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# Entanglement

In this video and the accompanying text, Rodney Van Meter introduces quantum entanglement, Einstein's spooky action at a distance.

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{|01rangle + |10rangle}{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{|01rangle+|10rangle}{sqrt{2}})
(|Psi^-rangle = frac{|01rangle+(pi)|10rangle}{sqrt{2}})
(|Phi^+rangle = frac{|00rangle+|11rangle}{sqrt{2}})
(|Phi^-rangle = frac{|00rangle+(pi)|11rangle}{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{|01rangle + |10rangle}{sqrt{2}})という状態もベル状態の一つでした。

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