So you want to learn quantum theory in ten minutes? Well I certainly can’t give you the full theory in all its wonder and all its gory detail in that time, but I can give you a light version of the quantum theory in about that time. And won’t that impress your friends!
To learn quantum theory you first need to learn classical theory. (Walk slowly little grasshopper.) What classical theory should we talk about: mechanics or general relativity or maybe electromagnetism? None of those! Those are crazy overwhelming and intimidating classical theories. Instead we can take a classical theory which is more appropriate for our computer literate age: the theory of classical information. Or at least a bastardized version of this theory (mmm, bastardized theories.)
So suppose you have a box (all good explanations begin with boxes!) When you open this box and look inside you see one of two things: either a turkey or a duck. Because we live in a digital age it is natural to call the turkey a 0 and the duck a 1 (we could do it the other way around, of course, but we will just pick a convention here, so turkey=0 and duck=1) Okay, you say, that is a kind of boring box. Indeed! (Although the turkey or duck may disagree with your assessment. They are perfectly happy being their own little selves, thankyouverymuch.)
But now suppose that we run an experiment where I get to play with the box and then you get the box back from me and open it up. Now I have some secret procedure for preparing the turkey or the duck in the box. So if we run this experiment multiple times you will get a turkey sometimes and a duck sometimes. Being a curious person you naturally tabulate these numbers. After many experiments you determine that 70% of the time I prepare a duck and 30% of the time I prepare a turkey inside of the box (are you hungry, yet?) Indeed this is exactly my preparation procedure (as a side note you really do have to trust that I am preparing the box this way: there is no way to force me to adhere to this procedure. But I’m an honest guy, so we can move on.)
Okay, now suppose you have the box in front of you and you know that I’ve done my preparation procedure. One way to describe the box is by those two percentages, i.e. by 30% and 70%. Indeed we might as well describe these two numbers, these probabilities, via a list (0.3,0.7). The first number is the probability of observing a 0=turkey and the second number is the probability of observing a 1=duck.
So far so good. Now suppose that you and I we arrange a deal such that you get to tell me what those percentages are. So you tell me, “Dave I want a box that has a probability of being a duck which is [insert your percentage here] and the box has a probability of being a turkey which is [insert your percentage here].” Of course those two percentages better add up to one hundred percent! And of course, since I’m such an honest guy I always deliver a box according to your prescription.
After a while with this setup you grow a little bored, and so you ask me what else could we do with the box. And I tell you about my friend, Fufufu, who will do interesting things to your box. I won’t tell you what he does, but I can tell you it is something interesting (well relatively interesting, this is a rather technical discussion, you know, so give me a break.) You then order up some boxes from me, with say 50% turkey and 50% duck and then give those boxes to my friend Fufufu who performs his magic on the boxes. You then open up the boxes, and low and behold the boxes are not 50% turkey and 50% duck, but instead are, you estimate, 25% turkey and 75% duck. What did Fufufu do?
Well you’re a curious person so you can run some more experiments. You order up a bunch of 100% turkey boxes and have them shipped off to Fufufu. When they come back you estimate that the boxes are 50% turkey and 50% duck. Curious. Next you order up a bunch of 100% duck boxes and have them shipped off to Fufufu. They come back 0% turkey and 100% duck. Aha! What is Fufufu doing? He is doing something which turns turkeys in the boxes into 50% turkey and 50% duck boxes and he is turning ducks in the boxes, into, well ducks.
So we can describe what Fufufu does by four percentages, the probability that Fufufu turns a duck into a duck, a duck into a turkey, a turkey into a duck, and a turkey into a turkey.
We have just described the classical theory of a probabilistic bit! A bit is a thing, which, when you look at it is either zero or one (turkey or duck.) Our description of this bit is given by two numbers, the probability that when we open the box we will see a zero (turkey) or the probability that we will see a one (duck.) Furthermore we can, instead of just immediately opening up the box, send the box off to someone like Fufufu who will carry out some procedure which changes the probabilities of the box being 0 or 1. In particular we can describe a general procedure by four probabilities, the probability that 0 goes to 1, 0 goes to 0, 1 goes to 0, and 1 goes to 1. In fact we can chain a bunch of these operations together. First send it to Fufufu, then send it to his friend Gugugu. The final description of our system by two probabilities can then be obtained by calculating the probability of the 0 or 1 after Fufufu does his magic followed by calculating what happens next when his friend Gugugu does his magic (he may have different probabilities than Fufufu for the four processes 0 goes to 0, 0 goes to 1, 1 goes to 0, 1 goes to 1.)
Short of the classical theory of a bit. Two states, 0 and 1. Description: two probabilities. Evolution of description: four probabilities describing transitions 0->0,0->1,1->0,1->1.
