Wikipedia:Reference desk/Archives/Science/2017 October 14

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October 14

Schroedinger's mathematician

I was just glancing at [1] - apparently you can entangle millions of atoms with a single photon. I'm not entirely sure, but I think they are boasting that the atoms are entangled in independent groups in this case, i.e. it is not a Schroedinger's cat state, but one where many pairs can be separately read without disturbing the others (I think!). But it intrigues me that they thought a Schroedinger's cat state was a possibility.

1) what is the closest we've actually come to Schroedinger's cat?

2) is it conceivable to genuinely make a macroscopic Schroedinger-cat state in space, with a well shielded capsule that would seem far from interacting with Earth? (Even if some bulk parameters remained observable, there might be multiple internal ways to explain them, for something less than a true superposition of all states but still useful?)

3) Is this a valid analogy for quantum computing? You send a mathematician up in one of those space capsules with a generous library, tell him to use a genuinely random number generator to decide what general methods to use to go about proving an unknown problem he has some interest in and what papers to read about it. The mission might be two months but he is instructed to break radio silence and phone home early if he has come up with a good proof. The theory is that the mathematician should be exceptionally lucky in his choices and phone home with unexpectedly high frequency, as viewed from Earth at least. OTOH I wonder if they could be an actual hazard for a mission to Mars if the probe called home early with bad news. But can the odds of calling home be changed by quantum effects at all? (I think they must be in QM computing?) Wnt (talk) 10:11, 14 October 2017 (UTC)

• 1,2) I'm no physics whiz but seems to me that Schroedinger's cat is more of a philosophical concept (Copenhagen interpretation) than a physical one, and that it's out of fashion these days. Certainly the classic experiment with a cat and a particle detector is buildable, though keeping it entangled with the whole world without quantum decoherence is beyond imagining. OTOH there is a developing theory[2] that entanglement explains spacetime, gravity, etc.

  3) I (maybe wrongly) think of QC like this: flip 1000 coins so each lands in a classical state <Heads|Tails> with 50-50 probability. The # of heads for the whole ensemble will follow a bell-shaped probability distribution (binomial distribution to be precise). Now quantum "bits" (qubits), as opposed to coins, rather than real-valued probabilities like 0.5, will have complex-valued probability amplitudes, so combining them will have wave-like behaviour showing constructive and destructive interference. A quantum computation is basically an experiment concocted to transform a state representing a problem, into a state where the potential solutions interfere in a way that the incorrect ones cancel each other out, leaving a measurable peak at the correct solution. Shor's algorithm for integer factoring is the most famous example.

  Scott Aaronson's book "Quantum Computing Since Democritus" should be a good semi-popular-level introduction (disclaimer: I haven't seen the actual book, just online excerpts, but I liked them). You might also like his blog, "Shtetl-Optimized", which discusses these subjects a lot. 173.228.123.121 (talk) 20:44, 14 October 2017 (UTC)

For what it's worth, I think that our current understanding of the collapse of superposition necessitating an observer is a load of codswhollop, at the very least fundamentally flawed. Think about it, if one of the central tenets of quantum mechanics - particle-wave duality, is correct, then every particle is acting as an "observer" of every other particle, and no part of the universe can be completely isolated from any other part (perhaps singularities are an exception). This means that superposition should be impossible in the first place, which is contrary to empirical evidence. The only reasonable conclusion that I can draw, is that collapse is not an absolute result, but is instead determined by statistical factors. Perhaps, the probability of collapse is determined by the butterfly effect - the greater the potential influence of a particular state on the environment, the more likely the collapse. However, I can't figure out how potential influence would be determined, unless the system somehow exchanges information with the future of every possible outcome, much like how a driver watches the road at night, only able to see the road directly in front, lit up by the headlights, deciding where to turn to keep on the road. Plasmic Physics (talk) 23:41, 14 October 2017 (UTC)

Ah, now you're getting to the advanced aspect of the model. The putative outcome is that we have a mathematician who comes up with a result thanks to many years of work ... done in a much shorter period of time. Did he experience this time? As who? The many-worlds interpretation is an obvious way to go, yet the premise is that somehow we get to pick the "right" world when the signal is sent, which seems absurd, if it weren't already being done by quantum computers.

