Can somebody explain a particular aspect of Quantum Mechanics to me?
In my readings of the Many Worlds Interpretation, which Eliezer fondly endorses in the QM sequence, I must have missed an important piece of information about when it is that amplitude distributions become separable in timed configuration space. That is, when do wave-functions stop interacting enough for the near-term simulation of two blobs (two “particles”) to treat them independently?
One cause is spatial distance. But in Many Worlds, I don’t know where I’m to understand these other worlds are taking place. Yes, it doesn’t matter, supposedly; the worlds are not present in this world’s causal structure, so an abstract “where” is meaningless. But the evolution of wavefunctions seems to care a lot about where amplitudes are in N-dimensional space. Configurations don’t sum unless they are the same spatial location and are representing the same quark type, right?
So if there’s another CoffeeStain that splits off based on my observation of a quantum event, why don’t the two CoffeeStains still interact, since they so obviously don’t? Before my two selves became decoherent with their respective quantum outcomes (say, of a photon’s path), the two amplitude blobs of the photon could still interact by the book, right? On what other axis has I, as a member of a new world, split off that I’m a sufficient distance from my self that is occupying the same physical location?
Relatedly, MWI answers “not-so-spooky” to questions regarding the entanglement experiment, but a similar confusion remains for me. Why, after I observe a particular polarization on my side of the galaxy and fly back in my spaceship to compare notes with my buddy on the other side of the galaxy, do I run into one version of him and not the other? They are both equally real, and occupying the same physical space. What other axis have the self-versions separated on?
Second: Suppose I want to demonstrate decoherence. I start out with an entangled state—two electrons that will always be magnetically aligned, but don’t have a chosen collective alignment. This state is written like |up, up> + |down, down> (the electrons are both “both up” and “both down” at the same time; the |> notation here just indicates that it’s a quantum state).
Now, before introducing decoherence, I just want to check that I can entangle my two electrons. How do I do that? I repeat what’s called a “Bell measurement,” which has four possible indications: (|up,up>+|down,down>) , (|up,up>-|down,down>) , (|up,down>+|down,up>) , (|up,down>-|down,up>).
Because my state is made of 100% Bell state 1, every time I make some entangled electrons and then measure them, I’ll get back result #1. This consistency means they’re entangled. If the quantum state of my particles had to be expressed as a mixture of Bell States, there might not be any entanglement—for example state 1 + state 2 just looks like |up,up>, which is boring and unentangled.
To create decoherence, I send the second electron to you. You measure whether it’s up or down, then re-magnetize it and send it back with spin up if you measured up, and spin down if you measured down. But since you remember the state of the electron, you have now become entangled with it, and must be included. The relevant state is now |up, up, saw up> + |down, down, saw down>.
This state is weird, because now you, a human, are in a superposition of “saw up” and “saw down.” But we’ll ignore that for the moment—we can always replace you with with a third electron if it causes philosophical problems :) The question at hand is: what happens when we try to test if our electrons are still entangled?
Again, we do this a bunch of times and do a repeated Bell measurement. If we get result #1 every time, they’re entangled just like before. To predict the outcome ahead of time, we can factor our state into Bell States, and see how much of each Bell State we have.
So we factor |up, up, saw up> into |(Bell state 1) + (Bell state 2), saw up>, and we factor |down, down, saw down> into |(Bell state 1) - (Bell state 2), saw down>.
Now, if that extra label about what you saw wasn’t here, the ups and the downs would be physically/mathematically equivalent and we could cancel terms to just get Bell state 1. But if any of the labels are different, you can’t subtract them to get 0 anymore. That is, they no longer interfere. And so you are just left with equal numbers of Bell state 1 and Bell state 2 terms. And so when we do the Bell measurement, we get results #1 and #2 with equal frequency, just like we would if the electrons were completely unentangled.
This is not to say they’re not entangled—they still are. But they can no longer be shown to be entangled by a two-particle test. They’re no longer usefully entangled. You need to collect all the pieces together before you can show that they’re entangled, now. And that gets awful hard once a macroscopic system like a human gets entangled with the electrons and starts radiating off still-entangled photons into the environment.
This is decoherence. I can have a nice entangled system, but if I let you peek at one of my electrons, you turn the state into into |(Bell state 1) + (Bell state 2), saw up> + |(Bell state 1) - (Bell state 2), saw down>, and they don’t behave in the entangled way they did anymore.
