Physicist and dabbler in writing fantasy/science fiction.
Ben
Karma: 2,285
I still find the effect weird, but something that I think makes it more clear is this phase space diagram:
We are treating the situation as 1D, and the circles in the x, p space are energy contours. Total energy is distance from the origin. An object in orbit goes in circles with a fixed distance from the origin. (IE a fixed total energy).
The green and purple points are two points on the same orbit. At purple we have maximum momentum and minimum potential energy. At green its the other way around. The arrows show impulses, if we could suddenly add momentum of a fixed amount by firing the rocket those are the arrows.
Its visually clear that the arrow from the purple point is more efficient. It gets us more than one whole solid-black energy contour higher, in contrast the same length of arrow at the green position only gets us to the dashed orbit, which is lower.
Visually we can see that if we want to get away from the origin of that x, p coordinate system we should shoot when out boost vector aligns with out existing vector.
A weird consequence. Say our spaceship didn’t have a rocket, but instead it had a machine that teleported the ship a fixed distance (say 100m). (A fixed change in position, instead of a fixed change in momentum). In this diagram that is just rotating the arrows 90 degrees. This implies the most efficient time to use the teleporting machine is when you are at the maximum distance from the planet (minimum kinetic energy, maximum potential). Mathematically this is because the potential energy has the same quadratic scaling as the kinetic. Visually, its because its where you are adding the new vector to your existing vector most efficiently.
I am not sure that is right. A very large percentage of people really don’t think the rolls are independent. Have you ever met anyone who believed in fate, Karma, horoscopes , lucky objects or prayer? They don’t think its (fully) random and independent. I think the majority of the human population believe in one or more of those things.
If someone spells a word wrong in a spelling test, then its possible they mistyped, but if its a word most people can’t spell correctly then the hypothesis “they don’t know the spelling’ should dominate. Similarly, I think it is fair to say that a very large fraction of humans (over 50%?) don’t actually think dice rolls or coin tosses are independent and random.
That is a cool idea! I started writing a reply, but it got a bit long so I decided to make it its own post in the end. ( https://www.lesswrong.com/posts/AhmZBCKXAeAitqAYz/celtic-knots-on-einstein-lattice )
I stuck to maximal density for two reaosns, (1) to continue the Celtic knot analogy (2) because it means all tiles are always compatible (you can fit two side by side at any orientation without loosing continuity). With tiles that dont use every facet this becomes an issue.
Thinking about it now, and without having checked carefully, I think this compatibilty does something topological and forces odd macrostructure. For example, if we have a line of 4-tiles in a sea of 6-tiles (4 tiles use four facets), then we cant end the line of 4 tiles without breaking continuity. So the wall has to loop, or go off the end. The ‘missing lines’ the 4 tiles lacked (that would have made them 6′s) would have been looped through the 4-wall. So having those tiles available is kind of like being able to delete a few closed loops from a 6s structure.
I might try messing with 4s to see if you are right that they will be asthetically useful.
That’s a very interesting idea. I tried going through the blue one at the end.
Its not possible in that case for each string to strictly alternate between going over and under, by any of the rules I have tried. In some cases two strings pass over/under one another, then those same two strings meet again when one has travelled two tiles and the other three. So they are de-synced. They both think its their turn to go over (or under).
The rules I tried to apply were (all of which I believe don’t work):
Over for one tile, under for the next (along each string)
Over for one collision, under for the next (0, 1 or 2 collisions, are possible in a tile)
Each string follows the sequence ‘highest, middle, lowest, highest, middle lowest...’ for each tile it enters.
My feeling having played with it for about 30-45 mins is that there probably is a rule nearby to those above that makes things nice, but I haven’t yet found it.
I wasn’t aware of that game. Yes it is identical in terms of the tile designs. Thank you for sharing that, it was very interesting and that Tantrix wiki page lead me to this one, https://en.wikipedia.org/wiki/Serpentiles , which goes into some interesting related stuff with two strings per side or differently shaped tiles.
Something related that I find interesting, for people inside a company, the real rival isn’t another company doing the same thing, but people in your own company doing a different thing.
