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2.11

## Keio University

Skip to 0 minutes and 4 seconds Measurement causes the collapse of the wave function. That’s what’s happening when we look at our qubit and find it to be in one particular state, whether that’s up or down or right or left. Of course, it could just be that we don’t know what the state of the two qubits is. We can take a couple of dice, throw them and cover them with a cup. We don’t yet know what number is up on each die, but there is definitely some number up. We wouldn’t say that the dice were in some sort of superposition while they were under the cup, we would just say that we haven’t looked yet.

Skip to 0 minutes and 37 seconds So why isn’t measuring our two qubits just like lifting the cup and looking? Recall that when we talked about measuring a single qubit, we described three ways to measure it. They correspond to the X, Y and Z axes on the Bloch sphere we saw back in the article on qubits. I’ve cut those axes into this ball as well to help us visualize it. To measure a qubit, you pick an axis through the middle of the sphere, and as you do the measurement you’ll find the qubit either on the near end or far end of that axis. For the Z axis, that’s the north pole and the south pole. For the X axis, it’s the left and right points.

Skip to 1 minute and 18 seconds Let’s call the place where my finger is the “plus one” result, and the place where my thumb is the “minus one” result. Let’s look at our entangled state. To make the math come out right, we’re going to rotate one of the states by pi, but you don’t need to worry about that for the moment. If both of the qubits are measured using the same axis, we will always find them pointing in opposite directions, one up and one down or one left and the other right. We’ll always have one plus result and one minus result. That much, we could fake with some prior secret agreement between our two qubits.

Skip to 1 minute and 52 seconds If we’re measured on the up-down axis, I’ll go up and you go down. If we’re measured on the left-right axis, I’ll go right and you go left but what if the left qubit is measured on the up-down axis, and the right qubit is measured on the left-right axis? Since our measurements don’t line up, they can’t quite agree. In fact, we’ll find that each one of them is fifty-fifty, and there’s no evidence that they ever agree. It’s starting to feel a little strange, isn’t it? It seems like maybe the two qubits somehow know something about how they are going to be measured.

Skip to 2 minutes and 28 seconds It doesn’t feel like normal classical probability, like, the dice did but we’re not quite to something yet where we can say this is definitely not classical. Let’s try a slightly more complicated experiment. Rather than always measuring the left qubit in one particular way and the right qubit in a different but also fixed way, what happens if we choose randomly how to measure the left and right qubits? Let’s give each measurer two choices for how to measure. For the left qubit, we’ll measure either on the vertical axis or the horizontal axis. For the right qubit, we’ll pick two axes, 45 degrees apart; one of them above the equator like this and the other one below the equator like that.

Skip to 3 minutes and 10 seconds Now, when we measure the two qubits will disagree with some probability and agree with some other probability. That is, instead of always one plus one and one minus one, we might get two plus ones or two minus ones. Whether they are more likely to agree or disagree depends on which axis is picked at both ends, not just one end. It turns out that they seem to know somehow whether to agree or disagree. If we change the angle between the two sets of measurement axes, we can actually change the probability that they will agree when we measure them. Weird. In fact, this correlation between the two qubits happens even when our two qubits are physically separated by an arbitrary distance.

Skip to 3 minutes and 58 seconds This is what bugged Einstein about quantum mechanics and he famously referred to it as “spooky action at a distance”, or spukhafte Fernwirkung in the original German. Think about it. By now, the hair on the back of your neck ought to be standing up. You’ve probably heard since you were an elementary school student that nothing can go faster than the speed of light and yet, somehow, the two qubits each seem to know what’s happening when the other is measured, even if that happens too far apart for a message to get there. The answer lies, at least in part, in the randomness of the results themselves.

Skip to 4 minutes and 37 seconds If we measure along the same axis, say, the up-down axis, we know that whatever we find, our partner will find the opposite but in the end, that’s really not very helpful. If you want to use this entanglement for something, in fact, you’re going to have to send classical messages back and forth. This saves the speed of light. It turns out that some sort of correlation can happen faster than light, but the correlation itself is useless. It’s just a set of random numbers, until we have additional data to back it up and help us interpret it. Saved!

Skip to 5 minutes and 9 seconds Entanglement is one of the weirdest and deepest aspects of quantum mechanics, and fully a hundred years after quantum mechanics was initially formulated, we are still finding out new things about how it behaves. In fact, it’s one of the keys to building a quantum computer.

# Measuring Entanglement

When we talked about measuring a single qubit, we described three ways to measure it. They correspond to the X, Y and Z axes on the Bloch sphere we saw back in the article on qubits. To measure a qubit, you pick an axis through the middle of the sphere, and as you do the measurement you’ll find the qubit either on the near end or far end of that axis. For the Z axis, that’s the north pole and the south pole. For the X axis, it’s the left and right points.

For a single qubit, it’s simple enough, if different from classical systems. What happens when we measure two entangled qubits? We will see the correlation between the two qubits happens even when our two qubits are physically separated by an arbitrary distance – the phenomenon known as “spooky action at a distance”.

## 3-D Printing

If you would like to print your own copy of the 3-D model of the Bloch sphere shown in this video, you can use either of these two files:

• STL file Most 3-D printing software will take this file for printing. You will probably want to adjust the scale to suit your printer. For your printer settings, we recommend printing this with a raft and support. You will need a small file to remove the supports once printed.
• OpenSCAD file If you would like to modify the shape and are willing to do a little programming, this was created in a language called OpenSCAD.

# 量子もつれを測定する

## 3Dプリント

このビデオで使ったブロッホ球の3Dモデルを印刷したい方は、次の2つのファイルのいずれかをご利用ください。

• STLファイル ほとんどの3Dプリントのソフトウェアで利用できるファイルです。 お使いのプリンタに合わせてスケールを調整してください。お使いのプリンタの設定では 、ラフト(床)とサポート(支え)を一緒に印刷することをお勧めします。 印刷後は、サポートを削除するために小さなファイルが必要になります。