation-independent axioms somehow become interpretation-dependent?

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Gold Member
Since ##\rho_A## doesn't change upon measurement, as proved from the interpretation-independent axioms of QM, it follows that ##\rho_A## is entangled after the measurement if and only if it was entangled prior to the measurement. Here, we apply the mathematical definition of entanglement: A state is entangled if it is not a product state.
I am pretty sure that there are some interpretations where any new information learned during measurement forces an update of the "global" state. As I wrote in that thread:
In the "standard" non-ensemble interpretation, you have to non-locally update the state upon learning new information about the state. And if you learn new information about the past, then a past state can "be updated" too ("in the past") based in new information in the present.

Mentor
In the thread https://www.physicsforums.com/threa...correlations-or-action-at-a-distance.1049354/, @PeterDonis claims that a certain mathematical derivation from the basic axioms of QM is an interpretation-dependent proposition.
No, that is not what I said.

What I said is that the interpretation of what the mathematical fact means is an interpretation-dependent proposition.

Please do not misrepresent what other people say. At the very least, if you are going to discuss what someone else has said, you should quote what they said instead of describing it in your own words, particularly if your description is wrong.

hutchphd and topsquark
Nullstein
I am pretty sure that there are some interpretations where any new information learned during measurement forces an update of the "global" state. As I wrote in that thread:
The transition ##\rho\rightarrow\rho^\prime## given above is an update to the global state. It is then a theorem that in this particular case, where the projections are of the form ##\mathbb 1\otimes P_i##, it is a theorem that ##\rho_A## is unaltered. Sure, some interpretations interpret more into the state vector than others, but they have to agree on the observable facts. The observable facts are fully described by ##\rho^\prime##, everything that goes beyond that is interpretation-dependent and unobservable.

Homework Helper
2022 Award
In the thread https://www.physicsforums.com/threa...correlations-or-action-at-a-distance.1049354/, @PeterDonis claims that a certain mathematical derivation from the basic axioms of QM is an interpretation-dependent proposition.
Either provide a direct citation that is explicitly on point or recapitulate in detail what was said. This is extraordinarilly sloppy and is not a sufficient description IMHO. I have no idea what you are refuting!
Or leave out the preamble

berkeman
Nullstein
No, that is not what I said.

What I said is that the interpretation of what the mathematical fact means is an interpretation-dependent proposition.
You said in post #54 that the statement I made in post #53 is interpretation dependent. It's not. What DrChinese said in post #38 is false, it contradicts the mathematical facts. His state does not describe the statistics of the full system after measurement correctly. It predicts different probabilities than the one I have given and a lab experiment would show that his state is incompatible with the recorded data.
Please do not misrepresent what other people say. At the very least, if you are going to discuss what someone else has said, you should quote what they said instead of describing it in your own words, particularly if your description is wrong.
I don't want to misrepresent you, but how else should I interpret your post #54?

Nullstein
Either provide a direct citation that is explicitly on point or recapitulate in detail what was said. This is extraordinarilly sloppy and is not a sufficient description IMHO. I have no idea what you are refuting!
Or leave out the preamble
I'm referring to post #54. I thought that was obvious, since it was the last post in the thread, but alright, I'll edit my OP.

Gold Member
The transition ##\rho\rightarrow\rho^\prime## given above is an update to the global state. It is then a theorem that in this particular case, where the projections are of the form ##\mathbb 1\otimes P_i##, it is a theorem that ##\rho_A## is unaltered.
So you just did the measurement, but didn't look at the result yet. That's what you meant by "if no conditional probabilities are taken". It boils down to the old question: "whose information?" The amplification process already happened, therefore the information is there in principle, but it is not yet mine or yours. So I have to admit that your update makes sense, for the typical version of those interpretations. There will be another update later, but the measurement itself just triggered your update.

Sure, some interpretations interpret more into the state vector than others, but they have to agree on the observable facts. The observable facts are fully described by ##\rho^\prime##, everything that goes beyond that is interpretation-dependent and unobservable.
Well, even if it would be "interpretation-dependent and unobservable," it could still be relevant for the discussion. Even what the observable facts are can be interpretation dependent, to a certain extent.

Nullstein
Nullstein
So you just did the measurement, but didn't look at the result yet. That's what you meant by "if no conditional probabilities are taken". It boils down to the old question: "whose information?" The amplification process already happened, so the information is there in principle, but it is not yet mine or yours. So I have to admit that your update makes sense, for the typical version of those interpretations. There will be another update later, but the measurement itself just triggered your update.
This is correct. And every interpretation must be able to calculate the state ##\rho^\prime##, since it is an actual distribution that shows up in the lab data, so it can't be an interpretation-dependent statement.
Well, even if it would be "interpretation-dependent and unobservable," it could still be relevant for the discussion. Even what the observable facts are can be interpretation dependent, to a certain extent.
Yes, it could be relevant for some aspects of the discussion, but my main goal was to debunk DrChinese's claim that entanglement swapping somehow enforces interpretations with non-local cause and effect relationships. So I was very careful to only make interpretation-independent statements. The math also proves that his understanding of monogamy of entanglement is fundamentally flawed since the math also unambiguously shows that ##\rho_A## isn't altered by the BSM and thus no entanglement is introduced in the 1&4 system, which his monogamy of entanglement argument was supposed to refute.

Mentor
You said in post #54 that the statement I made in post #53 is interpretation dependent.
No, that's not what I said in post #54 of the thread in question.

I don't want to misrepresent you
Then, as I've already said, you should quote me.

Nullstein
Mentor