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Does the constraint of pairwise interference imply another constraint on the nature of superposition?

No. It's not a constraint on the states, it's a constraint on possible rules for how to calculate probabilities. The constraint is consistent with the Born rule (standard quantum mechanics) or with classical probability theory (probabilities always additive).

I.e., are three-modal superpositions over A, B, and C, really just a mixture of pairwise suprepositions linking only---in any given coherent, single-shot experiment---modes A and B, or A and C, or B and C?

Superposition, addition, and mixture here all mean the same thing, and it's not meaningful to talk about a constraint on the types of sums by saying that they have to be a certain type of sum of sums. For example, I can write $f+g+h=(f+g/2)+(g/2+h)$, but this is possible for any linear combination of $f$, $g$, and $h$.

To use an anthropomorphic analogy, does the photon really "split" into three paths or does it only choose two paths at a time and completely ignore the third. (Of course, we cannot tell which two it chose.)

Well, this could get bogged down in words like "really" and "choose," which we clearly can't define here, but basically no, Sorkin is talking about generalizing quantum mechanics by modifying the probability measure, not the time evolution. The time evolution according to the Schrodinger equation is such that the photon passes through all three slits. (If we were to generalize the probability measure, and wanted to retain conservation of probability, we would have to modify the dynamics somehow, but Sorkin doesn't attempt that.)

I don't think the fact that the Born rule expands into a sum of pairwise interference terms is really all that mysterious. This is simply because probabilities in quantum mechanics are proportional to the square of a wavefunction, and when you square a sum, you get a sum of second-order terms. It seems much more mysterious to me how you could get a sensible third-order version of the Born rule. E.g., it seems like phases would become observable, which creates all kinds of craziness.

Does the constraint of pairwise interference imply another constraint on the nature of superposition?

No. It's not a constraint on the states, it's a constraint on possible rules for how to calculate probabilities. The constraint is consistent with the Born rule (standard quantum mechanics) or with classical probability theory (probabilities always additive).

I.e., are three-modal superpositions over A, B, and C, really just a mixture of pairwise suprepositions linking only---in any given coherent, single-shot experiment---modes A and B, or A and C, or B and C?

Superposition, addition, and mixture here all mean the same thing, and it's not meaningful to talk about a constraint on the types of sums by saying that they have to be a certain type of sum of sums. For example, I can write $f+g+h=(f+g/2)+(g/2+h)$, but this is possible for any linear combination of $f$, $g$, and $h$.

To use an anthropomorphic analogy, does the photon really "split" into three paths or does it only choose two paths at a time and completely ignore the third. (Of course, we cannot tell which two it chose.)

Well, this could get bogged down in words like "really" and "choose," which we clearly can't define here, but basically no, Sorkin is talking about generalizing quantum mechanics by modifying the probability measure, not the time evolution. The time evolution according to the Schrodinger equation is such that the photon passes through all three slits. (If we were to generalize the probability measure, and wanted to retain conservation of probability, we would have to modify the dynamics somehow, but Sorkin doesn't attempt that.)

Does the constraint of pairwise interference imply another constraint on the nature of superposition?

No. It's not a constraint on the states, it's a constraint on possible rules for how to calculate probabilities. The constraint is consistent with the Born rule (standard quantum mechanics) or with classical probability theory (probabilities always additive).

I.e., are three-modal superpositions over A, B, and C, really just a mixture of pairwise suprepositions linking only---in any given coherent, single-shot experiment---modes A and B, or A and C, or B and C?

Superposition, addition, and mixture here all mean the same thing, and it's not meaningful to talk about a constraint on the types of sums by saying that they have to be a certain type of sum of sums. For example, I can write $f+g+h=(f+g/2)+(g/2+h)$, but this is possible for any linear combination of $f$, $g$, and $h$.

To use an anthropomorphic analogy, does the photon really "split" into three paths or does it only choose two paths at a time and completely ignore the third. (Of course, we cannot tell which two it chose.)

Well, this could get bogged down in words like "really" and "choose," which we clearly can't define here, but basically no, Sorkin is talking about generalizing quantum mechanics by modifying the probability measure, not the time evolution. The time evolution according to the Schrodinger equation is such that the photon passes through all three slits. (If we were to generalize the probability measure, and wanted to retain conservation of probability, we would have to modify the dynamics somehow, but Sorkin doesn't attempt that.)

I don't think the fact that the Born rule expands into a sum of pairwise interference terms is really all that mysterious. This is simply because probabilities in quantum mechanics are proportional to the square of a wavefunction, and when you square a sum, you get a sum of second-order terms. It seems much more mysterious to me how you could get a sensible third-order version of the Born rule. E.g., it seems like phases would become observable, which creates all kinds of craziness.

Source Link
user4552
user4552

Does the constraint of pairwise interference imply another constraint on the nature of superposition?

No. It's not a constraint on the states, it's a constraint on possible rules for how to calculate probabilities. The constraint is consistent with the Born rule (standard quantum mechanics) or with classical probability theory (probabilities always additive).

I.e., are three-modal superpositions over A, B, and C, really just a mixture of pairwise suprepositions linking only---in any given coherent, single-shot experiment---modes A and B, or A and C, or B and C?

Superposition, addition, and mixture here all mean the same thing, and it's not meaningful to talk about a constraint on the types of sums by saying that they have to be a certain type of sum of sums. For example, I can write $f+g+h=(f+g/2)+(g/2+h)$, but this is possible for any linear combination of $f$, $g$, and $h$.

To use an anthropomorphic analogy, does the photon really "split" into three paths or does it only choose two paths at a time and completely ignore the third. (Of course, we cannot tell which two it chose.)

Well, this could get bogged down in words like "really" and "choose," which we clearly can't define here, but basically no, Sorkin is talking about generalizing quantum mechanics by modifying the probability measure, not the time evolution. The time evolution according to the Schrodinger equation is such that the photon passes through all three slits. (If we were to generalize the probability measure, and wanted to retain conservation of probability, we would have to modify the dynamics somehow, but Sorkin doesn't attempt that.)