The most accessible approach to conscious quantum parallel computing presently would seem to be precognition, a phenomenon that is at best difficult to control or study systematically. I would throw out an anecdote that "technical" precognition, like selecting which of a thousand files will be found to contain a keyword, or doing a web search using unrelated terms, seems to cause significant pain related to blood flow at the past end somewhere vaguely near Broca's area (this not being correlated to the level of detail or the time differential) ... but how to prove such a thing, and what collateral damage is done by the witch in the meanwhile? I suspect the essence of free will and qualia involve the choice of which solution to a causal loop "really" applies; it is thus an external interface for the universe. So ... if the mathematician is conscious, does this selection of a reality mean that he breaks the quantum parallel computing scheme? Well I just don't know. That's the fun stuff past the edge of the world we know. Wnt (talk) 03:45, 15 October 2017 (UTC)

Wnt, re the mathematician, I believe you're thinking of quantum postselection. Aaronson's informal description is: 1) write down your question (anything in NP. 2) Generate some random bits to guess a possible answer. 3) Check whether the answer is right. 4) If the answer is wrong, kill yourself! There will be a branch of the many-worlds interpretation in which you survived, which means in that universe you guessed the answer on the first try, lucky you! He joked about opening a crisis hotline for depressed complexity theorists, where he would explain to them that if suicide was really the answer to life's problems, that would give a way to solve NP-hard problems in polynomial time, which is widely thought to be impossible, so they shouldn't give up.

Plasmic Physics, the conscious observer theory is the Copenhagen interpretation which I think is now mostly thought of as quaint. See: interpretations of quantum mechanics. 173.228.123.121 (talk) 04:48, 15 October 2017 (UTC)

Precognition. Yes, however, in this instance it does not require sentience/conciousness. A superpositioned particle simply follows the timeline which results in the largest change in entropy of the system, as long as the change in entropy associated with that specific timeline is different enough. If the associated change in entropy is not sufficiently unique, then the particle has a higher chance of remaining in a superposition. Howver, the particle doesn't "know" which timeline gives the largest change in entropy, without actually having followed that timeline, which means that it has, is and will have travelled them all. Please forgive the confusing wording, I don't think that the correct paralance may have been invented yet. Plasmic Physics (talk) 05:45, 15 October 2017 (UTC)

Regarding 3), it won't quite work that way (the probability of a phone call won't increase by merely isolating the mathematician). But you should have a look at Grover's algorithm. The random number generator would initialize the system to the state $|s\rangle$ and the work of the mathematician corresponds to the operator $U_{\omega }$. Then a quantum circuit implementing $U_{s}$ should be applied to the mathematician's output (all while keeping the system, including the mathematician, isolated). You'd need to repeat that $O({\sqrt {N}})$ times as opposed to $O(N)$ times for the classical case.

Icek~enwiki (talk) 08:52, 15 October 2017 (UTC)

@Icek~enwiki: This is the real deal, but I have to admit I don't necessarily understand this operation or how to apply it to the mathematician. I would suppose N is all the ways that he can be given random cues to start the work. But I am not clear what $U_{s}$ is when you apply it to his output. Is it conceivable to apply it to an entire written proof he might generate, or is this just some kind of flag to indicate which initial conditions made him say he had the answer? And if you process the output this way, then reestablish contact with the mathematician, is the mathematician you contact the one who came up with the proof? Wnt (talk) 03:03, 16 October 2017 (UTC)

N is the number of possible random inputs as you say.