Now, onward and upward to quantum theory!
In quantum theory we have boxes, just like in classical theory. And when we open those boxes we see either a turkey or a duck. When we open a box we never see a half-turkey half-duck. Such monstrosities simply do not exist. (This does not imply that we can’t do crazy things in real life like make a turducken. I’m just saying that in our box, when we open it, you will either find a turkey or a duck.)
Okay well so far our quantum theory is just like our classical theory. But now there is a twist. Instead of describing our system by two probabilities, we need different numbers to describe our system. In particular we can again have only two numbers, but now we will allow these numbers to be negative (more generally we can allow these numbers to be complex, but this isn’t essential for understanding quantum theory right off the bat, so we’ll not make things more complex. Insert bad pun groan here.) Two negative numbers, you say? That’s just crazy talk! Certainly those numbers can’t represent the probabilities of the box containing a turkey or a duck?
Indeed these numbers do not represent probabilities! What, exactly, would a negative probability be!? However if we square these two numbers, then we do end up with numbers that will represent probabilities! Let’s do an example. We can describe our system by two numbers, for example they could be 3/5 for turkey and -4/5 for duck. If our description of the system is given by these two numbers, then the probability that, when we open the box, we will see a turkey is given by the square of the number we used to describe the turkey in the box: (3/5) times (3/5) which is 9/25, or 36 percent. Similarly the probability that when we open the box and we will see a duck is given by the square of the number we used to describe the duck in the box: (-4/5) times (-4/5) which is 16/25 or 64 percent. Notice that the probabilities still add up to one hundred percent (whew.)
But wait, you say. If we always square a number to get the probability of observing a turkey or a duck in the box, why do you need to do this silly description where you have a possibly negative number? Why couldn’t you just keep the square of those numbers? Well the reason is that we need them when we are going to talk about what a person like Quququ can do to the box. Previously we described what a person could do to change the description of the box by four different numbers, the probabilities of the processes 0->0, 0->1, 1->0, and 1->1. I didn’t mention it at the time, but there are some requirements on those numbers. First of all they had to be positive. Second of all the probability that a turkey turns into a duck plus the probability that a turkey turned into a turkey had better add up to 100 percent. Similarly the probability that a duck turns into a turkey plus the probability that a duck turns into a duck had better add up to 100 percent. In other words, the those pairs of numbers are probabilities.
Back to the discussion of what happens in the quantum world. Just like in the classical world we will have four numbers to describe the four processes that can occur to our box: we will have a number describing the transition from a duck to a turkey, from a duck to a duck, from a turkey to a duck, and a turkey to a turkey. But (and you could probably have predicted this) these numbers aren’t going to be like the positive probabilities in classical theory. In fact they are going to be numbers, but now they are allowed to be negative!
So lets talk about an example. My friend Quququ can perform the following: he can take a turkey and transform it into a system which is described by 3/5 duck and 4/5 turkey, and he can take a system which is a duck and transform it into a system which is described by 4/5 duck and -3/5 turkey.
Now suppose that you start with a box which I prepared for you whose description is 4/5 for the duck and -3/5 for the turkey. You give the box to Quququ. What will be your new description of the box? Well Quququ will take your 4/5 duck and transform it into 4/5 times 4/5 = 16/25 duck and 4/5 times -3/5 =-12/25 turkey. He will take your -3/5 turkey and transform it into -3/5 times 3/5 = -9/25 duck and -3/5 times 4/5 =-12/25 turkey. Thus after Quququ is done with the box, you will have a description which is 16/25-9/25=7/25 duck and -12/25-12/25=-24/25 turkey. So your new description is 7/25 duck and -24/25 turkey. If you were to now open the box you would obtain a duck with probability 7/25 times 7/25=49/625=7.84 percent and a turkey with probability -24/25 times -24/25=576/625=92.16 percent. Happy Thanksgiving!
Notice that in the above calculation, we ended up with two numbers which when we squared them added up to one hundred percent. In other words we started with a description whose sum of the square of the numbers added up to one hundred percent and after Quququ got done performing his magic on the box, we still had a description whose sum of the square of the numbers added up to one hundred percent. That’s a nice property to have. We might even call such sets of four transforms “valid.” In the classical theory we saw that for our four probabilities describing the four processes that could happen to our box, two of them had to really be probabilities in disguise. In the quantum world we have a similar requirement on what those four numbers can be. I won’t go into the details of these numbers as this would lead us too far astray. However I can tell you one simple way that you can check whether the set of four numbers is a transform which will never yield an description which yields probabilites which don’t sum to one hundred percent, given that you always start with descriptions which yields probabilities that sum to one hundred percent. Start with three different descriptions whose two numbers, when squared, sum to one hundred percent and which all yield different probabilities for either turkey or duck (like for instance (3/5,4/5),(4/5,-3/5), and (12/13,-5/12).) Then if you apply the transform to those three different descriptions, if you get descriptions which all sum to one hundred percent after the transform, then you have a valid transform.
