Transcript Slide 1

Group theoretic formulation of
complementarity
Joan Vaccaro
Centre for Quantum Dynamics,
Centre for Quantum Computer Technology
Griffith University
Brisbane
QCMC’06
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outline
waves & asymmetry
particles & symmetry
complementarity
Outline
Bohr’s complementarity of physical
properties
mutually exclusive experiments
needed to determine their values.
[reply to EPR PR 48, 696 (1935)]
Wootters and Zurek information theoretic formulation:
[PRD 19, 473 (1979)]
(path information lost)  (minimum value for given visibility)
Scully et al Which-way and quantum
erasure [Nature 351, 111 (1991)]
Englert distinguishability D of
detector states and visibility V
[PRL 77, 2154 (1996)]
D2  V 2  1
QCMC’06
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outline
waves & asymmetry
particles & symmetry
complementarity
Elemental properties of Wave - Particle duality
(1) Position probability density with spatial translations:
localised
de-localised
x
x
particles are “asymmetric”
waves are “symmetric”
(2) Momentum prob. density with momentum translations:
de-localised
localised
p
particles are “symmetric”
p
waves are “asymmetric”
Could use either to generalise particle and wave nature
– we use (2) for this talk. [Operationally: interference sensitive to ]
QCMC’06
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outline
waves & asymmetry
particles & symmetry
complementarity
In this talk
Tg
 discrete symmetry groups G = {Tg}
 measure of particle and wave nature is
information capacity of asymmetric and symmetric parts
of wavefunction
p
Tg
p
Tg
 balance between (asymmetry) and (symmetry)
wave
particle
Contents:
NW (  )  N P (  )  ln(D)
 waves and asymmetry
 particles and symmetry
 complementarity
QCMC’06
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outline
waves & asymmetry
particles & symmetry
complementarity
Waves & asymmetry
Waves can carry information in their translation:
group G = {Tg},
unitary representation: (Tg )1 = (Tg ) +
Tg
symbolically :
 g
 g = Tg  Tg+
p
Information capacity of “wave nature”:
Alice
Tg
000 001
...
…
Bob
101
...

g
1

[ ] 
T

T

g
g
O(G) g
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estimate parameter g
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outline
waves & asymmetry
particles & symmetry
complementarity
Waves & asymmetry
Waves can carry information in their translation:
unitary
representation:
{Tinterferometry
g for g  G}
Example:
single photon
group G = {g},
0
Tg
symbolically :
?

= photon in upper path
 g = Tg  Tg+
p
g
1
= photon in lower path
Information capacity of “wave
nature”:
particle-like states:
Alice
Tg
000 001
...

g
…
wave-like states:
101
group:
...
translation:
1

[ ] 
T

T
g g
O(G) gG
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 
0,1
0 1
2
Bob
,
 
0 1
2
G  {1,  z }
1    , z   
estimate parameter g
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outline
waves & asymmetry
particles & symmetry
complementarity
DEFINITION: Wave nature NW ()
NW () = maximum mutual information between Alice and
Bob over all possible measurements by Bob.
Tg
Alice
000 001
…
Bob
101
...
...
 g = Tg  Tg+
estimate parameter g
Holevo bound
S (  )  T r(  ln  )
NW (  )  S (
[  ])  S (  )
[ ] 
1

T

T
 g g
O(G) g
increase in entropy due to G
= asymmetry of  with respect to G
QCMC’06
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outline
waves & asymmetry
particles & symmetry
complementarity
Particles & symmetry
Particle properties are invariant to translations Tg  G
For “pure” particle state :   Tg Tg 
probability density unchanged
p
Tg
In general, however,
  Tg Tg  .
Q. How can Alice encode using particle nature part only?
1

[

]

T

T
A. She begins with the symmetric state
 g g
[  ] is invariant to translations Tg :
Tg’
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[  ] Tg’+ =
[ ]
O(G)
g
for arbitrary  .
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outline
waves & asymmetry
particles & symmetry
complementarity
DEFINITION: Particle nature NP()
NP () = maximum mutual information between Alice and Bob
over all possible unitary preparations by Alice
using [  ] and all possible measuremts by Bob.
[ ]
Alice
Uj
000 001
…
...
...
 j = Uj
Holevo bound
Bob
101
[  ]Uj+
estimate parameter j
dimension of state space
N P (  )  ln(D)  S ( [  ])
[ ] 
1

T

T
 g g
O(G) g
logarithmic purity of [  ]
= symmetry of  with respect to G
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outline
waves & asymmetry
particles & symmetry
complementarity
Complementarity
wave
particle
NW (  )  S ( [  ])  S (  )
N P (  )  ln(D)  S ( [  ])
sum
NW (  )  N P (  )  ln(D)  S (  )
Group theoretic complementarity - general
NW (  )  N P (  )  ln(D)  S (  )
asymmetry
NW  
symmetry
N P  
ln(D)
 S ( )
QCMC’06
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outline
waves & asymmetry
particles & symmetry
complementarity
Complementarity
wave
particle
NW (  )  S ( [  ])  S (  )
N P (  )  ln(D)  S ( [  ])
sum
NW (  )  N P (  )  ln(D)  S (  )
Group theoretic complementarity – pure states
NW (  )  N P (  )  ln(D)
asymmetry
NW  
symmetry
N P  
ln(D)
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outline
waves & asymmetry
particles & symmetry
complementarity
NW (  )  N P (  )  ln(D)
Englert’s single photon interferometry
[PRL 77, 2154 (1996)]
0 = photon in upper path
a single photon
is prepared by
some means

N P (  )  NW (  )  1
( D  2)
1 = photon in lower path
group: G  {1,  z }
particle-like states (symmetric):
0,
1,
wave-like states (asymmetric):
 
1
2
0
translation:
1   ,
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 1 ,
 
1
2
0
N P  1, NW  0
 1 , N P  0, NW  1
z   
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outline
waves & asymmetry
Bipartite system
particles & symmetry
complementarity
NW (  )  N P (  )  ln(D)  S (  )
a new application of particle-wave duality
0
2 spin- ½ systems ( D  4)
N P (  )  NW (  )  2  S (  )
1

group: G  1 1, 1  x , 1  y , 1  z
particle-like states (symmetric):
0 0  12 1,
1 1  12 1

N P  1, NW  0, S (  )  1
wave-like states (asymmetric):
N P  0,
   12  0 0  1 1 
translation:

  G
   ,   ,   ,  
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Bell
G

NW  2, S (  )  0
(superdense coding)
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Summary
 Momentum prob. density with momentum translations:
de-localised
localised
p
particle-like
p
wave-like
 Information capacity of “wave” or “particle” nature:
Alice
...
Bob
...
 Complementarity
asymmetry
NW  
estimate parameter
symmetry
N P  
NW (  )  N P (  )  ln(D)  S (  )
ln(D)
 S ( )
 New Application - entangled states are wave like
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