Let's write the initial state similar to the article:

$|s_{M}\rangle ={\frac {1}{\sqrt {N}}}\sum _{x=0}^{N-1}|x\rangle |M\rangle .$

Here $x$ is the random cue and $M$ is the mathematician (together with his library and everything he needs) in his initial state. The initial state is created by sending entangled photons (different polarization states stand for 0 and 1, and the bits make up the number $x$) to optical detectors attached to the isolated capsule.

Now the mathematician reads the random cue from a display inside his isolated capsule. Then he starts working on the problem, using the random cue as a guide. To make it simple, he has a clock that tells him when it's time to stop. If he has solved the problem within the time limit, he pushes a button which causes a mirror that causes a phase shift of 180 degrees to be placed in an optical path; if he hasn't solved the problem, he pushes a different button which causes a mirror that doesn't cause a phase shift to the placed in the same spot. A certain time after the mathematician got the stop signal from the clock (enough time to let the mathematician push the right button), the devices attached to the isolated capsule create photons with the same polarizations that have been measured just before the number $x$ was displayed for the mathematician. One of the photons is bounced off the mirror that is either phase-shifting or not. Then the state looks (slightly simplified; now there are actual photons somewhere representing $x$ while before the information was in the detectors or already on the display of the isolated capsule) like this:

$U_{\omega }|s_{M}\rangle ={\frac {1}{\sqrt {N}}}\sum _{x=0}^{N-1}\phi _{x}|x\rangle |M_{x}\rangle .$

Here $\phi _{x}=1$ if the mathematician hasn't been able to solve the problem with random cue $x$ and $\phi _{x}=-1$ if the mathematician has been able to solve it. $M_{x}$ symbolized the mathematician and his study after the experience of attempting to solve the problem with random cue $x$.

Now, applying the operator $U_{s}$ basically leaves the mathematician's state as it is and works only on the photons. We send the photons through the quantum circuit that implements $U_{s}$ and then we have as the new state:

$U_{s}U\omega |s_{M}\rangle ={\frac {1}{\sqrt {N}}}\sum _{x=0}^{N-1}\phi _{x}(2|s\rangle {\frac {1}{\sqrt {N}}}\sum _{x'=0}^{N-1}\langle x'|x\rangle -|x\rangle )|M_{x}\rangle ={\frac {1}{\sqrt {N}}}\sum _{x=0}^{N-1}\phi _{x}({\frac {2}{N}}\sum _{y=0}^{N-1}|y\rangle -|x\rangle )|M_{x}\rangle$

After this quantum circuit, the photons are sent to the optical detectors again, the mathematician gets a random cue again, performs his work and we get the state

$U_{\omega }U_{s}U_{\omega }|s_{M}\rangle ={\frac {1}{\sqrt {N}}}\sum _{x=0}^{N-1}\phi _{x}({\frac {2}{N}}\sum _{y=0}^{N-1}\phi _{y}|y\rangle |M_{xy}\rangle -\phi _{x}|x\rangle |M_{xx}\rangle )$

Here $M_{xy}$ symbolizes that the mathematician has attempted to do his work with cue $x$ and cue $y$, in that order.

After sending the photons through the quantum circuit again, and let the mathematician do his work again, we have the following state:

$U_{\omega }U_{s}U_{\omega }U_{s}U_{\omega }|s_{M}\rangle ={\frac {1}{\sqrt {N}}}\sum _{x=0}^{N-1}\phi _{x}({\frac {4}{N^{2}}}\sum _{y=0}^{N-1}\phi _{y}\sum _{z=0}^{N-1}\phi _{z}|z\rangle |M_{xyz}\rangle -{\frac {2}{N}}\sum _{y=0}^{N-1}\phi _{y}^{2}|y\rangle |M_{xyy}\rangle -{\frac {2}{N}}\sum _{y=0}^{N-1}\phi _{x}\phi _{y}|y\rangle |M_{xxy}\rangle +\phi _{x}^{2}|x\rangle |M_{xxx}\rangle )$

Going on with this iteration, we reach a point when the state of the photons is very close to $|a\rangle$, assuming there is only a single $a$ for which $\phi _{a}=-1$. The state state of the mathematician however is a superposition of having tried to solve the problem with various random cues.

