Physicist and dabbler in writing fantasy/science fiction.
Ben
Karma: 2,284
I think the reason is that in quantum physics we also have operators representing processes (like the Hamiltonian operator making the system evolve with time, or the position operator that “measures” position, or the creation operator that adds a photon), and the density matrix has exactly the same mathematical form as these other operators (apart from the fact the density matrix needs to be normalized).
But that doesn’t really solve the mystery fully, because they could all just be called “matrices” or “tensors” instead of “operators”. (Maybe it gets us halfway to an explanation, because all of the ones other than the density operator look like they “operate” on the system to make it change its state.)
Speculatively, it might be to do with the fact that some of these operators are applied on continuous variables (like position), where the matrix representation has infinite rows and infinite columns—maybe their is some technicality where if you have an object like that you have to stop using the word “matrix” or the maths police lock you up.
There are some non-obvious issues with saying “the wavefunction really exists, but the density matrix is only a representation of our own ignorance”. Its a perfectly defensible viewpoint, but I think it is interesting to look at some of its potential problems:
A process or machine prepares either |0> or |1> at random, each with 50% probability. Another machine prepares either |+> or |-> based on a coin flick, where |+> = (|0> + |1>)/root2, and |+> = (|0> - |1>)/root2. In your ontology these are actually different machines that produce different states. In contrast, in the density matrix formulation these are alternative descriptions of the same machine. In any possible experiment, the two machines are identical. Exactly how much of a problem this is for believing in wavefuntions but not density matrices is debatable—“two things can look the same, big deal” vs “but, experiments are the ultimate arbiters of truth, if experiemnt says they are the same thing then they must be and the theory needs fixing.”
There are many different mathematical representations of quantum theory. For example, instead of states in Hilbert space we can use quasi-probability distributions in phase space, or path integrals. The relevance to this discussion is that the quasi-probability distributions in phase space are equivalent to density matrices, not wavefunctions. To exaggerate the case, imagine that we have a large number of different ways of putting quantum physics into a mathematical language, [A, B, C, D....] and so on. All of them are physically the same theory, just couched in different mathematics language, a bit like say, [“Hello”, “Hola”, “Bonjour”, “Ciao”...] all mean the same thing in different languages. But, wavefunctions only exist as an entity separable from density matrices in some of those descriptions. If you had never seen another language maybe the fact that the word “Hello” contains the word “Hell” as a substring might seem to possibly correspond to something fundamental about what a greeting is (after all, “Hell is other people”). But its just a feature of English, and languages with an equal ability to greet don’t have it. Within the Hilbert space language it looks like wavefunctions might have a level of existence that is higher than that of density matrices, but why are you privileging that specific language over others?
In a wavefunction-only ontology we have two types of randomness, that is normal ignorance and the weird fundamental quantum uncertainty. In the density matrix ontology we have the total probability, plus some weird quantum thing called “coherence” that means some portion of that probability can cancel out when we might otherwise expect it to add together. Taking another analogy (I love those), the split you like is [100ml water + 100ml oil], (but water is just my ignorance and doesn’t really exist), and you don’t like the density matrix representation of [200ml fluid total, oil content 50%]. Their is no “problem” here per se but I think it helps underline how the two descriptions seem equally valid. When someone else measures your state they either kill its coherence (drop oil % to zero), or they transform its oil into water. Equivalent descriptions.
All of that said, your position is fully reasonable, I am just trying to point out that the way density matrices are usually introduced in teaching or textbooks does make the issue seem a lot more clear cut than I think it really is.
I just looked up the breakfast hypothetical. Its interesting, thanks for sharing it.
So, my understanding is (supposedly) someone asked a lot of prisoners “How would you feel if you hadn’t had breakfast this morning?”, did IQ tests on the same prisoners and found that the ones who answered “I did have breakfast this morning.” or equivalent were on average very low in IQ. (Lets just assume for the purposes of discussion that this did happen as advertised.)
It is interesting. I think in conversation people very often hear the question they were expecting, and if its unexpected enough they hear the words rearranged to make it more expected. There are conversations where the question could fit smoothly, but in most contexts its a weird question that would mostly be measuring “are people hearing what they expect, or what is being actually said”. This may also correlate strongly with having English as a second language.
I find the idea “dumb people just can’t understand a counterfactual” completely implausible. Without a counterfactual you can’t establish causality. Without causality their is no way of connecting action to outcome. How could such a person even learn to use a TV remote? Given that these people (I assume) can operate TV remotes they must in fact understand counterfactuals internally, although its possible they lack the language skills to clearly communicate about them.
