Back of the Envelope

Old Posts: In my previous posts on quantum computation, I explained what a qubit was and how you could use them for Quantum key distribution.

Entanglement is one of the keys to the power of quantum computation, and it is a very difficult concept to wrap your mind around. But I'll do my best to explain it in layman's terms.

Imagine you have a qubit that you put in the state a|0>+b|1>. Recall that a qubit can be in both the |0> and |1> state at the same time, but if measured, it will collapse to one state or the other, with a probability of |a|² that it will be measured in the 0 state and |b|² that it will be measured in the 1 state. Recall that both a and b are complex numbers, such that |a|²+|b|²=1. Now, let's say that we have another qubit, which is also in the a|0>+b|1> state, and we place them in a 2-bit register which can hold the values 00, 01, 10, or 11, where the first, higher-order bit is the value of the first qubit, and the second, lower-order bit is the value of the second qubit. Because both the bits in the register are really qubits, the register can hold all four values simultaneously, but when we measure it, we only get one value out. Which value? Well, it's a simple probability problem, where the probability of outcome A and outcome B equals the probability of A times the probability of B. The probability of the first qubit being measured as 0 is |a|², and the probability of the second qubit being measured as 0 is |a|², so the probability of measuring 00 is |a|²|a|². Then the probability of 01 is |a|²|b|², of 10 is |b|²|a|², and of 11 is |b|²|b|². This is the case as long as the two qubits are independent--they're not entangled.

Now let's do this a little differently. We'll start with the first qubit in a|0>+b|1>, and the second qubit in |0>. Then we'll apply an operation called CNOT. In classical language, CNOT, the controlled-not gate, inverts a target bit if and only if the control bit is in the 1 state. In quantum language, it's a bit more complicated, since the control qubit can be partially in the 1 state. So, it partially inverts the target qubit, right? Well, sort of. What it does is merge the two independent quantum systems into a single quantum system--what's called entangling the qubits. So let's consider our register again, and use the higher-order bit, which is in the a|0>+b|1> state, as the control qubit, and the lower-order bit, which is in the |0> state, as the target qubit, so the register begins in the state a|00>+b|10>. When we apply the CNOT, the state of the register becomes a|00>+b|11>. Measured independently, each qubit is in the state a|0>+b|1>. However, once you've measured one qubit, you've determined the state of the other, because the register is now one quantum system rather than two, so measuring a single qubit collapses the entire system. So your probability of measuring 00 is |a|², your probability of measuring 11 is |b|², and your probability of measuring 01 or 10 is 0. This is entanglement, and it should be obvious why it is useful. Without entanglement, you can create a superposition of one qubit, so it can be in both 0 and 1, but when it comes to two qubits, you're out of luck. You can place both of them in superpositions, but there's no way to have a superposition of 00 and 11 without also having a superposition of 01 and 10.

Entanglement gives rise to all sorts of interesting quantum behaviors, including what's called "spooky action-at-a-distance," which completely freaked Einstein out. Let's say that you have two entangled qubits, and you separate them by a distance of miles--heck, let's separate them by lightyears. Now you measure one, and at the same instance you make your measurement, you collapse the state of the other one as well, because they are still one quantum system. "Hey, wait a minute!" you say. "Einstein's theory of relativity won't let any effect take place faster than the speed of light." Funny, that's just what Einstein said. He came up with this theory, called EPR (Einstein-Podolsky-Rosen), precisely to show that quantum mechanics conflicted with special relativity and thus had to be wrong. He was the one who was wrong in this case, since it turns out that EPR works and can be tested in the laboratory. Fortunately, it turns out that EPR doesn't conflict with special relativity. Here's how you can do the experiment: In this case, our qubits are photons, whose states are represented by different linear polarizations. When you measure the state of one, you determine the state of the other as well, which collapses to a state orthogonal to the original. (Why orthogonal? It turns out that certain atoms, when they decay from a high energy state, naturally emit two photons in opposite directions with orthogonal polarizations.) Now, if you measure the two qubits at a reasonable distance apart, you can have electronics good enough to measure them more or less simultaneously (such that the time between the measurement of each qubit is less than the time it takes light to travel between the two experimental apparati), and then compare your answers, and determine that they are indeed orthogonal every time. Now at this point, you may be wondering whether you can be sure that you are indeed collapsing qubit states rather than just measuring some classical polarizations which just happen to be orthogonal every time. This is the "hidden variable" interpretation. The answer is you can be sure that the "hidden variable" interpretation doesn't work if your measurement apparati are set to measure not quite orthogonal states, so that your collapsed state is a superposition of two other states, and then you compare your statistical measurements with rules known as Bell's inequalities, but frankly, it's rather complicated and I'd have to look it up to make sure I've explained it right, and I'd rather not. So I'll just say that we do know we're collapsing superpositions.

So if this effect travels faster than light, how can it not conflict with special relativity? Well, when it comes right down to it, "effect" is probably the wrong word. When you measure one photon, there's really no way to tell whether the other photon has collapsed already, or whether you're just now causing it to collapse. All you know is that you measure it in a certain state which you knew there was some probability you'd measure it in anyway. Go back to our original CNOT operation and our a|00>+b|11> register. When you measure one, you get 0 with probability |a|² and 1 with probability |b|², the same as if they weren't entangled. You don't get any new information out due to their entanglement until you can compare them, and you can only compare them at the speed of light. And thus Einstein's special relativity remains intact, because there's no cause and effect taking place faster than the speed of light.

If this justification leaves you queasy, you're not the only one. I've been thinking about this for years and wondering whether there's some clever way to use this to communicate faster than the speed of light. It shouldn't be possible, not if special relativity remains intact, and lots of physicists smarter and more experienced than I have told me that again and again, but I can't help feeling that there's a violation going on someplace. I'll leave it as an exercise to the reader to prove either Einstein or quantum physics wrong.

Serge at Imago Dei has some thoughts on in vitro fertilization in a series of posts: Part 1, Part 2, and Part 3. I first thought seriously about in vitro fertilization when I was leading the Responsible Technology discussion group at MIT. At the time, I thought that while there is nothing inherently wrong with in vitro fertilization, there is something truly abhorrent in its wastefulness. Many more embryos are produced than are ever used, and this practice should be an outrage to those who believe life begins at conception. The sad truth is that the pro-life movement, while opposing the deliberate destruction of the embryos, makes little effort to address the practices that lead to it. And I cannot say that I myself have done any better, since I saw the practice as terrible, but at the time it was easier just not to think about it too much: I had more than enough things to become outraged about already. I'm glad to see that Serge has put some serious thought into the matter, going so far as to offer practical advice to thoughtful Christian couples whose fertility problems lead them to consider in vitro fertilization:

My advice for those who are having fertility problems would be the following:
1. Talk about how far you are willing to go for infertility treatments PRIOR to attempting to conceive a child. Clear thinking and a plan is important.
2. Clearly indicate your beliefs to the IVF clinic and ensure that they understand that you do not wish to have embryos killed in this process. If they do not want to play ball, go somewhere else.
3. Prayerfully determine whether you want to go through all of this, or to adopt a child that needs a loving Christian home.

In addition, you may have noticed that I've added Imago Dei to the blogroll. Serge's thoughtful essays on bioethics make him a must-read. Plus I noticed that I was on his blogroll. (While I do not have a reciprocity policy, I do tend to check out blogs that link to me, and I'll add them if I like what I see.)