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## Keio University

Skip to 0 minutes and 3 seconds So far, most of our discussion has involved waves and superposition and interference in ways that are almost entirely classical, except for measurement, where we talked about the state collapsing, and the idea that n qubits results in to entries in our state vector. Two qubits requires four of our dials, three of our qubits requires eight of our dials, and so on. Each of these dials represents our quantum amplitude for the corresponding state, so the absolute value of the square of the amplitude of each is the probability that we will measure that value if we measure the whole register. So far, so good. Now, we’re going to step out into the realm of really hair-raising quantum phenomena.

Skip to 0 minutes and 46 seconds We’re going to talk about “quantum entanglement”, the idea that worried Albert Einstein so much that he figured that maybe quantum mechanics as a theory wasn’t yet complete and correct. We’ll get to that in a minute. First, let’s assume we have quantum entanglement and talk about its effect on measuring quantum states. Then, we’ll talk a little about how we can make entanglement and how we can tell that it’s real. After you finish this video, there is more information in the accompanying article, covering the important concept of Bell pairs. There is also information on some of the most recent experimental demonstrations that quantum entanglement is real.

Skip to 1 minute and 23 seconds If you have two or more qubits, their states can be correlated in a way that classical systems can’t replicate. Let’s take an example. Suppose our two qubits have two states, which we’ll call “up” and “down.” Here is up. This can be our zero state, ket 0 and here is down – this can be our one state – ket one. If we have two qubits, for example, maybe one of them is always up and the other is down. Let me bring in my student, Shin Nishio, so we can demonstrate. One state would be this one. We can write that as ↑↓⟩ or 01⟩ and the other state is this one, naturally, we can write that as ↓↑⟩ or 10⟩.

Skip to 2 minutes and 14 seconds In fact, it doesn’t actually matter what write inside the ket. We could write a picture with our faces inside of it. Now let’s take those two states and put them in superposition. Remember, this is two qubits, but we have to normalize the state, just like we did with one qubit. We will take it and put it over a square root of 2 like in this equation. Now, each qubit is in superposition. The first qubit is in a state that is 50% zero and 50% one.

Skip to 2 minutes and 42 seconds The second qubit is also in a state that’s 50:50 but the states of the two quibits are random but not independent. This is critical. This is the essence of quantum entanglement. Okay, I can hear you thinking, how would this show up in the real world? How can we tell if this rather esoteric concept of entanglement is real? Earlier, we saw how to measure multi-qubits states. What happens with this entangled quantum state if we measure it? Of course, the system collapses into one of the states that has a probability that’s greater than zero. With two qubits, there are four possible states,

Skip to 3 minutes and 20 seconds just like with two classical bits: 00, 01, 10, and 11. In the case of our entangled state,

Skip to 3 minutes and 29 seconds only two of those four have probabilities that are non-zero: 01 and 10. When we measure the state, we will find either 01 or 10, but never 00 or 11,

Skip to 3 minutes and 40 seconds even though each qubit individually seems to be 50:50. This is what we mean when we say that they are random but not independent.

# Entanglement

So far, most of our discussion has involved waves and superposition and interference in ways that are almost entirely classical, except for measurement, and the idea that $$n$$ qubits results in $$2^n$$ entries in our state vector. Now we’re going to step out into the realm of really hair-raising quantum phenomena. We’re going to talk about quantum entanglement, the idea that worried Albert Einstein so much that he figured that maybe quantum mechanics as a theory wasn’t yet complete and correct.

## Bell states

In the video, we mentioned Bell states. Bell states are the most basic form of entangled state. You have already seen them, you just didn’t know it. The $$\frac{|01\rangle + |10\rangle}{\sqrt{2}}$$ state in the video is one of the Bell pairs.

Qubits have two characteristics: their value, and their phase. In this Bell pair, the value is always opposite (if one qubit is zero, the other is one), but the phase is always the same. Instead, we could have a Bell pair where the values are always the same and the phase is always the same, a Bell pair where the values are the same the phase is always opposite, or a Bell pair where the value is opposite and the phase is opposite. This gives us four types of Bell pair: $$|\Psi^+\rangle = \frac{|01\rangle+|10\rangle}{\sqrt{2}}$$
$$|\Psi^-\rangle = \frac{|01\rangle+(\pi)|10\rangle}{\sqrt{2}}$$
$$|\Phi^+\rangle = \frac{|00\rangle+|11\rangle}{\sqrt{2}}$$
$$|\Phi^-\rangle = \frac{|00\rangle+(\pi)|11\rangle}{\sqrt{2}}$$

## Experiments

An experimental demonstration of the existence of entanglement is called a Bell test or a Bell’s inequality violation, because it tests a particular equation that John Bell proposed. (We mostly use a different mathematical form of the test these days, but no matter.) The first demonstration was by Freedman and Clauser, in 1972, and a famous experiment was conducted by Aspect, Grangier, and Roger in 1981. Further experiments refined the test over the last three decades.