The question of “why should the observed frequencies of events be proportional to the square amplitudes” is actually one of the places where many people perceive something fishy or weird with many worlds. [https://www.sciencedirect.com/science/article/pii/S1355219809000306 ]
To clarify, its not a question of possibly rejecting the square-amplitude Born Rule while keeping many worlds. Its a question of whether the square-amplitude Born Rule makes sense within the many worlds perspective, and it if doesn’t what should be modified about the many worlds perspective to make it make sense.
I agree with this. Its something about the guilt that makes this work. Also the sense that you went into it yourself somehow reshapes the perception.
I think the loan shark business model maybe follows the same logic. [If you are going to eventually get into a situation where the victim pays or else suffers violence, then why doesn’t the perpetrator just skip the costly loan step at the beginning and go in threat first? I assume that the existence of loan sharks (rather than just blackmailers) proves something about how if people feel like they made a bad choice or engaged willingly at some point they are more susceptible. Or maybe its frog boiling.]
On the “what did we start getting right in the 1980′s for reducing global poverty” I think most of the answer was a change in direction of China. In the late 70′s they started reforming their economy (added more capitalism, less command economy): https://en.wikipedia.org/wiki/Chinese_economic_reform.
Comparing this graph on wiki https://en.wikipedia.org/wiki/Poverty_in_China#/media/File:Poverty_in_China.svg , to yours, it looks like China accounts for practically all of the drop in poverty since the 1980s.
Arguably this is a good example for your other points. More willing participation, less central command.
I don’t think the framing “Is behaviour X exploitation?” is the right framing. It takes what (should be) an argument about morality and instead turns it into an argument about the definition of the word “exploitation” (where we take it as given that, whatever the hell we decide exploitation “actually means” it is a bad thing). For example see this post: https://www.lesswrong.com/posts/yCWPkLi8wJvewPbEp/the-noncentral-fallacy-the-worst-argument-in-the-world. Once we have a definition of “exploitation” their might be some weird edge cases that are technically exploitation but are obviously fine.
The substantial argument (I think) is that when two parties have unequal bargaining positions, is it OK for the stronger party to get the best deal it can? A full-widget is worth a million dollars. I possess the only left half of a widget in the world. Ten million people each possess a right half that could doc with my left half. Those not used to make widgets are worthless. What is the ethical split for me to offer for a right half in this case?
[This is maybe kind of equivalent to the dating example you give. At least in my view the “bad thing” in the dating example is the phrase “She begins using this position to change the relationship”. The word “change” is the one that sets the alarms for me. If they both went in knowing what was going on then, to me, that’s Ok. Its the “trap” that is not. I think most of the things we would object to are like this, those Monday meetings and that expensive suit are implied to be surprises jumped onto poor Bob.]
The teapot comparison (to me) seems to be a bad. I got carried away and wrote a wall of text. Feel free to ignore it!
First, lets think about normal probabilities in everyday life. Sometimes there are more ways for one state to come about that another state, for example if I shuffle a deck of cards the number of orderings that look random is much larger than the number of ways (1) of the cards being exactly in order.
However, this manner of thinking only applies to certain kinds of thing—those that are in-principle distinguishable. If you have a deck of blank cards, there is only one possible order, BBBBBB.… To take another example, an electronic bank account might display a total balance of $100. How many different ways are their for that $100 to be “arranged” in that bank account? The same number as 100 coins labelled “1″ through “100”? No, of course not. Its just an integer stored on a computer, and their is only one way of picking out the integer 100. The surprising examples of this come from quantum physics, where photons act more like the bank account, where their is only 1 way of a particular mode to contain 100 indistinguishable photons. We don’t need to understand the standard model for this, even if we didn’t have any quantum theory at all we could still observe these Boson statistics in experiments.
So now, we encounter anthropic arguments like Doomsday. These arguments are essentially positing a distribution, where we take the exact same physical universe and its entire physical history from beginning to end, (which includes every atom, every synapse firing and so on). We then look at all of the “counting minds” in that universe (people count, ants probably don’t, aliens, who knows), and we create a whole slew of “subjective universes”, , , , , etc, where each of of them is atomically identical to the original but “I” am born as a different one of those minds (I think these are sometimes called “centred worlds”). We assume that all of these subjective universes were, in the first place, equally likely, and we start finding it a really weird coincidence that in the one we find ourselves in we are a human (instead of an Ant), or that we are early in history. This is, as I understand it, The Argument. You can phrase it without explicitly mentioning the different s, by saying “if there are trillions of people in the future, the chances of me being born in the present are very low. So, the fact I was born now should update me away from believing there will be trillions of people in the future”. - but the s are still doing all the work in the background.
