To state the obvious, a sensory stream generated by just re-sampling predicts that you’re constantly teleporting through the multiverse, and a sensory stream generated by putting a delta spike on the last state you sampled and then evolving that forward for a tick will… not yield good predictions (roughly, it will randomize all momenta).
You need to separate treating (1) something literally as a delta function, (2) treating something you have observed as having probability 1.0 for the purpose of further probability calculation.
If you are measuring something discrete like spin-up versus spin-down, it is completely standard to set the unobserved state to 0, effectively discarding it. The discarding (projection) is just as necessary a part of the procedure as the absolute-squaring (Born’s rule per se).
It’s not that no-one knows how to predict a series of observations with correct probabilities using QM, it is that the time-honoured method looks like Copenhagen.
I agree that there’s a difference between “put a delta-spike on the single classical state you sampled” and “zero out amplitude on all states not consistent with the observation you got from your sample”. I disagree that using the latter to generate a sensory stream from a quantum state yields reasonable predictions—eg, taken literally I think you’re still zeroing out all but a measure-zero subset of the position basis, and I expect the momenta to explode immediately. You can perhaps get this hypothesis (or the vanilla delta spike) hobbling by trying to smooth things out a bit (eg, keep a Gaussian centered on each classical state in which you made the sampled observation), but I still expect this to be experimentally distinguishable from what really happens (eg by way of some quantum-eraser-style hijinks or other sizeable entanglements), though I haven’t checked the details myself.
I disagree that using the latter to generate a sensory stream from a quantum state yields reasonable predictions—eg, taken literally I think you’re still zeroing out all but a measure-zero subset of the position basis
The observation you got from your sample is information. Information is entropy, and entropy is locally finite. So I don’t think it’s possible for the states consistent with the observation you got from your sample to have measure zero.
When you’re using TMs to approximate physics, you have to balance the continuity of physics against the discreteness of the machines somehow. The easy thing to do is to discuss the limiting behavior of a family of machines that perform the simulation at ever-finer fidelity. I was doing this implicitly, for lack of desire to get into details.
And as I’ve said above, I’m not attempting to suggest that these naive approaches—such as sampling a single classical state and reporting the positions of some things with arbitrary fidelity in the limit—are reasonable ideas. Quite the opposite. What I’m trying to point out is that if all you have is a quantum state and the Born rule, you cannot turn it into a hypothesis without making a bunch of other choices, for which I know of no consensus answer (and for which I have not seen proposals that would resolve the problem to my satisfaction, though I have some ideas).
I agree that the correct way of making these choices will almost surely not involve recording any observation with infinite precision (in the limit).
I agree that there’s a difference between “put a delta-spike on the single classical state you sampled” and “zero out amplitude on all states not consistent with the observation you got from your sample”. I disagree that using the latter to generate a sensory stream from a quantum state yields reasonable predictions—eg, taken literally I think you’re still zeroing out all but a measure-zero subset of the position basis
You have been assuming that all measurements are in the position basis, which is wrong. In particular, spin is its own basis.
If you make a sharp measurement in one basis, you have uncertainty or lack of information about the others. That does not mean the “momentum is randomised” in some catastrophic sense. The original position measurement was not deterministic, for one thing.
It is true that delta functions can be badly behaved. It’s also true that they can be used in practice … if you are careful. They are not an argument against discarding-and-renormalising , because if you don’t do that at all, you get much wronger results than the results you get by rounding off small values to zero, ie. using a delta to represent a sharp gaussian.
taken literally I think you’re still zeroing out all but a measure-zero subset of the position basis taken literally I think you’re still zeroing out all but a measure-zero subset of the position basis
That might be the case if you were making an infinitely sharp measurement of an observable with a real valued spectrum, but there are no infinitely sharp measurements, and not every observable is real-valued.
To be clear, the process that I’m talking about for turning a quantum state into a hypothesis is not intended to be a physical process (such as a measurement), it’s intended to be a Turing machine (that produces output suitable for use by Solomonoff induction).
That said, to be clear, I don’t think this is a fundamentally hard problem. My point is not “we have absolutely no idea how to do it”, it’s somehing more like “there’s not a consensus answer here” + “it requires additional machinery above and beyond [the state vector + the born rule + your home address]” + “in my experience, many (non-specialist-physicist) people don’t even register this as a problem, and talk about the Born rule as if it’s supposed to fill this gap”.
I agree that there are a bunch of reasonable additional pieces of machinery you can use to get the missing piece (such as choice of a “measurement basis”); my own suspicion is that the right answer looks a bit different from what I’ve seen others propose (and routes through, eg, machinery that lets you read off the remembered history, as opposed to machinery that picks some subbasis to retain); my guess is that there are experimental differences in theory but they’re probably tricky to create in practice, but I haven’t worked through the details myself.