So we have just described the quantum theory of a bit, which people call a qubit. A qubit is a thing, which, when you look at it is either zero or one (turkey or duck.) Our description of this qubit is given by two real numbers, which when we square these numbers and add them together we get one. These numbers can be negative! If we open the box, then the probability that we see a 0 (turkey) is the square of the number used in our description for the 0 (turkey), and the probability that we see a 1 (duck) is the square of the number used in our description for the 1 (duck.) Transformations on our box can be performed which are described by four numbers, again these numbers don’t have to be positive. The numbers describe the processes 0 goes to 0, 0 goes to 1, 1 goes to 0 and 1 goes to 1. The numbers can’t just be arbitrary, but satisfy a constraint which guarantees that if a description before hand yielded probabilities which summed to one when we squared the appropriate numbers, then the description after the process will also satisfy this condition that we get numbers whose sum of squares sum to one. We can, just like we did for our classical bit, string a bunch of transforms together and then we just need to do like we did before and calculate the new description at each step of a transform.
Notice that in all of the above discussion, when we did the transform, we didn’t look inside of the box. If we did, however, look inside the box, in either the classical or quantum case, we would see a duck or a turkey and we would immediately update our description to reflect this. This is called the “collapse postulate” and is the source of a great deal of bickering in the quantum world. In the classical world no one bats an eyelash at updating their description. Most physicists take the point of view that you shouldn’t bat your eyelash at the same process in quantum theory. (But not all physicists agree on this.) From a pragmatic point of view, you can use the above procedure without flinching.
So, now you’ve learned the basics of quantum theory. Was that ten minutes? The difference between the classical theory of a probabilistic bit and the theory of a quantum bit really aren’t that severe. Instead of there being probabilities to describe the system there are these other numbers which can be negative and which square to probabilities (these are called amplitudes by physicists.) Processes on the system change the description of the system in the classical case by probabilities of different transitions and in the quantum case by amplitudes which tell you how to update the quantum description. When we look inside of a box, in both cases we only see one of two outcomes and we then need to update our description appropriately.
Fromt his perspective what makes quantum theory so interesting is that you can have things which act like negative square roots of probabilities. There are classical analogies for these types of effects (for example water waves can be thought of as adding when they collide, and if you consider everything below a fixed level negative, then the math needed to describe this makes us add and subtract numbers.) Interestingly, however, these analogies are much harder to come by in the classical world when we insist that we be talking about probabilities and try to mimic these negative square roots of probabilities.
Of course there is much much more to quantum theory than our above quick lesson. Truely things get really interesting when you move from one quantum bit to two or more quantum bits. But I suspect that understanding the above could let you at least carry on a decent conversation with a theoretical physicists at a cocktail party. Well I guess that depends on whether the physicist has had too much to drink and is open to seeing turkeys and ducks…
O brave new quantum world!
Hey Julia, I’ve rewritten the offending paragraph. I knew when I was writing that section that it wasn’t coming out correctly but I forgot to go back and fix it.
Anyway, let me know if this helps or if it doesn’t and you have more questions.
Thanks Matt. It was fun to write, but I ran out of jokes at the end (and towards the beginning too 🙂 )
Wow, that’s even shorter than Polkinghorne’s book!
I need some help, please, following the sentence structure here.
I assume that in “three different descriptions which when we square the numbers and add them you get one,” the “we” and the “you” are actually the same person and “them” refers to the squared value of all numbers. Does “which” mean “such that”? Are “descriptions” and “numbers” synonyms?
So does “three different descriptions which when we square the numbers and add them you get one” mean the same as “three different descriptions such that the squares of the three numbers add up to one”?
And in the same way, does “transform descriptions which when we square the numbers and add them you get one” equate to “transform descriptions such that the squares of the numbers add up to one”? And am I right in assuming that “transform” in this particular sentence is used as a modifier for “descriptions” (as in “some transform-descriptions are . . .”), and not a predicate verb (as in “something transformed those descriptions”).
If I shorten “three different descriptions such that the squares of the three numbers add up to one” to X and shorten “transform descriptions such that the squares of the numbers add up to one” to Y, the sentence then says, “if you check whether X all turn into after the Y, then you are guaranteed you have a valid description for a valid transformation that can be enacted on the box.” I can’t make out the relationship between “whether” and “all turn into.” Would “what” instead of “whether” still express your meaning? Or could you maybe rephrase that?