So in the end, we get the answer $a$ for the right cue, and in the mathematician's history there will be in general various cues, but they include $a$. In fact, while doing his work, if the mathematician receives a cue that he received before and for which he solved the problem, he can use his time in other ways and just press the right button at the end.

Icek~enwiki (talk) 20:37, 16 October 2017 (UTC)

@Icek~enwiki: This is a great explanation, very clear. One thing that kind of amazes me about it though is that the mathematician sends out a known, non-quantum set of bits for a, which leaves his enclosure, then they get mirrored and come right back to him as qubits. And the qubits after a few iterations are (usually) going to provide the right conventional bits by incredible luck - even from the mathematician's point of view, I think! Now... is there anything about this process that requires that the mathematician is the one in the small enclosure and the $U_{s}$ part is in the rest of the world? Could you have a tiny perfectly isolated quantum device that implements U_s and an ordinary mathematician in an ordinary university sends his findings into it and gets oracular answers back based on the superposition of all possible worlds out here in the "normal" universe? Wnt (talk) 20:12, 17 October 2017 (UTC)

The point is that the photon detectors, the display, the mathematician and his library, the mirror and the photon emitter are in a superposition of states (maybe I wasn't clear about the last point when I wrote "the devices attached to the isolated capsule" - they should rather be inside, isolated as well). From inside it looks as if the photon emitter creates non-entangled photons with definite polarizations, and to the mathematician it must look like he's pretty lucky indeed (though he still needs to do $O({\sqrt {N}})$ attempts on average). But without the isolation, the wave function collapses and the probability of getting the right cue is just 1/N.

If there is spontaneous collapse of wave functions like in Ghirardi–Rimini–Weber theory, the system wouldn't work (unless the parts in superposition are small enough - if the spontaneous collapse is much rarer even than what Ghirardi, Rimini and Weber probably had in mind, then it still works).

Icek~enwiki (talk) 08:15, 18 October 2017 (UTC)

@Icek~enwiki: I was thinking the bit that does the U_s algorithm has to be outside the superposition. If everything is in the superposition then we can say the whole universe is an isolated system and may be superposed right now per the many-worlds idea. But the thing is, if the U_s calculator is outside the superposition, I'm thinking, what if the "outside" is a really really small area, a tiny electronic device, and the rest of the world is the superposition? Essentially we could use this design you proposed to communicate with parallel universes who are doing different work with different random seeds as they diverge. Indeed, I'm not sure this needs to be mathematical: if we could set the bits of a to be a message and phase-shift it depending on whether it is 'really important' (I am still a little iffy on the phase shift when I think of it) could we simply exchange random greetings through the device as our histories diverge? Wnt (talk) 11:53, 18 October 2017 (UTC)

You are right to be iffy about the phase shift; I overlooked something: One cannot measure the phase of a single photon, and so the photon detector in the isolated capsule cannot. There would be a random phase factor between the detected photons and the photons that are emitted after the mathematician pushes the button.

To prevent this, we need a quantum nondemolition measurement (that measures the polarization of the photons without destroying them, perhaps like this), and we need to keep the photons around while the mathematician does his work. Then, one of the photons that are still around is reflected off the phase-shifting or non-phase-shifting mirror, as before.

After that, it doesn't really matter whether the quantum circuit that implements $U_{s}$ is inside or outside the isolated capsule (because it's a quantum circuit, what happens to the photons inside it is well isolated from the rest of the universe).

But if we manage to do all that, what you suggest should work if there is no spontaneous collapse. We would need no isolation of the mathematician at all, the whole universe would be in a superposition.