The conclusion depends on treating all those different subscripted s as distinguishable, like we would for cards that had symbols printed on them. But, if all the cards in the deck are identical there is only one sequence possible. I believe that all of the , , , ’s etc are identical in this manner. By assumption they are atomically identical at all times in history, they differ only by which one of the thinking apes gets assigned the arbitrary label “me”—which isn’t physically represented in any particle. You think they look different, and if we accept that we can indeed make these arguments, but if you think they are merely different descriptions of the same exact thing then the Doomsday argument no longer makes sense, and possibly some other anthropic arguments also fall apart. I don’t think they do look different, if every “I” in the universe suddenly swapped places—but leaving all memories and personality behind in the physical synapses etc, then, how would I even know it? I would be a cyborg fighting in WWXIV and would have no memories of ever being some puny human typing on a web forum in the 21s Cent. Instead of imaging that I was born as someone else I could imagine that I could wake up as someone else, and in any case I wouldn’t know any different.
So, at least to me, it looks like the anthropic arguments are advancing the idea of this orbital teapot (the different scripted s, although it is, in fairness, a very conceptually plausible teapot). There are, to me, three possible responses:
1 - This set of different worlds doesn’t logically exist. You could push this for this response by arguing “I couldn’t have been anyone but me, by definition.” [Reject the premise entirely—there is no teapot]
2 - This set of different worlds does logically make sense, and after accepting it I see that it is a suspicious coincidence I am so early in history and I should worry about that. [accept the argument—there is a ceramic teapot orbiting Mars]
3 - This set of different worlds does logically make sense, but they should be treated like indistinguishable particles, blank playing cards or bank balances. [accept the core premise, but question its details in a way that rejects the conclusion—there is a teapot, but its chocolate, not ceramic.].
So, my point (after all that, Sorry!) is that I don’t see any reason why (2) is more convincing that (3).
[For me personally, I don’t like (1) because I think it does badly in cases where I get replicated in the future (eg sleeping beauty problems, or mind uploads or whatever). I reject (2) because the end result of accepting it is that I can infer information through evidence that is not causally linked to the information I gain (eg. I discover that the historical human population was much bigger than previously reported, and as a result I conclude the apocalypse is further in the future than I previously supposed). This leads me to thinking (3) seems right-ish, although I readily admit to being unsure about all this.].
I found this post to be a really interesting discussion of why organisms that sexually reproduce have been successful and how the whole thing emerges. I found the writing style, where it switched rapidly between relatively serious biology and silly jokes very engaging.
Many of the sub claims seem to be well referenced (I particularly liked the swordless ancestor to the swordfish liking mates who had had artificial swords attached).
“Stock prices represent the market’s best guess at a stock’s future price.”
But they are not the same as the market’s best guess at its future price. If you have a raffle ticket that will, 100% for definite, win $100 when the raffle happens in 10 years time, the the market’s best guess of its future price is $100, but nobody is going to buy it for $100, because $100 now is better than $100 in 10 years.
Whatever it is that people think the stock will be worth in the future, they will pay less than that for it now. (Because $100 in the future isn’t as good as just having the money now). So even if it was a cosmic law of the universe that all companies become more productive over time, and everyone knew this to be true, the stocks in those companies would still go up over time, like the raffle ticket approaching the pay day.
Toy example:
1990 - Stocks in C cost $10. Everyone thinks they will be worth $20 by the year 2000, but 10 years is a reasonably long time to wait to double your money so these two things (the expectation of 20 in the future, and the reality of 10 now) coexist without contradiction.
2000 - Stocks in C now cost $20, as expected. People now think that by 2010 they will be worth $40.
Other Ant-worriers are out there!
“”it turned out this way, so I guess it had to be this way” doesn’t resolve my confusion”
Sorry, I mixed the position I hold (that they maybe work like bosons) and the position I was trying to argue for, which was an argument in favor of confusion.
I can’t prove (or even strongly motivate) my “the imaginary mind-swap procedure works like a swap of indistinguishable bosons” assumption, but, as far as I know no one arguing for Anthropic arguments can prove (or strongly motivate) the inverse position—which is essential for many of these arguments to work. I agree with you that we don’t have a standard model of minds, and without such a model the Doomsday Argument, and the related problem of being cosmically early might not be problems at all.