Sorry for the questions, but I really, really do want to learn something about quantum theory even if it takes a little more than ten minutes. All the other turkey and duck things are amazingly clear, but I just can’t seem to understand this one sentence because I can’t grasp how it is structured.
Great post. Thanks a lot!
Ah! Much clearer! Thanks. Very interesting post.
Sorry buddy but Feynman does it better.
Gee what a stunning insult.
Hey Robin,
10 minute contextuality is my new challenge. Of course the three qubit mermin contextuality argument is probably the simplest.
Sorry buddy but Feynman does it better.
QFT in an hour would be awesome. I don’t think I could do it, but still…
Very nice! But… can I pose a question/suggestion? Okay, thanks!
After explaining the classical bit, you open the discussion of qubits by saying, “Now, for a quantum bit we need to use negative amplitudes, etc, etc.” (well, not in so many words, but…). How about, instead, opening with “Now, if this is a quantum bit, then there’s an experiment that shows this data… and we see that there’s no way to explain that using the 4 transition probabilities… so how can we explain it? Well, if we use these negative amplitudes, etc, etc.”
This strategy makes it clear that there’s a problem, which the theory of amplitudes solves, rather than just positing it axiomatically.
I have a very little bit of experience indicating that your original approach works better with eager physics students (who are happy to be told that nature is weird), and that the latter approach works better with normal people (who tend to ask “What? Why the hell would we do that?”).
Any thoughts?
Okay, one more question. Suppose you had a real skeptic in your class, who said “Okay, I’m convinced that your quantum birdbit is not just a classical birdbit… but that doesn’t mean it’s anything special. I think it’s just got extra, hidden degrees of freedom! It’s really a classical dial, or something.”
Of course, the correct answer to this is “Brilliant point! See me after class.” However, if you did take up the Quixotic challenge to answer this skeptic on the spot, do you think it’s possible to demonstrate contextuality — preferably involving edible fowl — in another 10 minutes?
Just curious!
Dave,
Cool. I’m looking forward to the 3-turkey paradox! Or, heck, maybe I should just read Mermin’s argument… ’cause I’m embarrassed to admit I don’t know it.
Feynman was never my favorite.
Keeping those C-numbers out of it seems like a nice move. Although I think it needs a bit of work to be publishable, have you considered Am.J.Phys?
I look forward to the Elfbit version in a month’s time. Moving on to Eggbits?
Please can we also have QFT in an hour, maybe interacting QFT in a couple of days?
Dave says: QFT in an hour would be awesome. I don’t think I could do it, but still…
Feynman wrote an unusual little book titled QED, the Strange Theory of Light and Matter (if I remember the title right) that makes a pretty good attempt at it …
thanks for the article – fun to read.
about half way down you have this:
“So your new description is 7/25 duck and -24/25 turkey. If you were to now open the box you would obtain a duck with probability 7/25 times 7/25=24/625=7.84 percent and a turkey with probability -24/25 times -24/25=576/625=92.16 percent. Happy Thanksgiving!”
should be 7/25*7/25 = 49/625
Doh! Thank meicheni. I fixed it.
I’m sorry, but I hope that very few people read this, because you have written an explanation that confuses things more than it clarifies them.
You start with a long example of what is very clearly and explicitly a hidden variable situation. Your reader’s state of knowledge of what is in the boxes may be statistical in nature, but, what is in the box is in fact a turkey, or else it is in fact a duck. There’s a hidden variable that you don’t know, but it actually is a chicken, or it’s a duck, whether you know it or not.
Having now spent a page or two getting your readers to visualize hidden-variables clearly and explicitly, there’s pretty much no possible way to move to quantum mechanics, since quantum mechanics is not a hidden variable theory. Unless possibly by saying “Now, to understand quantum mechanics, simply forget everything I just told you.”
More explicitly, in terms of your analogy, the possiblity that you could ever explain quantum mechanics vanishes when you get to this statement: “When we open a box we never see a half-turkey half-duck. Such monstrosities simply do not exist. ” The whole point of quantum mechanics is that such linear superpositions do exist. A state that is, say, (1/SQRT(2)) |spin up>+|spin down>) is every bit as real as a spin up or a spin down state. Once you’ve told the reader that a superposition states “simply do not exist”, you’ve essentially frozen them in the classical world. They can, and do, exist, and understanding that they exist is, more than any other point, the key to understanding quantum mechanics.
There’s much truth in what Geoffrey A. Landis wrote. What I want to know is whether in generalized Bell inequalities, elliptically polarized turducken disproves reality, as well as locality.
The whole point of quantum mechanics is that such linear superpositions do exist.
I completely and totally disagree and believe that teaching students that wave functions are “real” is one of the biggest disservices ever rendered on physics students. Indeed I would claim it stopped my field, quantum computing, from every even being contemplated by at least twenty years.
