# 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 weโre going to step out into the realm of really hair-raising quantum phenomena. Weโ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 wasnโ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 didnโ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 Bellโ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 Bobโ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 canโ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 โ almost like an encryption key โ 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 wonโt have any further need for teleportation, but its importance canโt be overstated.

# ้ๅญใใคใ

ใใใพใงใฏๆณขใจใใฎ้ใญๅใใใๅนฒๆธใชใฉใๆธฌๅฎใฎ่ฉฑใ้คใใฐใฉใใๅคๅธ็ฉ็ใฎ่ฉฑ้กใซใคใใฆๅญฆ็ฟใใฆใใพใใใใใใฆn้ๅญใใใใฏ2nใฎใจใณใใชใฎ็ถๆใใฏใใซใง่กจใใใใจใใฟใฆใใพใใใใใใใใฏใใใใใใใฃใจ้ฉใใใใช้ๅญ็พ่ฑกใซใคใใฆใฟใฆใใใใจใซใใพใใใใใพใใฏ้ๅญใใคใใจใใ็พ่ฑกใซใคใใฆ็ดนไปใใฆใใใพใใๅฎใฏใใฎ็พ่ฑกใๅทกใฃใฆ้ๅปใซใฏใใใฎใขใซใใผใใขใคใณใทใฅใฟใคใณใ้ๅญๅๅญฆใฎไธๅฎๅจๆงใไธปๅผตใใใปใฉๅฝผใๆฉใพใใ็่ซใจใใฆใๆๅใงใใ

## ใใซ็ถๆ

ใใฎๅ็ปใงใฏใใซ็ถๆใจใใๆฆๅฟตใซใคใใฆ่ฉณใใ็ดนไปใใฆใใพใใใใซ็ถๆใจใใใฎใฏ้ๅญใใคใใฎ็ถๆใฎไธญใงใใฃใจใๅบๆฌ็ใฏ็ถๆใฎใใจใใใใพใใๅฎใฏไปฅๅ็ดนไปใใ(frac{|01rangle + |10rangle}{sqrt{2}})ใจใใ็ถๆใใใซ็ถๆใฎไธใคใงใใใ

้ๅญใใใใฏๆฐๅคใจไฝ็ธใฎ๏ผใคใฎ็นๅพดใซใใฃใฆๆฑบใพใใพใใไธ่จใฎใใซ็ถๆใฏๆฐๅคใฏ้(็ๆนใ0ใชใใฐใใไธๆนใฏ1)ใงไฝ็ธใฏๅใใงใใใใใซๅฏพใใฆใๆฐๅคใไฝ็ธใๅใใๆฐๅคใฏๅใใ ใไฝ็ธใ้ใจใใ็ถๆใใใกใใๅญๅจใใพใใๆด็ใใใจใใซใใขใฏไปฅไธใฎ๏ผ็จฎ้กใซๅใใใใจใใงใใพใใใใฎ๏ผ็จฎ้กใฏใใใใไปฅไธใฎใใใซ่กจ็พใใใพใใ

[|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}}]

## ๅฎ่จผๅฎ้จ

้ๅญใใคใใฎๅญๅจใ็ขบ่ชใใๅฎ้จใฏJohn Bellใซใใฃใฆๆๅฑใใใๆน็จๅผใซใกใชใใงใใใซใในใใใใซใฎไธ็ญๅฎ้จใชใฉใจๅผใฐใใฆใใพใ(ๆ่ฟใฏใใใจใฏ็ฐใชใๆฐ็ใงๅฎ้จใใใใใจใๅคใใงใ)ใๆๅใฎๅฎ่จผๅฎ้จใฏFreedmanใจClauserใซใใฃใฆ1972ๅนดใซ่กใใใพใใใไปใซใ1981ๅนดใซAspectใGrangierใRogerใ่กใฃใๅฎ้จใๆๅใงใใ้ๅป30ๅนดใฎ้ใง็นฐใ่ฟใใใใใๅฎ้จใจๆนๅใ่กใใใฆใใพใใใ

2015ๅนดใซใฏๆงใใชๅคงๅญฆใง็ด ๆดใใใๆๆใๅ ฑๅใใใฆใใพใใไปฅไธใซๅฎ้จใซใคใใฆ็ดนไปใใ่จไบใฎใชใณใฏใๆฒ่ผใใฆใใใพใใ

ใฆใฃใญใใใฃใขใซใ้ๅป40ๅนด้ใซๆธกใใใซใในใใซใคใใฆ่ฉณใใๆฒ่ผใใใฆใใพใใ

## ใใฌใใผใใผใทใงใณ

้ๅญใใคใใฎๆใ้่ฆใชๅฟ็จใฎไธใคใซ้ๅญใใฌใใผใใผใทใงใณใจใใใใฎใใใใพใใใใฎใใฌใใผใใผใทใงใณใจใใใฎใฏๅฎ้ใซ็ฉ่ณชใ็ฌ้็งปๅใใใฎใงใฏใชใใใใๅฐ็นใง้ๅญใใใใๅดฉๅฃใใๅฅใฎๅฐ็นใงๅใๆๅ ฑใๆใฃใ้ๅญใใใใจใใฆๅ็ๆใใใใจใใๅ็ใงใใใใซ้ๅญใใคใใ็จใใใใฆใใพใใ