Interestingly, I don’t think the weird boson argument actually does anything for worries about whether we are simulations, or Boltzmann brains—those fears (I think) survive intact.
I suspect there is a large variation between countries in how safely taxi drivers drive relative to others.
In London my impression is that the taxis are driven more safely than non-taxis. In Singapore it appears obvious to casual observation that taxis are much less safely driven than most of the cars.
At least in my view, all the questions like the “Doomsday argument” and “why am I early in cosmological” history are putting far, far too much weight on the anthropic component.
If I don’t know how many X’s their are, and I learn that one of them is numbered 20 billion then sure, my best guess is that there are 40 billion total. But its a very hazy guess.
If I don’t know how many X’s will be produced next year, but I know 150 million were produced this year, my best guess is 150 million next year. But is a very hazy guess.
If I know that the population of X’s has been exponentially growing with some coefficient then my best guess for the future is to infer that out to future times.
If I think I know a bunch of stuff about the amount of food the Earth can produce, the chances of asteroid impacts, nuclear wars, dangerous AIs or the end of the Mayan calendar then I can presumably update on those to make better predictions of the number of people in the future.
My take is that the Doomsday argument would be the best guess you could make if you knew literally nothing else about human beings apart from the number that came before you. If you happen to know anything else at all about the world (eg. that humans reproduce, or that the population is growing) then you are perfectly at liberty to make use of that richer information and put forward a better guess. Someone who traces out the exponential of human population growth out to the heat death of the universe is being a bit silly (lets call this the Exponentiator Argument), but on pure reasoning grounds they are miles ahead of the Doomsday argument, because both of them applied a natural, but naïve, interpolation to a dataset, but the exponentiator interpolated from a much richer and more detailed dataset.
Similarly to answer “why are you early” you should use all the data at your disposal. Given who your parents are, what your job is, your lack of cybernetic or genetic enhancements, how could you not be early? Sure, you might be a simulation of someone who only thinks they are in the 21st centaury, but you already know from what you can see and remember that you aren’t a cyborg in the year 10,000, so you can’t include that possibility in your imaginary dataset that you are using to reason about how early you are.
As a child, I used to worry a lot about what a weird coincidence it was that I was born a human being, and not an ant, given that ants are so much more numerous. But now, when I try and imagine a world where “I” was instead born as the ant, and the ant born as me, I can’t point to in what physical sense that world is different from our own. I can’t even coherently point to in what metaphysical sense it is different. Before we can talk about probabilities as an average over possibilities we need to know if the different possibilities are even different, or just different labelling on the same outcome. To me, there is a pleasing comparison to be made with how bosons work. If you think about a situation where two identical bosons have their positions swapped, it “counts as” the same situation as before the swap, and you DON’T count it again when doing statistics. Similarly, I think if two identical minds are swapped you shouldn’t treat it as a new situation to average over, its indistinguishable. This is why the cyborgs are irrelevant, you don’t have an identical set of memories.
I remember reading something about the Great Leap Forward in China (it may have been the Cultural Revolution, but I think it was the Great Leap Forward) where some communist party official recognised that the policy had killed a lot of people and ruined the lives of nearly an entire generation, but they argued it was still a net good because it would enrich future generations of people in China.
For individuals you weigh up the risk/rewards of differing your resource for the future. But, as a society asking individuals to give up a lot of potential utility for unborn future generations is a harder sell. It requires coercion.
I think we might be talking past each other. I will try and clarify what I meant.
Firstly, I fully agree with you that standard game theory should give you access to randomization mechanisms. I was just saying that I think that hypotheticals where you are judged on the process you use to decide, and not on your final decision are a bad way of working out which processes are good, because the hypothetical can just declare any process to be the one it rewards by fiat.
Related to the randomization mechanisms, in the kinds of problems people worry about with predictors guessing your actions in advance its very important to distinguish between [1] (pseudo-)randomization processes that the predictor can predict, and [2] ones that it cannot.
[1] Randomisation that can be predicted by the predictor is (I think) a completely uncontroversial resource to give agents in these problems. In this case we don’t need to make predictions like “the agent will randomise”, because we can instead make the stronger prediction “the agent will randomize, and the seed of their RNG is this, so they will take one box” which is just a longer way of saying “they will one box”. We don’t need the predictor to show its working by mentioning the RNG intermediate step.
[2] Randomisation that is beyond the predictor’s power is (I think) not the kind of thing that can sensibly be included in these thought experiments. We cannot simultaneously assume that the predictor is pretty good at predicting our actions and useless at predicting a random number generator we might use to choose our actions. The premises: “Alice has a perfect quantum random number generator that is completely beyond the power of Omega to predict. Alice uses this machine to make decisions. Omega can predict Alice’s decisions with 99% accuracy” are incoherent.
