EPR

parad o x

Bell inequalities

1

Can quantum-mechanical description of p h ysical r eality be conside r ed complete?

• A. Einstein, B. P odolsk y , and N. Rosen, P h ys. Re v . 47, 777 - 780 (1935)

In a complete theor y ther e is an element cor r esponding to each element of r ealit y . A sufficient condition f or the r eality of a ph ysical quantity is the possibility of pr edicting it with cer taint y , without disturbing the system. In quantum mechanics in the case of tw o ph ysical quantities described b y non- comm uting operators, the kno wledge of one pr ecludes the kno wledge of the othe r . Then either (1) the description of r eality giv en b y the wa v e function in quantum mechanics is not complete or (2) these tw o quantities cannot ha v e sim ultaneous r ealit y . Consideration of the pr oblem of making pr edictions concerning a system on the basis of measur ements made on another system that had pr e viousl y interacted with it leads to the r esult that if (1) is false then (2) is also false . One is thus led to conclude that the description of r eality as giv en b y a wa v e function is not complete .

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2

Entangled Pair

• Pr epar e the state = ( 01 + 10 ) / p 2 of tw o identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

3

Entangled Pair

• Pr epar e the state = ( 01 + 10 ) / p 2 of tw o identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

4

Entangled Pair

• Pr epar e the state = ( 01 + 10 ) / p 2 of tw o identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

5

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

6

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

7

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

8

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

9

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

10

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

11

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

12

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

13

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

14

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

15

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

16

Entangled Pair

• Pr epar e the state i = ( | 01 i + | 10 i ) / p 2 of tw o

identical par ticles (spins)

• Par ticles mo v e to Alice and Bob , that measur e their angular momentum S z , obtaining either +1 or -1

A B

• Experiment r epeated man y times: perf ect anti- cor r elation

17

T r a v eler pair

• Anti-cor r elation also in “classical” experiment

• T w o tr a v elers with balls inside tw o luggages

Image s by MIT OpenCourseWare.

• Alice and Bob check the luggages: perf ect anti- cor r elation

18

T r a v eler pair

• Anti-cor r elation also in “classical” experiment

• T w o tr a v elers with balls inside tw o luggages

Image s by MIT OpenCourseWare.

Alice and Bob check the luggages: perf ect anti- cor r elation

19

T w o p r oper ties of balls

• No w assume that the balls can be r ed or gr een and matte or shin y .

• Anti-cor r elation also f or the pr oper ty of gloss

20

T w o a x es

• In QM, the 2 pr oper ties ar e the spin along 2 a x es:

( x σ z

• W e can r e write the state in ( x basis,

i = ( | 01 i + | 10 i ) / p 2 = ( | + -i + | - + i ) / p 2

• thus measuring ( x Alice and Bob obtain same anti-cor r elation

21

Classical h ypothesis

Realism & Locality

22

Realism

• At pr eparation, par ticles a and b possess both the pr oper ties

(color and gloss f or the classical balls

σ x , σ z , with σ x,z = ±1 f or the quantum par ticles)

23

Locality

• When I measur e par ticle a , I cannot modify instantaneousl y the r esult of measuring par ticle b .

Ther e is no action at distance (faster than light)

24

EPR Parado x

• An y complete description of the w orld m ust r espect local r ealism

• Local r ealism is violated b y quantum mechanics

➜ Quantum mechanism is not a complete description of the w orld

25

Bell Inequalities

Quantitativ e measur e of violation of local r ealism

26

Cor r elation f or a n y a x es

• Assume tw o spins in Bell State ✓

i = ( | 01 i + | 10 i ) / p 2

• Alice measur e a A obtaining a , while Bob measur e

a B = cos ✓a B + s i n ✓a B getting b ∈ {+1,-1} .

b z x

• What is the cor r elation

h ab i = h a A a B i ?

z b

27

Some calculations...

h a A a B i 1 A B A B

z b =

2 ( h 01 | a z a b | 01 i + h 01 | a z a b | 10 i +

+ h 10 | a A a B | 01 i + h 10 | a A a B | 10 i )

z b z b

= 1 h 0 | a A | 0 ih 1 | a B | 1 i + h 0 | a A | 1 ih 1 | a B | 0 i +

2 z b z b

+ h 1 | a A | 1 ih 0 | a B | 0 i + h 1 | a A | 0 ih 0 | a B | 1 i

z b z b

= 1 h 1 | a B | 1 i - h 0 | a B | 0 i = - cos ✓

2 b b

28

Bell experiment

• Alice measur es along either a or a’

• Bob measur es along either b or b’

a’

a b b = cos ✓ a c a 0 = cos ¢ b b’

b c b 0 = cos ¢ a d 0 b 0 = cos ✓

29

Cor r elations

• The cor r elations among the measur ements ar e

then:

h ab i = h a 0 b 0 i = - cos ✓

h a 0 b i = - cos( ✓ - ¢ ) h ab 0 i = - cos( ✓ + ¢ )

• W e want to calculate 〈 S 〉

h S i = h ab i + h a 0 b 0 i + h ab 0 i - h a 0 b i

30

Locality + Realism

• At each measur ement, I should be able to calculate the value of the operator :

S k = ( σ A σ B ) k + ( σ A σ B ) k + ( σ A σ B ) k — ( σ A σ B ) k

a b a 0 b 0

a b 0

a 0 b

• The expectation value is then h S i = l i m 1 S

N !1 N

k

31

Realism

• Ev en if I measur e the spin along a , the pr oper ty

spin along a’ is re a l (that is, it has a definite value)