ใใๆๅ ฑใใใฌใใผใใใใใใใซใฏใใพใๆๅ ฑใฎ้ๅไฟกใใใใ2ไบบใฎ้ใงใใซ็ถๆใฎ้ๅญใใใใใใๅฑๆใใฆใใๅฟ่ฆใใใใพใ(ใใฎ2ไบบใฏๆฃ็ฟ็ใซใขใชในใจใใใจๅผใฐใใพใ)ใใใใฆใขใชในใฏ้ไฟกใใใๆๅ ฑใจใใฆใฎ้ๅญใใใใๅฅใงๆใฃใฆใใพใใ

ใขใชในใฏๅฝผๅฅณใๆใฃใฆใใ2ใคใฎ้ๅญใใใ(้ไฟกใใใใใผใฟใจใใฆใฎ้ๅญใใใใจใใใจใใใใใๅฑๆใใฆใใใใใซใใข้ๅญใใใ)ใใใซๆธฌๅฎใจใใ็นๅฅใชๆนๆณใงไธ็ทใซๆธฌๅฎใใพใใๆธฌๅฎใซใใฃใฆๅฝ็ถใขใชในใๆใฃใฆใใใใซใใข้ๅญใใใใ้ไฟกใใใใใผใฟใจใใฆใฎ้ๅญใใใใฎ้ใญๅใใ็ถๆใๅฃใใฆใใพใใพใใ

ใใฎใใซๆธฌๅฎใซใใฃใฆใขใชในใฏ2ใใใใฎๆๅ ฑใๅพใใใจใใงใใพใใไธๆนใงใใใฎใใซใใข้ๅญใใใใฎใใไธๆนใๆฎใใใฆใใ็ถๆใซใชใใพใใใขใชในใๅพใ2ใใใใฎๆๅ ฑใๅคๅธ้ไฟกใชใฉใงใใใซ้ไฟกใใใใใฏใใฎๆๅ ฑใซๅบใฅใใฆๅฝผใฎๆใคใใซใใข้ๅญใใใใซใใๅฆ็ใๆฝใใจใใชใใจใใฎ้ๅญใใใใฏใขใชในใ้ไฟกใใใใฃใใใผใฟ้ๅญใใใใจๅใ็ถๆใซๅๆงๆใใใใฎใงใใใใใใฆใใผใฟใฏใขใชในใใใใใฎๅใธใใฌใใผใใใใใจใซใชใใพใใ

ๅ็ปใฎไธญใงใ้ๅญใใคใ็ถๆใฎ้ๅญใใใใ้ ่ท้ขใงใๆๅ ฑใๅฑๆใใ(ใใ ใๅ้ไปฅไธใง)ใจใใๆง่ณชใซใคใใฆ็ดนไปใใฆใใพใใใชใใใฎใใใซๅคๅธใใผใฟใ้ไฟกใใๅฟ่ฆใใใใใชใใใใฎๅดใงใใผใฟใๅๆงๆใใใฎใซใใฎใใผใฟใไฝฟใใชใใใฐใชใใชใใฎใงใใใใ๏ผไปฎใซใใใใขใชในใใใฎๅคๅธ2ใใใใใฎๅไฟกใๅพใใใซ่ชๅใฎใใซใใข้ๅญใใใใๆธฌๅฎใใฆใใพใฃใๅ ดๅใๅใซใใฎ้ๅญใใใใฎๅญๅจ็ขบ็ใใๅพใใใ็ตๆใซใชใฃใฆใใพใใพใใๅไฟกใใ(ๆๅทใ่คๅใใใใใฎ้ตใฎใใใชๅฝนๅฒใใใค)2ใใใใฎๆๅ ฑใๅใซๅฆ็ใๆฝใใใจใงๅใใฆใใใฎๆใค้ๅญใใใใฏใขใชในใ้ไฟกใใใใฃใใใผใฟใจใใฆๆฏใ่ใใใใซใชใใพใใ

้ๅญใใฌใใผใใผใทใงใณใฏ้ๅญใใใใฏใผใฏใซใใใๆใๅบ็ค็ใชๆ่กใงใใใใใ็นๅฎใฎ็จฎ้กใฎ้ๅญใณใณใใฅใผใใฃใณใฐใซใใฃใฆๅทใ่กใใใพใใใใฎใณใผในๅใงใฏใใฌใใผใใผใทใงใณใซใคใใฆใฎ็ฅ่ญใฏใใฎใใใใงๅๅใงใใใใใฎๆ่กใฎๅใพใใใๅฐใใงใใ็่งฃใใใ ใใใงใใใใใ