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.https://trolonphilciapred.cf/1310-top-site-rencontre.php
A Beginners Guide to Quantum Mechanics
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. 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. 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. 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 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 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. You give the box to Quququ. What will be your new description of the box? 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.
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That's a nice property to have. We might even call such sets of four transforms "valid.
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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.
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.
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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 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. 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.
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.
Quantum Physics - A Beginner- s Guide
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. 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. Thanks Matt. It was fun to write, but I ran out of jokes at the end and towards the beginning too :.
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. How about, instead, opening with "Now, if this is a quantum bit, then there's an experiment that shows this data 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?
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Why the hell would we do that? 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 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. Of course the three qubit mermin contextuality argument is probably the simplest. I'm looking forward to the 3-turkey paradox!
Or, heck, maybe I should just read Mermin's argument 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. 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 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. 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. 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:.
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! Rob Spekkens has done some interesting work, making a list of phenomena in quantum information theory that can and cannot resp.
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 I don't think I am -- the problem is that the word "exist" is ill-defined in this context . Dave is, I think, restricting "existence" to measurable properties like "is it a turkey? 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.
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. I February Richard Hamming , 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 , 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. 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? 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? So by definition, of course that's not what you will ever see.
And this is of course true in the case of a box containing a turkey or a duck. 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. Does this mean diagonally polarized photons can't exist? Certainly not. Most certainly superposition states do exist and by this I mean in the real, physical sense, not the abstract philosophical sense.
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 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. My discomfort with the second statement stems from the fact that I think this is quantum mechanics -- i. If we could, we wouldn't see the monster Dave described 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. 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. Cleese, M. Palin, E. Idle, G. Chapman, T. Jones, T. Gilliam, C. Cleveland, J. A great deal of research has been and is currently being conducted in order to earn quantum mechanics operational in our day-to-day lives.
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