2015 was an especially good year for Bell tests. Here are links to some popular science reports for experiments conducted at universities throughout the world.

Wikipedia has an article devoted specifically to tracking Bell inequality violations going back four decades.

## Teleportation

One of the most important uses of entanglement is quantum teleportation. In teleportation, the state of one qubit is destroyed in one place and resurrected in another. This is accomplished by using entanglement.

To teleport data, begin with a Bell pair shared between two people (always called Alice and Bob in quantum discussions). Alice also has the qubit she wishes to teleport to Bob.

Alice measures her two qubits (the data qubit and the Bell pair qubit) together, in a special way known as Bell state analysis. This measurement destroys the entanglement in the Bell pair, and also forces the collapse of any superposition in the data qubit.

This Bell state analysis gives Alice two classical bits of information, and leaves Bob’s half of the Bell pair in an ambiguous state. If Alice sends her two classical bits to Bob, he can use those to turn his qubit into the state of the original qubit. The data has been teleported from Alice to Bob!

We saw in the video that the entanglement behaves like there is some communication between the qubits over a distance, but it can’t be used to transmit data faster than the speed of light. The need for the transmission of the classical data and its use in order to recreate the data qubit is why this is so. If Bob measures the qubit without waiting for the data from Alice, he will get only random numbers that are of no use. It is only once the bits arrive, and tell him how to interpret that data – almost like an encryption key – that his data becomes useful.

Teleportation is a fundamental primitive in quantum networking, and in executing certain types of quantum computing. In this course, we won’t have any further need for teleportation, but its importance can’t be overstated.

# 量子もつれ

ここまでは波とその重ね合わせや干渉など、測定の話を除けばどれも古典物理の話題について学習してきました。そしてn量子ビットは2nのエントリの状態ベクトルで表せることもみてきました。ここからはいよいよ、あっと驚くような量子現象についてみていくことにしましょう。まずは量子もつれという現象について紹介していきます。実はこの現象を巡って過去には、かのアルバートアインシュタインが量子力学の不完全性を主張したほど彼を悩ませた理論としても有名です。

## ベル状態

この動画ではベル状態という概念について詳しく紹介しています。ベル状態というのは量子もつれの状態の中でもっとも基本的は状態のことをさします。実は以前紹介した$$\frac{|01\rangle + |10\rangle}{\sqrt{2}}$$という状態もベル状態の一つでした。

$|\Psi^+\rangle = \frac{|01\rangle+|10\rangle}{\sqrt{2}}$ $|\Psi^-\rangle = \frac{|01\rangle+(\pi)|10\rangle}{\sqrt{2}}$ $|\Phi^+\rangle = \frac{|00\rangle+|11\rangle}{\sqrt{2}}$ $|\Phi^-\rangle = \frac{|00\rangle+(\pi)|11\rangle}{\sqrt{2}}$

## 実証実験

2015年には様々な大学で素晴らしい成果が報告されています。以下に実験について紹介した記事のリンクを掲載しておきます。

ウィキペディアにも過去40年間に渡るベルテストについて詳しく掲載されています。

## テレポーテーション

ある情報をテレポートさせるためには、まず情報の送受信をしたい2人の間でベル状態の量子をあらかじめ共有しておく必要があります(この2人は慣習的にアリスとボブと呼ばれます)。そしてアリスは送信したい情報としての量子ビットを別で持っています。

アリスは彼女が持っている2つの量子ビット(送信したいデータとしての量子ビットとボブとあらかじめ共有しておいたベルペア量子ビット)をベル測定という特別な方法で一緒に測定します。測定によって当然アリスが持っていたベルペア量子ビットも送信したいデータとしての量子ビットの重ね合わせ状態も壊れてしまいます。

このベル測定によってアリスは2ビットの情報を得ることができます。一方でボブのベルペア量子ビットのもう一方が残されている状態になります。アリスが得た2ビットの情報を古典通信などでボブに送信し、ボブはその情報に基づいて彼の持つベルペア量子ビットにある処理を施すと、なんとこの量子ビットはアリスが送信したかったデータ量子ビットと同じ状態に再構成されるのです。こうしてデータはアリスからボブの元へテレポートしたことになります。