(Warning: I am not a physicist; I learnt a bit of about QM from my physics classes, the Sequences, Feynmann Lectures on Physics, and Good and Real, but I don’t claim to even understand all that’s in there)
I’m not sure I totally understand your question, but I’ll take a stab at answering:
The important thing is configuration space, and spatial distance is just one part of that; there is just one configuration space over which the quantum wave-function is defined, and points in configuration space correspond to “universe states” (the position, spin, etc. of all particles).
So two points in configuration space A and B “interfere” if they are similar enough that both can “evolve” into state C, i.e. state C’s amplitude will be function of A and B’s amplitudes. The more different A and B are, the less likely they are to have shared “descendant states” (or more precisely, descendant states of non-infinitesimal amplitude), so the more they can be treated like “parallel branches of the universe”. Differences between A and B can be in psychical distance of particles, but also of polarity/spin, etc. - as long as the distance is significant on one axis (say spin of a single particle), physical distance shouldn’t matter.
I think spin could be an example of “another axis” you’re looking for (though even thinking in terms of Axis may be a bit misleading, since all the attributes aren’t nice and orthogonal like positions in cartesian space).
This is pretty much correct, but to be more general and not just restrict yourself to the position basis, you can talk about the wavefunction in general, in terms of the eigenvector basis.
Two states ‘strongly interact’ if they share many of their high-amplitude eigenvectors. This is because eigenvectors evolve independently, and so if you have two states that do not share many eigenvectors, they will also evolve independently.
In the position basis, this winds up being much the same as having particles far from each other. In the momentum basis, it’s less intuitive. You can have states with very similar representations in this basis but nevertheless very different eigenvector expansions.
I must admit I have very little understanding of how eigenvectors fit in with QM. I’ll have to read up more on that, thanks for pointing out holes in my knowledge (though in the domain of QM, there are a lot of holes).
Can somebody explain a particular aspect of Quantum Mechanics to me?
In my readings of the Many Worlds Interpretation, which Eliezer fondly endorses in the QM sequence, I must have missed an important piece of information about when it is that amplitude distributions become separable in timed configuration space. That is, when do wave-functions stop interacting enough for the near-term simulation of two blobs (two “particles”) to treat them independently?
One cause is spatial distance. But in Many Worlds, I don’t know where I’m to understand these other worlds are taking place. Yes, it doesn’t matter, supposedly; the worlds are not present in this world’s causal structure, so an abstract “where” is meaningless. But the evolution of wavefunctions seems to care a lot about where amplitudes are in N-dimensional space. Configurations don’t sum unless they are the same spatial location and are representing the same quark type, right?
So if there’s another CoffeeStain that splits off based on my observation of a quantum event, why don’t the two CoffeeStains still interact, since they so obviously don’t? Before my two selves became decoherent with their respective quantum outcomes (say, of a photon’s path), the two amplitude blobs of the photon could still interact by the book, right? On what other axis has I, as a member of a new world, split off that I’m a sufficient distance from my self that is occupying the same physical location?
Relatedly, MWI answers “not-so-spooky” to questions regarding the entanglement experiment, but a similar confusion remains for me. Why, after I observe a particular polarization on my side of the galaxy and fly back in my spaceship to compare notes with my buddy on the other side of the galaxy, do I run into one version of him and not the other? They are both equally real, and occupying the same physical space. What other axis have the self-versions separated on?
First: check this out.
Second: Suppose I want to demonstrate decoherence. I start out with an entangled state—two electrons that will always be magnetically aligned, but don’t have a chosen collective alignment. This state is written like |up, up> + |down, down> (the electrons are both “both up” and “both down” at the same time; the |> notation here just indicates that it’s a quantum state).
Now, before introducing decoherence, I just want to check that I can entangle my two electrons. How do I do that? I repeat what’s called a “Bell measurement,” which has four possible indications: (|up,up>+|down,down>) , (|up,up>-|down,down>) , (|up,down>+|down,up>) , (|up,down>-|down,up>).
Because my state is made of 100% Bell state 1, every time I make some entangled electrons and then measure them, I’ll get back result #1. This consistency means they’re entangled. If the quantum state of my particles had to be expressed as a mixture of Bell States, there might not be any entanglement—for example state 1 + state 2 just looks like |up,up>, which is boring and unentangled.