Imagine you work at Microsoft in the AI research team in 2021. Management want to cut R&D spending, so either your lot or the team doing quantum computer research are going to be redundant soon. Then, the timeline splits. In one universe, Open AI release Chat GPT, in the other PsiQuantum do something super impressive with quantum stuff. In which of those universes do the Microsoft AI team do well? In one, promotions and raises, in the other, redundancy.
People recognise this instinctively. Changing companies is much faster and easier than changing specialities. So people care primarily about their speciality doing well, their own specific company is a secondary concern.
A fusion expert can expert at a different fusion company way faster and more easily than they can become an expert in wind turbines. Therefore, to the fusion expert all fusion companies are on the same side against the abominable wind farmers. I suspect this is also true of most people in AI, although maybe when talking to the press they will be honour bound to claim otherwise.
I wonder if any of the perceived difference between fusion and AI might be which information sources are available to you. It sounds like you have met the fusion people, and read their trade magazines, and are comparing that to what mainstream news says about AI companies (which won’t necessarily reflect the opinions of a median AI researcher.).
I think economics should be taught to children, not for the reasons you express, but because it seems perverse that I spent time at school learning about Vikings, Oxbow lakes, volcanoes, Shakespeare and Castles, but not about the economic system of resource distribution that surrounds me for the rest of my life. When I was about 9 I remember asking why ‘they’ didn’t just print more money until everyone had enough. I was fortunate to have parents who could give a good answer, not everyone will be.
Stock buybacks! Thank you. That is definitely going to be a big part f the “I am missing something here” I was expressing above.
I freely admit to not really understanding how shares are priced. To me it seems like the value of a share should be related to the expected dividend pay-out of that share over the remaining lifetime of the company, with a discount rate applied on pay-outs that are expected to happen further in the future (IE dividend yields 100 years from now are valued much less than equivalent payments this year). By this measure, justifying the current price sounds hard.
Google says that the annual dividend on Nvidia shares is 0.032%. (Yes, the leading digits are 0.0). So, right now, you get a much better rate of return just leaving your money in your bank’s current account. So, at least by this measure, Nvidia shares are ludicrously over-priced. You could argue that future Nvidia pay outs might be much larger than the historical ones due to some big AI related profits. But, I don’t find this argument convincing. Are future pay outs going to be 100x bigger? It would require a 100-fold yield increase for it to just be competitive with a savings account. If you time discount a little (say those 100-fold increases don’t materialise for 3 years) then it looks even worse.
Now, clearly the world doesn’t value shares according to the same heuristics that make sense to a non-expert like me. For example, the method “time integrate future expected dividend pay outs with some kind of time discounting” tells us that cryptocurrencies are worthless, because they are like shares with zero dividends. But, people clearly do put a nonzero value on bitcoin—and there is no plausible way that many people are that wrong. So they are grasping something that I am missing, and that same thing is probably what allows company shares to be prices so high relative to the dividends.
That is very interesting! That does sound weird.
In some papers people write density operators using an enhanced “double ket” Dirac notation, where eg. density operators are written to look like |x>>, with two “>”’s. They do this exactly because the differential equations look more elegant.
I think in this notation measurements look like <<m|, but am not sure about that. The QuTiP software (which is very common in quantum modelling) uses something like this under-the-hood, where operators (eg density operators) are stored internally using 1d vectors, and the super-operators (maps from operators to operators) are stored as matrices.
So structuring the notation in other ways does happen, in ways that look quite reminiscent of your tensors (maybe the same).
Yes, in your example a recipient who doesn’t know the seed models the light as unpolarised, and one who does as say, H-polarised in a given run. But for everyone who doesn’t see the random seed its the same density matrix.
Lets replace that first machine with a similar one that produces a polarisation entangled photon pair, |HH> + |VV> (ignoring normalisation). If you have one of those photons it looks unpolarised (essentially your “ignorance of the random seed” can be thought of as your ignorance of the polarisation of the other photon).