The technical challenge here is to keep the photons coherent for all the time. If you can make them reflect off mirrors all the time, make sure that the distance between the mirror doesn't change depending on what the mathematician does. A tiny amount of change of distance will cause a relevant phase shift when the time is long enough, so it's better to have something like spins of particles in a magnetic field as qubits, at least for the long time that the mathematician needs to do his work.

It is weird that this should actually work, even though there is no way to communicate with other worlds if the many-worlds interpretation is correct.

By the way, I don't see how the many-worlds interpretation actually explains anything - I don't see that one could deduce the Born rule (the probability of a given state is equal to the square of the absolute value of its amplitude) from it. But if such an experiment succeeds in getting the right answers as predicted, then it would pretty much rule out spontaneous collapse.

Icek~enwiki (talk) 18:42, 18 October 2017 (UTC)

As a matter of sheer stupid speculation, I find myself suspecting the old mystical-like phrasing of the Copenhagen interpretation might have been literally correct. I am suspicious that consciousness is a paranormal phenomenon and like all such phenomena is a function of precognition; in particular, that free will and qualia are time inverses of one another and result from causality paradoxes where a boundary condition is arbitrarily imposed on the universe. So I'm suspicious that it is quite literally the conscious observer that collapses the wavefunction, and thus the mathematician would not work as hypothesized... and that it would be possible to test a cat or a brain-dead woman carrying a fetus in it to directly measure whether these things are conscious. Yeah, that's nuts, but I think we parted ways with sanity some time ago here. ;) Wnt (talk) 22:49, 18 October 2017 (UTC)

colds and endogenous opiates

One of the painful lessons I learned about colds when I was young is that they seem -- for me -- to involve production of a fairly large amount of some kind of endogenous opiate during the "sniffle" phase. I concluded this because back then I suffered some very sore throats after colds on account of burning myself with hot soup without realizing it. More recently, though exerting extra care with hot food, I managed to do something to a back muscle that left it sore for a couple of days when that phase of a cold ended. And, of course, there is the observation that for a few days it is possible to sniffle huge amounts of crap and feel/hear wheezing in the lungs with no instinctive reaction to cough, which suggests a powerful cough suppressant activity.

But for something that seems so important to the course of the cold, I didn't quickly find much discussion of it (mostly gene therapy papers with endogenous opiates in adenoviruses!), though I understand endorphins are part of a generalized stress response, so I just wanted to check if there is a medical term for this sort of suppressive effect I'm not thinking of. I mean, do other people even have this response? Wnt (talk) 22:24, 14 October 2017 (UTC)

When you feel pain in one place, it does seem to make less severe pains elsewhere even less noticeable, but I'm not sure of the mechanism. It might be entirely within the brain, which has a limited ability to pay attention to different pains. I suppose that makes sense, as there's little survival advantage to being worried about your aching back whilst being mauled by wild animals. Best to only concern yourself with the most immediate threat, which presumably is causing the most severe pain. (Note that there's no reason to assume that this phenomenon is exclusive to humans.) StuRat (talk) 23:47, 14 October 2017 (UTC)

Stress-induced analgesia sounds like what you're describing, a form of hypoalgesia. StuRat is suggesting counterstimulation (a page that could do with some work!). Klbrain (talk) 23:58, 14 October 2017 (UTC)

I think I've experienced this - I remember being momentarily freaked out by a prank back at an undergrad and not noticing I'd misplaced a bit of skin on one malleolus the size of a dime until I happened to spot the blood. But the sort of hypoalgesia during a "fight or flight" seems hard to relate mentally to the first days of a common cold type infection. Hypoalgesia seemed like a great keyword ... but didn't bring anything up with rhinovirus or coronavirus or "common cold". I should point out though that none of the generic mechanisms seem to match what happens with a cold as I experience it -- because the cough suppression and hypoalgesia stop early on during the cold, even though there are still plenty of distracting, unpleasant stimuli, and not obviously less stress. Wnt (talk) 03:01, 15 October 2017 (UTC)

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