Superpositions do not exist except on the pieces of paper we write down to describe a quantum system. Teaching students to believe that there is a wave function which is no different from the electromagnetic field instills into them all sorts of bad misconceptions.
And point of fact, there does exist a local hidden variable theory for a single qubit (contextual, of course.)
Of course I’m sure I won’t win you over: you sound like an old school physicist 🙂 But if I can’t win you over then maybe at least I could point you to something interesting to read which might make you yell at me less:
http://netlib.bell-labs.com/who/cafuchs/VaccineQPH.pdf
or perhaps even David Mermin’s new quantum computing book.
Certainly superposition states exist; if nothing else, the K0_long / K0_short particle pair experimentally demonstrates that.
(I apologize here that It’s hard to discuss QM without implicitly falling into an interpretation, because the language varies depending on the interpretation used, although the underlying mathematics is the same. I’ll do my best to use interpretation-free language, but don’t count on it) Basically, when you make a measurement (look inside the box, in your example), the probability of the measurement giving a result R is the projection of the initial state into the eigenstates defined by the measurement operator; but once you have made the measurement, the probability of the system being in that state is unity. Right so far? Now the reason that you never open the box and see “turkey plus duck/root(2)” or “turkey-duck/root(2)” is that these aren’t eigenvalues of your observation protocol. So by definition, of course that’s not what you will ever see. I don’t even have a clue what an instrument that would differentiate a “turkey+duck/root(2)” from a “turkey-duck/root(2)” would look like. Saying that you will never see a “turkey+duck/root(2)” is trivial. But you are guiding the reader into thinking that they will never see “turkey+duck/root(2)” because such a thing can’t exist, and inside the box (the hidden variable) it must therefore be turkey, or duck, but couldn’t be something between. And this is of course true in the case of a box containing a turkey or a duck.
But it’s not true in quantum mechanics.
If you don’t like K0 mesons, the trivial (in fact, nearly classical) case is a polarized photon. Light that is polarized at a 45 degree angle is a linear superposition of X and Y polarization. If you measure the polarization using, say, an X polarizer, you will determine 100% of the time that the photon is either X polarized, or Y polarized, but no photon you ever measure will be (X+Y)/root(2). Does this mean diagonally polarized photons can’t exist? Certainly not. You can’t measure them with X or Y polarizers, you measure them with +45 and -45 degree polarizers.
Most certainly superposition states do exist (and by this I mean in the real, physical sense, not the abstract philosophical sense.)
Nicely done, Dave!
Geoffrey,
You’re right that Dave’s description doesn’t rule out hidden variables, and therefore fails to illustrate noncontextuality. But — is contextuality the most important thing about QM? It’s certainly one of the most deep and puzzling things about QM, and every student should realize at some point that QM is not compatible with hidden variables.
A much more deadly fallacy, to me, is your conflation of “superpositions exist” (with which I agree!) with Dave’s “half-turkey half-duck[s]… do not exist.” Dave is absolutely, 100%, without-doubt correct that we never open boxes and see turduckens. Superposition states definitely exist [1]; half-turkey/half-duck states don’t.
A superposition of A and B is not “both A and B at once” — when we open the box, we see either A or B. Schroedinger’s cat is not “dead and alive at the same time” — there’s no logical connection between the state (|dead>+|alive>)/sqrt(2) and the propositions |dead><dead| and |alive><alive|.
Rob Spekkens has done some interesting work, making a list of phenomena in quantum information theory that can and cannot (resp.) be reproduced in a simple noncontextual hidden variable theory. It turns out that the vast majority of them can; they depend only on (roughly speaking) information-disturbance relations. It’s quite hard to come up with a task that specifically requires contextuality.
I also think that contextuality is hard to explain unless you’ve already spent some time thinking about hidden variables. So, if I only had 10 minutes to explain quantum, I’d probably skip contextuality. On the other hand, if I had another 10 minutes, I’d cover it, using… um, so… how’re those notes coming, Dave? 🙂
[1] I appear to be flat-out disagreeing with Dave here, by saying that superpositions definitely exist. I don’t think I am — the problem is that the word “exist” is ill-defined in this context [2]. Dave is, I think, restricting “existence” to measurable properties like “is it a turkey?”. I’m just saying that superpositions of |0> and |1> exist in every sense that |0> and |1> themselves exist.
[2] If you doubt me, ponder the difference between “This table exists”, and “The real numbers exist”. Already, “exist” is getting ambiguous. It gets worse with quantum mechanics.
The hidden variable issue was deepened by Bell’s Inequality and Aspect’s experiment, and then deepened again, as I hinted, with the elliptical polarization theory that undercuts Bell but threatened “realism” in QM.