To be clear, the process that I’m talking about for turning a quantum state into a hypothesis is not intended to be a physical process (such as a measurement), it’s intended to be a Turing machine (that produces output suitable for use by Solomonoff induction).
Then you run into the basic problem of using SI to investigate MW: SI’s are supposed to output a series of definite observations. They are inherently “single world”
If the program running the SWE outputs information about all worlds on a single output tape, they are going to have to be concatenated or interleaved somehow. Which means that to make use of the information, you have to identify the subset if bits relating to your world. That’s extra complexity which isn’t accounted for because it’s being done by hand, as it were.
In particular, if you just model the wave function, the only results you will get represent every possible outcome. In order to match observation , you will have to keep discarding unobserved outcomes and renormalising as you do in every interpretation. It’s just that that extra stage is performed manually, not by the programme.
To get an output that matches one observers measurements, you would need to simulate collapse somehow. You could simulate collapse with a PRNG, but it won’t give you the right random numbers.
Or you would need to keep feeding your observations back in so that the simulator can perform projection and renormalisation itself. That would work, but that’s a departure from how SI’s are supposed to work.
Meta: trying to mechanise epistemology doesn’t solve much , because mechanisms still have assumptions built into them.
Yeah yeah, this is the problem I’m referring to :-)
I disagree that you must simulate collapse to solve this problem, though I agree that that would be one way to do it. (The way you get the right random numbers, fwiw, is from sample complexity—SI doesn’t put all its mass on the single machine that predicts the universe, it allocates mass to all machines that have not yet erred in proportion to their simplicity, so probability mass can end up on the class of machines, each individually quite complex, that describe QM and then hardcode the branch predictions. See also the proof about how the version of SI in which each TM outputs probabilities is equivalent to the version where they don’t.)
SI doesn’t put all its mass on the single machine that predicts the universe, it allocates mass to all machines that have not yet erred in proportion to their simplicity,
If your SI can’t make predictions ITFP, that’s rather beside the point. “Not erring” only has a straightforward implementation if you are expecting the predictions to be deterministic. How could an SI compare a deterministic theory to a probablistic one?
How could an SI compare a deterministic theory to a probablistic one?
The deterministic theory gets probability proportional to 2^-length + (0 if it was correct so far else -infty), the probabilistic theory gets probability proportional to 2^-length + log(probability it assigned to the observations so far).
That said, I was not suggesting a solomonoff inductor in which some machines were outputting bits and others were outputting probabilities.
I suspect that there’s a miscommunication somewhere up the line, and my not-terribly-charitable-guess is that it stems from you misunderstanding the formalism of Solomonoff induction and/or the point I was making about it. I do not expect to clarify further, alas. I’d welcome someone else hopping in if they think they see the point I was making & can transmit it.
The only reason that sort of discarding works is because of decoherence (which is a probabilistic, thermodynamic phenomenon), and in fact, as a result, if you want to be super precise, discarding actually doesn’t work, since the impact of those other eigenfunctions never literally goes to zero.
You need to separate treating (1) something literally as a delta function, (2) treating something you have observed as having probability 1.0 for the purpose of further probability calculation.
If you are measuring something discrete like spin-up versus spin-down, it is completely standard to set the unobserved state to 0, effectively discarding it. The discarding (projection) is just as necessary a part of the procedure as the absolute-squaring (Born’s rule per se).
It’s not that no-one knows how to predict a series of observations with correct probabilities using QM, it is that the time-honoured method looks like Copenhagen.
I agree that there’s a difference between “put a delta-spike on the single classical state you sampled” and “zero out amplitude on all states not consistent with the observation you got from your sample”. I disagree that using the latter to generate a sensory stream from a quantum state yields reasonable predictions—eg, taken literally I think you’re still zeroing out all but a measure-zero subset of the position basis, and I expect the momenta to explode immediately. You can perhaps get this hypothesis (or the vanilla delta spike) hobbling by trying to smooth things out a bit (eg, keep a Gaussian centered on each classical state in which you made the sampled observation), but I still expect this to be experimentally distinguishable from what really happens (eg by way of some quantum-eraser-style hijinks or other sizeable entanglements), though I haven’t checked the details myself.
The observation you got from your sample is information. Information is entropy, and entropy is locally finite. So I don’t think it’s possible for the states consistent with the observation you got from your sample to have measure zero.
When you’re using TMs to approximate physics, you have to balance the continuity of physics against the discreteness of the machines somehow. The easy thing to do is to discuss the limiting behavior of a family of machines that perform the simulation at ever-finer fidelity. I was doing this implicitly, for lack of desire to get into details.
