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…