“If you doubt me, ponder the difference between “This table exists”, and “The real numbers exist”. Already, “exist” is getting ambiguous. It gets worse with quantum mechanics.”
Robin Blume-Kohout brings in the difficult questions of the metaphysics of Mathematics, the metaphysics of Physics, and Wigner’s paper on the “the unreasonable efficacy of mathematics in explaining the physical world.” Following the Wikipedia entry on that phrase:
“The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960).
Richard Hamming (1980), who was neither a physicist nor a philosopher of mathematics but an applied mathematician and a founder of computer science, reflects on and extends Wigner’s Unreasonable Effectiveness, mulling over four “partial explanations” for it.
A different response, advocated by Physicist Max Tegmark (2007), is that physics is so successfully described by mathematics because the physical world is completely mathematical, isomorphic to a mathematical structure, and that we are simply uncovering this bit by bit. In this interpretation, the various approximations that constitute our current physics theories are successful because simple mathematical structures can provide good approximations of certain aspects of more complex mathematical structures. In other words, our successful theories are not mathematics approximating physics, but mathematics approximating mathematics.
I think that Tegmark is intentionally Over The Top in arguing beyond Platonic Idealism and saying that we ACTUALLY live inside a mathematical object. But, of course, in other parts of the Multiverse I agree with him. In yet others, I am him. But do I exist somewhere as a superposition of me and him? Of me and you?
“exist” and “reality” are words that, in our geek culture, always have implicit quotation marks around them.
Geoffrey,
I agree with almost everything you said in the long paragraph — put concisely, the measurement protocol determines the set of things you could possibly see when you open the box. For turkeys and ducks, the only measurement protocol we know (or, for that matter, can imagine) reveals |turkey> or |duck>. For photon polarization, we can easily design protocols that measure in one basis or in its conjugate.
However, I’m not comfortable with two phrases:
1. “it must therefore be turkey, or duck, but couldn’t be something in between”
2. “But it’s not true in quantum mechanics”
My objection to the first one is the implication that a superposition of |turkey> and |duck> would be “something in between” the two. Consider L/R circular polarization, for instance — these states are qualitatively different from linear polarization, and don’t appear physically “between” (which carries strong connotations of convex combination) |H> and |V>. I believe that Dave’s 10-minute lecture was intended to convey the [valuable & true] fact that when you observe an L or R polarization state in the H/V basis, you see either H or V, not something “in between”… which, of course, is basically what you said above.
My discomfort with the second statement stems from the fact that I think this is quantum mechanics — i.e. we live in a quantum world, which looks classical because there’s a heck of a lot of decoherence going on. So I actually don’t agree that (|turkey>+|duck>)/sqrt(2) can’t be prepared; I just think we don’t have a protocol to measure it in an arbitrary basis. If we could, we wouldn’t see the monster Dave described… we’d “see” something at least as qualitatively different from turkeys & ducks as circular polarization is different from H/V polarization.
Yes, I agree. Well said.
I used the phrase ‘something in between’ because English just doesn’t have any simple, accurate words for superposition states of macroscopic objects. But you are quite right.
As Robin pointed out, one point which my little story didn’t try to convey was that of contextuality. For qubits, unless you allow generalized measurements, not only is there a hidden variable theory, but there is a noncontextual hidden variable theory. So saying that |H>+|V> “exists” isn’t so bad. But this fails when you go to higher dimensional quantum systems or allow generalized measurements. The Kochen-Specker theorem says, in quite a real sense, that if you think about superpositions as “existing” independent of the context of the measurement, you will get yourself into trouble. Thus, in a real sense (haha), I think that emphasizing that superpositions “exist” is a dangerous game which, only for qubits, doesn’t get you into dangerous territory.
Now, though, you seem to be implying that some states exist independent of context, but superpositions of them don’t.
Well I certainly failed at communicating then. I was trying to say that for qubits, since there is a noncontextual hidden variable theory for projective measurements on qubits, then you believe this hidden variable theory then you really can say |H>+|V> exists independent of context. But this doesn’t hold for higher dimensional quantum theory. So I think the |H>+|V> case is a misleading example.
But seriously, every mixed state can be thought of as arising from a pure state on a larger Hilbert space. This special version of Stinespring’s theorem is usually called the GNS construction of quantum states, after Gelfand and Naimark, and Segal.