32

Calculate outcome of S k

• Re write as

S k = ( A ( ( B + ( B ) k - ( A ( ( B - ( B ) k

a b b 0 a 0 b b 0

• b b 0

• b b 0 b b 0

• Then S k = ± 2 a a or S k = ± 2 a 0

33

Locality

• The fact of measuring b or b’ does not change the value of a or a’ (that h a v e outcomes +/-1 ). Then:

S k = ± 2

• The expectation value of S is then

— 2 < ⟨ S ⟩ < +2

34

Bell inequality

• If | h S i | > 2 at least one of the tw o h ypothesis (locality or r ealism) is not true

35

Bell inequality

• Choose a=z; a’=x; b=-x+z; b’=x+z;

ab i = h a 0 b 0 i = - cos ✓ ab = - 1 / p 2 b

ab 0 i = - cos ✓ ab 0 = - 1 / p 2

a 0 b i = - cos ✓ a 0 b = + 1 / p 2

• W e obtain

a b’

a’

h S i = h ab i + h a 0

b 0 i + h ab 0

i - h a

4

b i = - p 2

= - 2 p 2 < - 2

36

Ref e r ences

• J. S. Bell, On the Einstein P odolsk y Rosen P ar ado x , P h ysics 1, 195-200 (1964)

• Alain Aspect, Philippe Grangie r , and Gerar d Roge r , P h ys. Re v . Lett. 47, 460 - 463 (1981)

Exper imental T ests of Realistic Local Theor ies via Bell's Theor em

37

State: inf ormatio or object?

• No-g o theor em: if the quantum state mer el y r epr esents inf ormation about the r eal ph ysical state of a system, then experimental pr edictions ar e

obtained that contradict those of quantum theor y .

38

Non- Locality

• If locality is lost, can it be used f or action at distance?

• T elepor tation?

39

Quantum T elepor tation

• Alice has a qubit in a state

| i = α | 0 i + β | 1 i

but does not kno w an ything about it.

• As obser ving the state destr o ys it, Alice can ’ t measur e the qubit and tell the ans w er to Bob

Alice ca n ’t giv e the state to Bob b y classical mea ns.

• No-cloning theor em, no quantum channel.

40

Entangled pair

• Assume Alice and Bob shar e a pair of qubits that is pr epar ed in an entangled state .

• Alice and Bob each h a v e access to one Qubit.

| ϕ i =

1

p 2 | 0 i A | 0 i B +

1

p 2 | 1 i A | 1 i B

• The full-state then is the pr oduct of Alice ’ s Qubit and the shar ed r egister :

1 1 1 1

| i | ϕ i = α p 2 | 000 i + p 2 β | 100 i + p 2 α | 011 i + p 2 β

| 111 i

41

Comm unication Scheme

Classical Channel

Alice

(Internet)

Copied State

| i = α | 0 i + β | 1 i

Initial State

| i = α | 0 i + β | 1 i

Bob

Entangled Sour ce

1 1

| ϕ i =

p 2 | 0 i A | 0 i B + p 2 | 1 i A | 1 i B

42

Alg orithm

• Alice then perf orms a CN O T on her half of the r egiste r , using her m yster y bit as the contr ol.

1 1 1 1

| ϕ i | i = α p 2 | 000 i + p 2 β | 110 i + p 2 α | 011 i + p 2 β | 101 i

• She then applies the Hadamard gate t o | i A

1

| ϕ i | i =

2

1

+

2

| 00 i ( α | 0 i + β | 1 i ) +

| 10 i ( α | 0 i — β | 1 i ) +

1

| 01 i ( α | 1 i + β | 0 i )

2

1

| 11 i ( α | 1 i — β | 0 i )

2

43

Measu r ement

• Alice then measur es her 2 qubits and tells Bob

• Note that this destr o ys her original state .

• The outcome of this obser vation is unpr edictable .

• If Alice measur es 00, then Bob has the original state . Otherwise , Bob has some other state .

• The state is kno wn , so Bob can perf orm a kno wn operation to r etrie v e the original state

σ x

1 1 1 | 00 i ( α | 0 i + β | 1 i ) + 1 | 01 i ( α | 1 i + β | 0 i )

2 2

1 1

+ | 10 i ( α | 0 i — β | 1 i ) + | 11 i ( α | 1 i — β | 0 i )

σ z 2 2 σ y

44

Quantum T elepor tation

• This pr ocedur e r elied on superposition and entanglement.

• It was necessar y to account f or the pr obabilistic natur e of QM b y giving Bob par ticular actions to ta k e , depending on the (unpr edictable) outcome of the obser vation.

• Theor y: C . H. Bennett, G. Brassar d, C . Crépeau, R. J ozsa, A. P er es, W . K. W ootters, T elepor ting an Unkno wn Quantum State via Dual Classical and Einstein-P odolsky- Rosen Channels, Ph ys. Re v . Lett. 70, 1895-1899 (1993)

• Experiments:

D . Bouwmeeste r , .., A. Zeilinge r , Experimental Quantum T elepor tation, Natur e 390, 575 (1997)

M. D . Bar r ett, …., D . J. Wineland, Deterministic Quantum T elepor tation of Atomic Qubits, Natur e 429, 737 (2004).

S. Olmschenk, … C . Monr oe , Quantum T elepor tation betw een Distant Matter Qubits, Science 323, 486 (2009)

45

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Fall 2012

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