If someone else (possibly outside your light cone) measures the other photon in the HV basis then half the time they will project your photon into |H> and half the time into |V>, each with 50% probability. This 50⁄50 appears in the density matrix, not the wavefunction, so is “ignorance probability”.
In this case, by what I understand to be your position, the fact of the matter is either (1) that the photon is still entangled with a distant photon, or (2) that it has been projected into a specific polarisation by a measurement on that distant photon. Its not clear when the transformation from (1) to (2) takes place (if its instant, then in which reference frame?).
So, in the bigger context of this conversation,
OP: “You live in the density matrices (Neo)”
Charlie :”No, a density matrix incorporates my own ignorance so is not a sensible picture of the fundamental reality. I can use them mathematically, but the underlying reality is built of quantum states, and that randomness when I subject them to measurements is fundamentally part of the territory, not the map. Lets not mix the two things up.”
Me: “Whether a given unit of randomness is in the map (IE ignorance), or the territory is subtle. Things that randomly combine quantum states (my first machine) have a symmetry over which underlying quantum states are being mixed that looks meaningful. Plus (this post), the randomness can move abruptly from the territory to the map due to events outside your own light cone (although the amount of randomness is conserved), so maybe worrying too much about the distinction isn’t that helpful.
What is the Bayesian argument, if one exists, for why quantum dynamics breaks the “probability is in the mind” philosophy?
In my world-view the argument is based on Bell inequalities. Other answers mention them, I will try and give more of an introduction.
First, context. We can reason inside a theory, and we can reason about a theory. The two are completely different and give different intuitions. Anyone talking about “but the complex amplitudes exist” or “we are in one Everett branch” is reasoning inside the theory. The theory, as given in the textbooks, is accepted as true and interpretations built on.
However, both historically and (I think) more generally, we should also reason about theories. This means we need to look at experimental observations, and ask questions like “what is the most reasonable model?”.
Many quantum experiments give random-looking results. As you point out, randomness is usually just “in the mind”. Reality was deterministic, but we couldn’t see everything. The terminology is “local hidden variable”. For an experiment where you draw a card from a deck the “local hidden variable” was which card was on top. In a lottery with (assumedly deterministic) pinballs the local hidden variable is some very specific details of the initial momentums and positions of the balls. In other words the local hidden variable is the thing that you don’t know, so to you it looks random. Its the seed of your pseudorandom number generator.
Entanglement—It is possible to prepare two (or more) particles in a state, such that measurements of those two particles gives very weird results. What do I mean by “very weird”. Well, in a classical setting if Alice and Bob are measuring two separate objects then there are three possible (extremal) situations (1): Their results are completely uncorrelated, for example Alice is rolling a dice in Texas and Bob is rolling a different dice in London. (2) Correlated, for example, Alice is reading an email telling her she got a job she applied for, and Bob is reading an email telling him he failed to get the same job. (4) Signalling (we skipped 3 on purpose, we will get to that). Alice and Bob have phones, and so the data they receive is related to what the other of them is doing. Linear combinations of the above (eg noisy radio messages, correlation that is nor perfect etc) are also possible.
By very weird, I mean that quantum experiments give rise (in the raw experimental data, before any theory is glued on) to a fourth type of relation; (3): Non-locality. Alice and Bob’s measurement outcomes (observations) are random, but the correlation between their observation’s changes depending on the measurements they both chose to make (inputs). Mathematically its no more complex than the others, but its fiddly to get your head around because its not something seen in everyday life.
An important feature of (3) is that it cannot be used to create signalling (4). However, (3) cannot be created out of any mixture of (1) and (2). (Just like (4) cannot be created by mixing (1) and (2)). In short, if you have any one of these 4 things, you can use local actions to go “down hill” to lower numbers but you can’t go up.
Anyway, “hidden variables” are shorthand for “(1) and (2)” (randomness and correlation). The “local” means “no signalling” (IE no (3), no radios). The reason we insist on no signalling is because the measurements Alice and Bob do on their particles could be outside one another’s light cones (so even a lightspeed signal would not be fast enough to explain the statistics). The “no signalling” condition might sound artificial, but if you allow faster than light signalling then you are (by the standards of relativity) also allowing time travel.
