- #1

Nullstein

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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?

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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