I. M. Gelfand, M. A. Naimark: On the Imbedding of Normed Rings into the Ring of Operators in Hilbert space; Mat. Sb. 12 (1943) 197
I. E. Segal: Irreducible Representations of Operator Algebras; Bull. Math. Soc. 61 (1947) 69
http://www.quantiki.org/wiki/index.php/The_Church_of_the_larger_Hilbert_space
Well, there was the paper by the University of Monty Python Research Group [“E8, a Simple Theory of Everything, Writers as Bosons, and Episode 6”, J. Cleese, M. Palin, E. Idle, G. Chapman, T. Jones, T. Gilliam, C. Cleveland, J. Blog Phys.], which identified the macroscopic superposition of “yes” and “no” as follows:
Fifth Writer: Sir, I don’t know how to say this but I got to be perfectly frank. I really and truly believe this story of yours is the greatest story in motion-picture history.
Larry: Get out!
Fifth Writer: What?
Larry: If there’s one thing I can’t stand, it’s a yes-man! Get out! (fifth writer leaves very fast, the others go very quiet) I’ll see you never work again. (to sixth writer) What do you think?
Sixth Writer: Well… I…
Larry: Just because I have an idea it doesn’t mean it’s great. It could be lousy.
Sixth Writer: It could?
Larry: Yeah! What d’ya think?
Sixth Writer: It’s lousy.
Larry: There you are, you see, he spoke his mind. He said my idea was lousy. It just so happens my idea isn’t lousy so get out you goddam pinko subversive, get out! (sixth writer exits) You… (looking straight at fourth writer)
Fourth Writer: Well … I think it’s an excellent idea.
Larry: Are you a yes-man?
Fourth Writer: No, no, no, I mean there may be things against it.
Larry: You think it’s lousy, huh?
Fourth Writer: No, no, I mean it takes time.
Larry (really threatening): Are you being indecisive?
Fourth Writer: Yo. Nes. Perhaps. (runs out)
Larry: I hope you three gentlemen aren’t going to be indecisive! (they try to hide under the table) What the hell are you doing under that table?
First Writer: We dropped our pencils.
Larry: Pencil droppers, eh?
Writers: No, no, no, no, no!
Larry Right. Now I want your opinion of my idea … (pointing at first writer) You…
First Writer (quaking): Oh…
First writer looks around and then faints.
Larry: Has he had a heart attack?
Second and Third Writers: Er…
Larry If there’s one thing I can’t stand, it’s people who have heart attacks.
First Writer (recovering immediately): I feel fine now.
Larry: Well, what do you think?
Writers: Oh! Eh! You didn’t ask me you asked him. He didn’t ask me, he asked him. No, him.
Larry: I’ve changed my mind. I’m asking you, the one in the middle.
Second Writer: The one in the middle?
Larry: Yes, the one in the middle. (the phone rings) Hello, yes, yes, yes, yes, yes, yes, Dimitri … (all jockey for position desperately trying to put the others in the middle and finish sitting on one chair) What the hell are you doing?
Second Writer: I’m thinking.
Larry: Get back in those seats immediately. (back to phone) Yes… (second writer is grabbed by the others and held in the middle chair; Larry finishes with the phone) Right you. The one in the middle, what do you think?
Second Writer (panic): Er… er…
Larry Come on!
Second Writer: Splunge.
Larry: Did he say splunge?
First and Third Writers: Yes.
Larry: What does splunge mean?
Second Writer: It means … it’s a great-idea-but-possibly-not-and-I’m-not-being-indecisive!
Larry: Good. Right . .. (to third writer) What do you think?
Third Writer: Er. Splunge?
Larry: OK…
First Writer: Yeah. Splunge for me too.
Larry: So all three of you think splunge, huh?
Writers: Yes!
Larry: Well now we’re getting somewhere.
Er… Dave, you just lost me. Sorry. I really meant to stop commenting, but now I’m baffled.
You wrote “The Kochen-Specker theorem says, in quite a real sense, that if you think about superpositions as `existing’ independent of the context of the measurement, you will get yourself into trouble.”
I assumed, up until then, that you were speaking from the Fuchs-Caves-Schack-Spekkens-Leifer-etc “Quantum States are States of Knowledge” soapbox[es]… which I respect even though I don’t agree with it. Now, though, you seem to be implying that some states exist independent of context, but superpositions of them don’t.
I still don’t know which definition of “exist” you’re adopting… but even allowing for that confusion, I can’t figure out what you’re saying. It sounds like you’re saying that contextuality selects a preferred basis — but now I have to assign 99% probability to “I’m misunderstanding you.”
Help?
BTW, I noticed something odd about the commenting system. Since the comments interpret HTML, you can’t use less-than and greater-than signs in the text without triggering HTML badness. The solution is to use “>” and “<”.
However, a funny thing happens on the way to the forum. When I hit “Preview”, I get a good preview, but in my text-entry box, my HTML magic codes get replaced by plain old > and < characters. Which trigger HTML badness errors the next time around!