And as I’ve said above, I’m not attempting to suggest that these naive approaches—such as sampling a single classical state and reporting the positions of some things with arbitrary fidelity in the limit—are reasonable ideas. Quite the opposite. What I’m trying to point out is that if all you have is a quantum state and the Born rule, you cannot turn it into a hypothesis without making a bunch of other choices, for which I know of no consensus answer (and for which I have not seen proposals that would resolve the problem to my satisfaction, though I have some ideas).
I agree that the correct way of making these choices will almost surely not involve recording any observation with infinite precision (in the limit).
You have been assuming that all measurements are in the position basis, which is wrong. In particular, spin is its own basis.
If you make a sharp measurement in one basis, you have uncertainty or lack of information about the others. That does not mean the “momentum is randomised” in some catastrophic sense. The original position measurement was not deterministic, for one thing.
It is true that delta functions can be badly behaved. It’s also true that they can be used in practice … if you are careful. They are not an argument against discarding-and-renormalising , because if you don’t do that at all, you get much wronger results than the results you get by rounding off small values to zero, ie. using a delta to represent a sharp gaussian.
That might be the case if you were making an infinitely sharp measurement of an observable with a real valued spectrum, but there are no infinitely sharp measurements, and not every observable is real-valued.
To be clear, the process that I’m talking about for turning a quantum state into a hypothesis is not intended to be a physical process (such as a measurement), it’s intended to be a Turing machine (that produces output suitable for use by Solomonoff induction).
That said, to be clear, I don’t think this is a fundamentally hard problem. My point is not “we have absolutely no idea how to do it”, it’s somehing more like “there’s not a consensus answer here” + “it requires additional machinery above and beyond [the state vector + the born rule + your home address]” + “in my experience, many (non-specialist-physicist) people don’t even register this as a problem, and talk about the Born rule as if it’s supposed to fill this gap”.
I agree that there are a bunch of reasonable additional pieces of machinery you can use to get the missing piece (such as choice of a “measurement basis”); my own suspicion is that the right answer looks a bit different from what I’ve seen others propose (and routes through, eg, machinery that lets you read off the remembered history, as opposed to machinery that picks some subbasis to retain); my guess is that there are experimental differences in theory but they’re probably tricky to create in practice, but I haven’t worked through the details myself.
Then you run into the basic problem of using SI to investigate MW: SI’s are supposed to output a series of definite observations. They are inherently “single world”
If the program running the SWE outputs information about all worlds on a single output tape, they are going to have to be concatenated or interleaved somehow. Which means that to make use of the information, you have to identify the subset if bits relating to your world. That’s extra complexity which isn’t accounted for because it’s being done by hand, as it were.
In particular, if you just model the wave function, the only results you will get represent every possible outcome. In order to match observation , you will have to keep discarding unobserved outcomes and renormalising as you do in every interpretation. It’s just that that extra stage is performed manually, not by the programme.
To get an output that matches one observers measurements, you would need to simulate collapse somehow. You could simulate collapse with a PRNG, but it won’t give you the right random numbers.
Or you would need to keep feeding your observations back in so that the simulator can perform projection and renormalisation itself. That would work, but that’s a departure from how SI’s are supposed to work.
Meta: trying to mechanise epistemology doesn’t solve much , because mechanisms still have assumptions built into them.
Yeah yeah, this is the problem I’m referring to :-)
I disagree that you must simulate collapse to solve this problem, though I agree that that would be one way to do it. (The way you get the right random numbers, fwiw, is from sample complexity—SI doesn’t put all its mass on the single machine that predicts the universe, it allocates mass to all machines that have not yet erred in proportion to their simplicity, so probability mass can end up on the class of machines, each individually quite complex, that describe QM and then hardcode the branch predictions. See also the proof about how the version of SI in which each TM outputs probabilities is equivalent to the version where they don’t.)
If your SI can’t make predictions ITFP, that’s rather beside the point. “Not erring” only has a straightforward implementation if you are expecting the predictions to be deterministic. How could an SI compare a deterministic theory to a probablistic one?
The deterministic theory gets probability proportional to 2^-length + (0 if it was correct so far else -infty), the probabilistic theory gets probability proportional to 2^-length + log(probability it assigned to the observations so far).
That said, I was not suggesting a solomonoff inductor in which some machines were outputting bits and others were outputting probabilities.
I suspect that there’s a miscommunication somewhere up the line, and my not-terribly-charitable-guess is that it stems from you misunderstanding the formalism of Solomonoff induction and/or the point I was making about it. I do not expect to clarify further, alas. I’d welcome someone else hopping in if they think they see the point I was making & can transmit it.
The only reason that sort of discarding works is because of decoherence (which is a probabilistic, thermodynamic phenomenon), and in fact, as a result, if you want to be super precise, discarding actually doesn’t work, since the impact of those other eigenfunctions never literally goes to zero.
Maybe, but decoherence doesn’t imply MW.