To create decoherence, I send the second electron to you. You measure whether it’s up or down, then re-magnetize it and send it back with spin up if you measured up, and spin down if you measured down. But since you remember the state of the electron, you have now become entangled with it, and must be included. The relevant state is now |up, up, saw up> + |down, down, saw down>.
This state is weird, because now you, a human, are in a superposition of “saw up” and “saw down.” But we’ll ignore that for the moment—we can always replace you with with a third electron if it causes philosophical problems :) The question at hand is: what happens when we try to test if our electrons are still entangled?
Again, we do this a bunch of times and do a repeated Bell measurement. If we get result #1 every time, they’re entangled just like before. To predict the outcome ahead of time, we can factor our state into Bell States, and see how much of each Bell State we have.
So we factor |up, up, saw up> into |(Bell state 1) + (Bell state 2), saw up>, and we factor |down, down, saw down> into |(Bell state 1) - (Bell state 2), saw down>.
Now, if that extra label about what you saw wasn’t here, the ups and the downs would be physically/mathematically equivalent and we could cancel terms to just get Bell state 1. But if any of the labels are different, you can’t subtract them to get 0 anymore. That is, they no longer interfere. And so you are just left with equal numbers of Bell state 1 and Bell state 2 terms. And so when we do the Bell measurement, we get results #1 and #2 with equal frequency, just like we would if the electrons were completely unentangled.
This is not to say they’re not entangled—they still are. But they can no longer be shown to be entangled by a two-particle test. They’re no longer usefully entangled. You need to collect all the pieces together before you can show that they’re entangled, now. And that gets awful hard once a macroscopic system like a human gets entangled with the electrons and starts radiating off still-entangled photons into the environment.
This is decoherence. I can have a nice entangled system, but if I let you peek at one of my electrons, you turn the state into into |(Bell state 1) + (Bell state 2), saw up> + |(Bell state 1) - (Bell state 2), saw down>, and they don’t behave in the entangled way they did anymore.
Not to undermine your point, but |up, up> + |down, down> is perfectly oriented in the X direction.
What works better for this is that you indicate that the state is A |up, up> + B|down, down>, and you don’t know A and B.
Nay. (|up>+|down>)(|up>+|down>) is oriented in the X-direction.
Hmmm… Yes.
I’m used to people forgetting that every single-particle spinor maps onto a single direction. Then I forget spinor addition. Oops.
(Warning: I am not a physicist; I learnt a bit of about QM from my physics classes, the Sequences, Feynmann Lectures on Physics, and Good and Real, but I don’t claim to even understand all that’s in there)
I’m not sure I totally understand your question, but I’ll take a stab at answering:
The important thing is configuration space, and spatial distance is just one part of that; there is just one configuration space over which the quantum wave-function is defined, and points in configuration space correspond to “universe states” (the position, spin, etc. of all particles).
So two points in configuration space A and B “interfere” if they are similar enough that both can “evolve” into state C, i.e. state C’s amplitude will be function of A and B’s amplitudes. The more different A and B are, the less likely they are to have shared “descendant states” (or more precisely, descendant states of non-infinitesimal amplitude), so the more they can be treated like “parallel branches of the universe”. Differences between A and B can be in psychical distance of particles, but also of polarity/spin, etc. - as long as the distance is significant on one axis (say spin of a single particle), physical distance shouldn’t matter.
I think spin could be an example of “another axis” you’re looking for (though even thinking in terms of Axis may be a bit misleading, since all the attributes aren’t nice and orthogonal like positions in cartesian space).
This is pretty much correct, but to be more general and not just restrict yourself to the position basis, you can talk about the wavefunction in general, in terms of the eigenvector basis.
Two states ‘strongly interact’ if they share many of their high-amplitude eigenvectors. This is because eigenvectors evolve independently, and so if you have two states that do not share many eigenvectors, they will also evolve independently.
In the position basis, this winds up being much the same as having particles far from each other. In the momentum basis, it’s less intuitive. You can have states with very similar representations in this basis but nevertheless very different eigenvector expansions.
I must admit I have very little understanding of how eigenvectors fit in with QM. I’ll have to read up more on that, thanks for pointing out holes in my knowledge (though in the domain of QM, there are a lot of holes).