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quantum mysteries again!
‘ quantum mechanics is weird”
N. Bohr
 classical vs. quantum correlations
 Bell’s inequality? QM VIOLATES IT!
D. Mermin, Am. J. Phys. 49, 940 (1981)
1
singlet state (EPR pair)

sin g

  
2
take two spins and move them apart (no common
preparation or exchange of signals between them)
and measure them in various directions (settings).
What are the results? always opposite!
EPR paradox (1935) or quantum non-locality?
“strange action at a distance” or common state?
2
quantum vs. classical
correlations
what are correlations due to?
 fast communication (via exchange of messages)
or
 common preparation (via hidden variables)
3
2 spins in the singlet state
 if spin 1 is up & spin 2 is down in the z-dir
 if spin 1 is up in the z-dir the spin 2 is down in the θ
n-direction with angle θ
  c( n )  n  c( n ) 
n


  
  
 cos  
 sin   
 0  (n) 
 2    c(n) 
2 
    c



 i



  S
1
i

 
S z (1)  1
e
sin

e
cos
 
   z (1)  1


 2 
 2   S  1


S ( 2)  1
 ( 2)
 
 
2 
2 
 c( n )  sin  , c( n )   cos 

P

sin
,
P

cos


 


2
2
2
2
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quantum correlation function
 measure the spins in two directions with angle θ θ
C ( )  S1( z ) S 2 ( )
P   sin
2
P   cos


P  (1)( 1)  P  (1)( 1)  P  (1)( 1)  P  (1)( 1)
P   P   P   P 

P   P   P   P 
P   P   P   P 
2
2
P   cos 2
P   sin 2

2

2

2



2  cos 2  sin 2 
2
2
 
  cos 



2 sin 2  cos 2 
2
2

S z (1)  1

 S ( 2 )  1
e.g. C (0)  S1( z ) S 2( z )
1
 (1)( 1)  (1)( 1) 
2
 1  cos 0
remember, the mean value of SzSθ is
taken on the singlet (entangled) state
5
classical correlation function
 suppose spins have definite (if unknown) values,
then the orientation of spin is random (of course
the spins are opposite to each other) S  1
z (1)

P  

θ
θ

P   1 


P   1 


P  

Cclas ( )  S1( z ) S 2( )

 S ( 2 )  1
clas
P   P   P   P 
P   P   P   P 
1   
   1 
2  
2
  
  1 
  
 

 




1

6
quantum vs. classical
Cqua ( )  S1( z ) S 2( )
1
qua
  cos 
Cclas ( )  S1( z ) S 2 ( )
-1
2
clas

1

quantum correlations are stronger than classical
(Bell showed QM can go out of mathematical
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limits!!!)
measure one spin
• measure the spin in various directions (settings) with
results  1 (in units of  2 )
a  1
a  1
in z-dir
in n-dir (θ)
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…measure both spins (in a singlet)
• measure the spin in various directions (settings) with
results  1 (in units of  2 ) at locations A and B
 location A
a  1
a  1
in z-dir
in n-dir (θ)
 location B
b  1
b  1
in z-dir
in n-dir (-θ)
consider now the linear combination of correlations
g  ab  ab' a' b  a' b'
 a(b  b' )  a' (b  b' )
how many possible results? 16 what are they?g= +2 or -2
9
Bell-CHSH inequality
λ: hidden variable
mean correlations
ab   a b   d
S  g  ab  ab'  a' b  a' b'  2
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quantum correlation function
violates
it!
   
 
ab  sin g a    b   sin g  a  b   cos 
Bell- CHSH inequality:
ab  ab'  a' b  a' b'  2
  cos 0  cos   cos   cos2   2
 1  2 cos   cos2   2
violation of the inequality at π/3: |1+2(1/2)-(-1/2)|=2.5>2!
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violation of Bell’s inequality
S  1  2 cos( )  cos(2 )
2
S 2
0
π/3 π/2
maximum violation at π/3!
π
θ
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remember! Bell’s inequality
is only maths!
 physics (QM) often violates it!
13
quantum mysteries for everybody!
 D. Mermin, Am. J. Phys. 49, 940 (1981)
14
pedestrian’s set up!
entangled particle source and A & B detectors:
public language: three settings (1,2,3) & two flash Red or Green
our language: dirs of measurement (0, -π/3,+π/3) & up or down)
1

sin g

  
1
2
e particle source
2
3
2
3
three settings: 1,2,3 and two results: Red or Green
15
classical correlations
hidden variables: particles carry identical instruction sets
(eight possibilities)
RRR, RGG, GRG,GGR, GRR, RGR, RRG, GGG
SAME  e.g. if RRR then: for 12 RR, for 23 RR, for 13 RR
the same are 100% of the time
(TWO)
DIFFR  e.g. if RGG then: for 12 RG, for 23 GG, for 13 RG
(SIX) prob to be the same =1/3 (prob no smaller than 1/3)
prob to flash same colour can never be smaller than 1/3
this is Bell’s inequality
16
quantum correlations
entangled particles have prob=cos2 ( θ/2) to flash the
same colour (why?), for θ=0, -120, 120 we have
prob=1, ¼, ¼ to flash the same colour, respectively
1 2 3
1
2
3
1
1
4
1
4
1
4
1
1
4
1
4
1
4
1
the quantum prob=1/4 is smaller than 1/3
violating classical statistics!
17
our world is non-local!
 Einstein: quantum physics is incomplete (EPR
paradox)
 Bell: quantum physics violates mathematical
inequalities (Bell’s inequalities)
experiment showed: Bell is right!
(non-local quantum correlations exist)
A B
superposition & entanglement
A1 A2  B1B2
A1 A2 A3  B1B2 B3
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end of lecture
quantum mysteries revisited:
 quantum correlations: violate Bell’s inequalities
(neither fast communication nor common preparation)
 quantum world: neither deterministic nor local!
 entanglement is the key!
 superposition of distant states
non-locality was verified in experiments
via violation of Bell’s inequalities
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