So I don’t see how randomization helps. The first kind, [1] doesn’t change anything, and the second kind [2], seems like it cannot be consistently combined with the premise of the question. Perfect predictors and perfect random number generators cannot exist in the same universe.
Their might be interesting nearby problems where you imagine the predictor is 100% effective at determining the agents algorithm, but because the agent has access to a perfect random number generator that it cannot predict their actions. Maybe this is what you meant? In this kind of situation I am still much happier with rules like “It will fill the box with gold if it knows their is a <50% chance of you picking it”, [the closest we can get to “outcomes not processes” in probabilistic land], (or perhaps the alternative “the probability that it fills the box with gold is one-minus the probability with which it predicts the agent will pick the box”.). But rules like “It will fill the box with gold if the agents process uses either randomisation or causal decision theory” seem unhelpful to me.
I see where you are coming from. But, I think the reason we are interested in CDT (for any DT) in the first place is because we want to know which one works best. However, if we allow the outcomes to be judged not just on the decision we make, but also on the process used to reach that decision then I don’t think we can learn anything useful.
Or, to put it from a different angle, IF the process P is used to reach decision X, but my “score” depends not just on X but also P then that can be mapped to a different problem where my decision is “P and X”, and I use some other process (P’) to decide which P to use.
For example, if a student on a maths paper is told they will be marked not just on the answer they give, but the working out they write on the paper—with points deducted for crossings outs or mistakes—we could easily imagine the student using other sheets of paper (or the inside of their head) to first work out the working they are going to show and the answer that goes with it. Here the decision problem “output” is the entire exame paper, not just the answer.
I like this framing.
An alternative framing, which I think is also part of the answer is that some art is supposed to hit a very large audience and give each a small amount of utility, and other art is supposed to hit a smaller, more specialized, audience very hard. This framing explains things like traditional daytime TV, stuff that no one really loves but a large number of bored people find kind of unobjectionable. And how that is different from the more specialist TV you might actually look forward to an episode off but might hit a smaller audience.
(Obviously some things can hit a big audience and be good, and others can be bad on both counts. But the interesting quadrants two compare are the other two).
Random thoughts. You can relatively simply get a global phase factor at each timestep if you want. I don;t think a global phase factor at each step really counts as meaningfully different though. Anyway, as an example of this:
So that, at each (classical) timestep every single element of the CA tape just moves one step to the right. (So any patterns of 1′s and 0′s just orbit the tape in circles forever, unchanging.). Its quite a boring CA, but a simple example.
We can take the quantum CA that is exactly the same, but with some complex phase factor:
Where the delta function is saying “1 iff , else 0.”
This is exactly the same as the old classical one (everything moves on step to the right), but this time we also have a global phase factor applied to the total system. The total phase factor is , where N is the total number of cells on the tape.
Tiny bit more interesting:
Now we only gain phase factors on values of 1, so the global phase depends on the total number of 1′s on the tape, rather than its length.
To get proper quantum stuff we need phase factors that are not global. (IE some relative phases). I feel like this equation below is a reasonable kind of place to start, but I have run out of time for now so might return to this later.
After finding a Unitary that comes from one of your classical Cellular Automata then any power of that unitary will also be a valid unitary. So for example in classical logic their is a the “swap” gate for binary inputs, but in quantum computing the “square-root swap” gate also exists.
So you can get one of your existing unitary matrices, and (for example) take its square root. That would kind of be like a quantum system doing the classical Cellular Automata, that is interrupted halfway through the first step. (Because applying the root matrix twice is the same as applying the matrix). Similarly you can look at the 1/3rd step by applying the cube root of the matrix.
So would you consider the square root matrix a quantum elementary CA? Its not exactly equivalent to anything classical, because classically you can’t look “between the steps”.
[This is a long winded way of me saying that I don’t “get” the question. You want a unitary, U, of the form given in that equation for <y|U|x>, but you also don’t want U to be “basically equivalent” to a classical CA. How are you defining “basically equivalent”, is anything satisfying your equation automatically “basically equivalent”?]
You are completely correct in the “how does the machine work inside?” question. As you point out that density matrix has the exact form of something that is entangled with something else.
I think its very important to be discussing what is real, although as we always have a nonzero inferential distance between ourselves and the real the discussion has to be a little bit caveated and pragmatic.