Measure of existence? (Stanford Encyclopedia of Philosophy)

[Sayestu]

TL;DR Summary

Will someone please put this excerpt of the Stanford Encyclopedia's entry on the MWI in layperson's terms?

I'm reading the article on the Many Worlds Interpretation in the Stanford Encyclopedia of Philosophy. I'm keeping up well, but this excerpt uses things I'm very unfamiliar with:

There are many worlds existing in parallel in the Universe. Although all worlds are of the same physical size (this might not be true if we take into account the quantum aspects of early cosmology), and in every world sentient beings feel as “real” as in any other world, there is a sense in which some worlds are larger than others. Vaidman 1998 describes this property as the measure of existence of a world.

There are two aspects of the measure of existence of a world. First, it quantifies the ability of the world to interfere with other worlds in a gedanken experiment, as expounded at the end of this section. Second, the measure of existence is the basis for introducing an illusion of probability in the MWI as described in the next chapter. The measure of existence is the parallel of the probability measure discussed in Everett 1957 and pictorially described in Lockwood 1989 (p. 230).

Given the decomposition (2), the measure of existence of the world i� is μi=|αi|2.��=|��|2. It can also be expressed as the expectation value of Pi��, the projection operator on the space of quantum states corresponding to the actual values of all physical variables describing the world i�:

(3)μi=⟨ΨUNIVERSE∣Pi∣ΨUNIVERSE⟩.(3)��=⟨ΨUNIVERSE∣��∣ΨUNIVERSE⟩.
Note, that although the measure of existence of a world is expressed using the quantum state of the Universe (2), the concept of measure of existence, as the concept of a world belongs to part (ii) of the MWI, the bridge to our experience.

“I” also have a measure of existence. It is the sum of the measures of existence of all different worlds in which I exist. Note that I do not directly experience the measure of my existence. I feel the same weight, see the same brightness, etc. irrespectively of how tiny my measure of existence might be.

My current measure of existence is relevant only for gedanken situations like Wigner’s friend Wigner 1961 (recently revived by Frauchiger and Renner 2018) which demonstrates the meaning of the measure of existence of a world as a measure of its ability to interfere with other worlds. If I am a friend of Wigner, a gedanken superpower who can perform interference experiments with macroscopic objects like people, and I perform an experiment with two outcomes A and B such that two worlds will be created with different measures of existence, say 2μA=μB2��=��, then there is a difference between Lev A and Lev B in how Wigner can affect their future through the interference of worlds. Both Lev A and Lev B consider performing a new experiment with the same device. Wigner can interfere the worlds in such a way that Lev A (the one with a smaller measure of existence) will not have the future with result A of the second experiment. However, Wigner cannot prevent the future result A from Lev B, see Vaidman 1998 (p. 256).

I guess some characters weren't recognized. It's Section 3.6 here. I'm somewhat familiar with Wigner's Friend, but that's about it here. Would someone please explain this in terms for someone with a basic understanding of quantum mechanics?

 

[PeterDonis]

Moderator's note: Thread moved to interpretations subforum.

 

[Sayestu]

Moderator's note: Thread moved to interpretations subforum.

Didn't know that existed. Thanks!

 

[PeterDonis]

Would someone please explain this in terms for someone with a basic understanding of quantum mechanics?

In the MWI, or at least this version of it, the amplitude in front of each branch of the wave function is referred to as the "measure of existence" of that branch. For example, if you measure spin-x on a spin-z up qubit, you end up with two branches, one in which the result is "x-spin up" and everything else is consistent with that, and one in which the result is "x-spin down" and everything else is consistent with that. The "measure of existence" of each of these branches would be its amplitude, which in this case is $1 / \sqrt{2}$, corresponding to a probability of $1/2$ for each result.

I'm somewhat familiar with Wigner's Friend

IMO the use of Wigner's Friend type scenarios here is highly problematic. The modern understanding of the MWI is that "branching" of worlds occurs when decoherence occurs, i.e., when a measurement result is irreversibly recorded through spreading of entanglement among a very large number of untrackable degrees of freedom in the environment. But once that happens, the different branches cannot interfere with each other; that is what decoherence means.

Wigner's Friend scenarios contradict that by supposing that, under certain conditions, different branches, such as the different possible measurement results observed by the friend, can interfere with each other, since Wigner is assumed to be capable of making a measurement that shows such interference.

 

[Sayestu]

Would you please elaborate on this, and is it possible, even if unsupported?

 

[Sayestu]

under certain conditions, different branches, such as the different possible measurement results observed by the friend, can interfere with each other

This, sorry.

 

[PeterDonis]

Would you please elaborate on this, and is it possible, even if unsupported?

This, sorry.

In a Wigner's Friend experiment, the friend makes a measurement in his lab that has two possible results (such as measuring the spin of a qubit) and observes a result. This requires decoherence to happen, which means that the two different branches of the wave function that correspond to the two different possible results cannot interfere with each other.

Then Wigner is claimed to make a measurement that shows interference between the two possible results that the friend could have gotten. For example, Wigner could make a measurement with two possible results that correspond to $\ket{\text{friend measured spin-up}} + \ket{\text{friend measured spin-down}}$ and $\ket{\text{friend measured spin-up}} - \ket{\text{friend measured spin-down}}$. But for Wigner to make such a measurement would be impossible if decoherence occurred when the friend made his measurement. And if decoherence did not occur when the friend made his measurement, then "measurement" would be the wrong term for what the friend did since the friend could not have observed a result.

 

[Sayestu]

Thanks!