Why is Preview prematurely substituting characters in the entry box?
Why is Preview prematurely substituting characters in the entry box?
Hm, that’s a strange one. Let me experiment a bit…
I’m not sure if I can answer this without simply repeating what I already said. In the physical, real world in which we live, superposition states exist– the neutral K meson, if nothing else, clearly demonstrates this– and quantum mechanics cannot be explained by a simple local hidden-variable theory. There’s little more to say than this.
I think that there may be a difference in viewpoint between garden-variety physicists, who think of quantum mechanics as a way of understanding and calculating the energy and time-evolution of real particles in the real world, and information theorists, who use quantum mechanics to calculate vastly simpler systems of qubits.
In some problems– say, calculating the scattering of an electron of given incident k-vector from a spherically-symmetrical potential– I suppose you can say “oh, although you may use superposition states internally in the calculation to decompose plane waves in the basis set consisting of spherical harmonics, the spherical harmonic states aren’t “real”, they’re just a calculational tool.” But ultimately you will end up disregarding most of the real world. How do you calculate a sp3 hybrid orbital– the bonding state of diamond– if you don’t accept the reality of the superposition of s and p orbitals? For that matter, how can you show that the quantum harmonic oscillator reproduces the classical harmonic oscillator at high quantum numbers without superimposing the harmonic eigenfunctions (which are time independent!) into coherent quasiclassical states. Are the quasiclassical states– the ones which exhibit the classical omega-t oscillation that any student with a spring and a weight observes– the ones that aren’t “real”?
Furthermore, a state that looks like a pure state in one reference frame looks like a superposition in another. Ignoring the reality of superposition basically requires you to have a reference frame fixed by God. The trivial example of this is rotational reference frames; a spin that is pointed “up” in a reference frame defined by Galactic North is a superposition in a reference frame defined by the Earth’s rotation. But this applies to more abstract basis vectors as well.
Thanks! Two comments, though:
— Would be good to re-order the description of Quququ so that it matches the picture more closely.
— Where could I learn more? 🙂 [I mean, what would *you* recommend?]
mrG: What I’m really saying is that God plays Duck, Duck, Turkey.
SR: Scott Aaronson’s lecture notes are a bit much, but very fun to read: http://www.scottaaronson.com/democritus/
Years ago I encountered a book which explained quantum theory a lot like the above description. For the life of me, however, I haven’t been able to track down the book (I read it sometime in the late 80s.)
this is taking more than 10 minutes to digest here, but just tell me this much: Are you saying that God plays with farmyard animals?
better than dice, I suppose, but I fear still may earn you a fatwa. And besides, you didn’t mention mystics, not even once 😉
Thanks, Dave, for attempting to “explain” what is basically unexplainable. Since most of us non-physicists can’t do the math, any attempt to “learn” QM or cosmology or what-have-you is futile, since the core language of these disciplines is math. (I don’t read music, so jazz theory would have about the same impact on me. I love jazz nonetheless!) All we can get is cocktail party chit-chat. Then again, appreciation for the the kinds of problems you QPs tackle, IS valuable. And thus we get the social necessity of “QM for Dummies”: an enlarged appreciation of the nature of science and its relationship to other fields like philosophy. The fact that QPs have fundamental disagreements about the nature of Nature and the “reality” of the various constructs you work with is, frankly, cool. You’d be damn boring if you all agreed! As far as Nature goes, I subscribe to the idea that humans improve their approximations as our science advances. I’m not mystical enough to believe we are uncovering any sort of Truth. I’m happy with this generation’s ‘truth’ being a little b it better than the previous generation’s version.
Mark O’Connor: Feynman laughed when I made up and told him this Jazz-Quantum Mechanics joke.
Q: Why did Feynman play jazz bongo drums while submitting “A Relativistic Cut-Off for Classical Electrodynamics” to Phys Rev?
A: Because he wanted to Be Bopp.
Unfortunately, to get the joke, you need to know:
Phys. Rev. 74, 939 – 946 (1948)
[Issue 8, October 1948]
A Relativistic Cut-Off for Classical Electrodynamics
R. P. Feynman
Cornell University, Ithaca, New York
Received 8 June 1948
Ordinarily it is assumed that interaction between charges occurs along light cones, that is, only where the four-dimensional interval s^2=t^2-r^2 is exactly zero. We discuss the modifications produced if, as in the theory of F. Bopp, substantial interaction is assumed to occur over a narrow range of s^2 around zero. This has no practical effect on the interaction of charges which are distant from one another by several electron radii. The action of a charge on itself is finite and behaves as electromagnetic mass for accelerations which are not excessive. There also results a classical representation of the phenomena of pair production in sufficiently strong fields.
(c)1948 The American Physical Society