entanglement presentation

7/30/2019 entanglement presentation

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History Entanglement Generation Application

Outline

1 HistoryEPR ParadoxBells Theorem

2 Entanglement

Fabric of the ComosEntangled States

3 Generation of Entangled particlesAtomic Cascades in Calcium

Parametric down-conversionAspects experiment

4 Application of EntanglementQuantum Cryptography


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Outline


2 Entanglement






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EPR Paradox

Figure : Einstein, A; B Podolsky, N Rosen (1935-05-15). CanQuantum-Mechanical Description of Physical Reality be ConsideredComplete?. Physical Review 47 (10): 777780.


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Outline


2 Entanglement





Hi E l G i A li i


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Bells Theorem

Bell Inequality

|P (a, b ) P (a, c )| 1 + P (b,c ) (1)

Hi t E t gl t G ti A li ti


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Bells Theorem

Bell Inequality

|P (a, b ) P (a, c )| 1 + P (b,c ) (1)CHSH(John Clauser, Michael Horne, Abner Shimony and RichardHolt) Inequality

2 S 2 (2)where,

S = E (a , b ) + E (a , b ) + E (a , b ) E (a , b ) (3)



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Bells Theorem

Derivation of CHSH InequalityAlice can measure either one of two observablesa and a , whileBob can measure either b or b .



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Bells Theorem


If a , a =

1, it follows that either a + a = 0, in which case

a a = 2, or else a a = 0, in which case a + a = 2; thereforeS = ( a + a )b + ( a a )b = 2 (4)



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Bells Theorem


If a , a =

1, it follows that either a + a = 0, in which case

a a = 2, or else a a = 0, in which case a + a = 2; thereforeS = ( a + a )b + ( a a )b = 2 (4)

EvidentlyS = | d ()S ()| 2 (5)

since the average of a sum is the sum of the averages,

| S | = |E (a , b ) + E (a , b ) + E (a , b ) E (a , b )| 2 (6)



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Outline


2 Entanglement







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y g pp

Fabric of the Comos

Explanation of Entanglement

http://youtu.be/Nv1_YB1IedE?t=25m53s


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Outline


2 Entanglement







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Entangled States

Bell statesFor two qubits, the Bell states are

| = 1 2(| | ) (7)

| =1 2(| | ) (8)

These four pure states are all maximally entangled



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Entangled States

Violation Bell Inequalitylet a , a denote the Hermitian operators

a = A a , a = A

a (9)acting on a qubit in Alices possession, Similarly denote

b = B b , b = B

b (10)acting on Bobs qubit.



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Entangled States

Violation Bell Inequalitylet a , a denote the Hermitian operators

a = A a , a = A

a (9)acting on a qubit in Alices possession, Similarly denote

b = B b , b = B

b (10)acting on Bobs qubit.

| A

a B

b | = a b = cos (11)where is the angle between a and b .



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Entangled States

Consider the case where a , a , b , b are coplanar and separated bysuccessive 45 angles. the quantum mechanical predictions are

a

b

=a

b

= a

b

= cos4 =

1

2 (12)a b = cos

34

=1 2 (13)



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Entangled States

Consider the case where a , a , b , b are coplanar and separated bysuccessive 45 angles. the quantum mechanical predictions are

a

b

=a

b

= a

b

= cos4 =

1

2 (12)a b = cos

34

=1 2 (13)

S = 2 2 (14)which violates the CHSH inequality



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Entangled States

Greenberger

Horne

Zeilinger(GHZ) state

|GHZ = | M + |

M

2 (15)



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Outline


2 Entanglement







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Atomic Cascades in Calcium

Figure : Radiative cascade of calcium(Adapted from Advances In Atomic

Physics, Claude Cohen-Tannoudji, David Guery-Odelin)

State of the two photon emitted

| =

1

2[

|1 , +

|2 , +

|1 ,

|2 , + ] (16)



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Outline


2 Entanglement







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Parametric down-conversion

Figure : Schematic representation of a down-conversion process within anonlinear crystal.(Adapted from Quantum Optics, An Introduction, MarkFox)



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Conservation of energy and momentum, respectively, require thatPhase-matching conditions

0 = 1 + 2 (17)

k 0 = k 1 + k 2 (18)

The down-conversion process is called degenerate when1 = 2 = 0 / 2



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Conservation of energy and momentum, respectively, require thatPhase-matching conditions

0 = 1 + 2 (17)

k 0 = k 1 + k 2 (18)

The down-conversion process is called degenerate when1 = 2 = 0 / 2

Type-II phase matching

| =1 2[| 1 , 2 + e

i | 1 , 2 ] (19)



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Outline


2 Entanglement







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Aspects experiment

Figure : Schematic diagram of the apparatus for the third Aspectexperiment. The correlated photon pairs were generated by a calciumcascade source S. An acousto-optical switch (AOS) was added on each

side of the apparatus to deect the beam towards different polarizerswith axes a or a and b or b as appropriate. The short switching time of the AOS ensured that the polarization detection angle was being changedfaster than any information-carrying signals could pass between thedetectors. (Adapted from Quantum Optics, An Introduction, Mark Fox)



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Outline


2 Entanglement







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Quantum Cryptography

Figure : Pictorial representation of encrypted communication



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Figure : Bob and Alice will have two groups of measurement, they willreveal only the rst group which is used to establish the value of S. Thisassures them that the measurements obtain within the second group areanti-correlated which can be converted to useful information.



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Figure : A single run of the experiment may look like above



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Figure : A single run of the experiment may look like above

Figure : Resultant table after Alice and Bob discard all resultscorresponding to instances in which they used different machines.



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Figure : check whether the two strings are indeed perfectly correlated by

comparing randomly selected entries in public (shown grey in the tablebelow)



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Figure : check whether the two strings are indeed perfectly correlated by

comparing randomly selected entries in public (shown grey in the tablebelow)

Figure : The publicly revealed entries are discarded and the remainingresults are kept as the key.



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The End

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