Bell inequality experiments have been done. They measure result (3). (3) cannot be made out of ordinary “ignorance” probabilities (cannot be made from (2)). (3) could be made out of (4) (faster than light signalling), but we don’t see the signalling itself, and assuming it exists entails time travel.
So, if we reject signalling, we know that whatever it is that is happening in a Bell inequality experiment it can’t be merely apparent randomness due to our ignorance. We also know the individual results collected by Alice and Bob look random (but not the correlations between the results), this backs us into the corner of accepting that the randomness is somehow an intrinsic feature of the world, even the photon didn’t “know” if it would go through the polariser until you tried it.
The wiki article on Bell inequalities isn’t very good unfortunately.
Just the greentext. Yes, I totally agree that the study probably never happened. I just engaged with the actualy underling hypothesis, and to do so felt like some summary of the study helped. But I phrased it badly and it seems like I am claiming the study actually happened. I will edit.
I thought they were typically wavefunction to wavefunction maps, and they need some sort of sandwiching to apply to density matrices?
Yes, this is correct. My mistake, it does indeed need the sandwiching like this .
From your talk on tensors, I am sure it will not surprise you at all to know that the sandwhich thing itself (mapping from operators to operators) is often called a superoperator.
I think the reason it is as it is is their isn’t a clear line between operators that modify the state and those that represent measurements. For example, the Hamiltonian operator evolves the state with time. But, taking the trace of the Hamiltonian operator applied to the state gives the expectation value of the energy.
The way it works normally is that you have a state , and its acted on by some operator, , which you can write as . But this doesn’t give a number, it gives a new state like the old but different. (For example if a was the anhilation operator the new state is like the old state but with one fewer photons). This is how (for example) an operator acts on the state of the system to change that state. (Its a density matrix to density matrix map).
In dimensions terms this is: (1,1) = (1, 1) * (1,1)
(Two square matrices of size N multiply to give another square matrix of size N).
However, to get the expected outcome of a measurement on a particular state you take : where Tr is the trace. The trace basically gets the “plug” at the left hand side of a matrix and twists it around to plug it into the right hand side. So overall what is happening is that the operators and , each have shapes (1,1) and what we do is:
Tr( (1,1) * (1,1)) = Tr( (1, 1) ) = number.
The “inward facing” dimensions of each matrix get plugged into one another because the matrices multiply, and the outward facing dimensions get redirected by the trace operation to also plug into one another. (The Trace is like matrix multiplication but on paper that has been rolled up into a cylinder, so each of the two matrices inside sees the other on both sides). The net effect is exactly the same as if they had originally been organized into the shapes you suggest of (2,0) and (0,2) respectively.
So if the two “ports” are called A and B your way of doing it gives:
(AB, 0) * (0, AB) = (0, 0) IE number
The traditional way:
Tr( (A, B) * (B, A) ) = Tr( (A, A) ) = (0, 0) , IE number.
I haven’t looked at tensors much but I think that in tensor-land this Trace operation takes the role of a really boring metric tensor that is just (1,1,1,1...) down the diagonal.
So (assuming I understand right) your way of doing it is cleaner and more elegant for getting the expectation value of a measurement. But the traditional system works more elegantly for applying an operator too a state to evolve it into another state.
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.
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.
That is a nice idea. The “two sides at 180 degrees” only occurred to me after I had finished. I may look into that one day, but with that many connections is needs to be automated.
In the 6 entries/exits ones above you pick one entry, you have 5 options of where to connect it. Then, you pick the next unused entry clockwise, and have 3 options for where to send it, then you have only one option for how to connect the last two. So its 5x3x1 = 15 different possible tiles.
With 14 entries/exits, its 13x11x9x7x5x3x1 = 135,135 different tiles. (13!!, for !! being double factorial).
You also have (I think) 13+12+11+10+… = 91 different connection pieces.
One day, I may try and write a code to make some of those. I strongly suspect that they won’t look nice, but they might be interesting anyway.