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1QCMC’06
Joan VaccaroCentre for Quantum Dynamics,
Centre for Quantum Computer Technology
Griffith University
Brisbane
Group theoretic formulation of complementarity
Group theoretic formulation of complementarity
2QCMC’06
outline waves & asymmetry particles & symmetry complementarity
OutlineOutline
Bohr’s complementarity of physical properties mutually exclusive experiments needed to determine their values.
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)]
[reply to EPR PR 48, 696 (1935)]
122 VD
3QCMC’06
outline waves & asymmetry particles & symmetry complementarity
Elemental properties of Wave - Particle duality
x x
localised de-localised
particles are “asymmetric” waves are “symmetric”
(1) Position probability density with spatial translations:
(2) Momentum prob. density with momentum translations:
pp
localisedde-localised
particles are “symmetric” waves are “asymmetric”
Could use either to generalise particle and wave nature – we use (2) for this talk. [Operationally: interference sensitive to ]
4QCMC’06
outline waves & asymmetry particles & symmetry complementarity
In this talk
discrete symmetry groups G = {Tg}
measure of particle and wave nature is information capacity of asymmetric and symmetric parts of wavefunction
balance between (asymmetry) and (symmetry) wave particle Contents: waves and asymmetry particles and symmetry complementarity
)ln( )()( DNN PW
p pTg
Tg
Tg
5QCMC’06
outline waves & asymmetry particles & symmetry complementarity
Waves can carry information in their translation:
Waves & asymmetryWaves & asymmetry
Tg
Information capacity of “wave nature”:
group G = {Tg}, unitary representation: (Tg )1 = (Tg )+
g
g = Tg Tg+
000 001 … 101
symbolically :
Alice Bob
. . .. . .
gg
g TTGO
)(
1][
g
p
estimate parameter g
Tg
6QCMC’06
outline waves & asymmetry particles & symmetry complementarity
Tg
Tg
Waves can carry information in their translation:
Waves & asymmetryWaves & asymmetry
Information capacity of “wave nature”:
group G = {g}, unitary representation: {Tg for g G}
g
g = Tg Tg+
000 001 … 101
symbolically :
AliceBob
. . .. . .
gGg
g TTGO
)(
1][
p
estimate parameter g
0
1
wave-like states:
2
10,
2
10
group: },{ zG 1
Example: single photon interferometry
particle-like states:
?
translation: z,1
= photon in upper path
= photon in lower path
1,0
g
7QCMC’06
outline waves & asymmetry particles & symmetry complementarity
DEFINITION: Wave nature NW () NW () = maximum mutual information between Alice and Bob over all possible measurements by Bob.
)(])[()( SSNW
increase in entropy due to G= asymmetry of with respect to G
)ln(Tr)( SHolevo bound
000 001 … 101 Alice Bob
. . .. . .
estimate parameter g g = Tg Tg+
Tg
g
gg TTGO
)(
1][
8QCMC’06
outline waves & asymmetry particles & symmetry complementarity
Tg’ Tg’+ = for arbitrary .
Particles & symmetryParticles & symmetryParticle properties are invariant to translations Tg G
probability density unchanged
gg TT
g
gg TTGO
)(
1][
For “pure” particle state :
A. She begins with the symmetric state
p
In general, however,
Q. How can Alice encode using particle nature part only?
][ is invariant to translations Tg :
][ ][
. gg TT
Tg
9QCMC’06
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.
])[()ln()( SDNP
logarithmic purity of= symmetry of with respect to G
Holevo bound
000 001 … 101 Alice Bob
. . .
Uj
estimate parameter j j = Uj Uj+
][
][
. . .][
][
dimension of state space
g
gg TTGO
)(
1][
10QCMC’06
outline waves & asymmetry particles & symmetry complementarity
ComplementarityComplementarity
])[()ln()( SDNP )(])[()( SSNW
)()ln()()( SDNN PW
waveparticle
sum
)()ln()()( SDNN PW Group theoretic complementarity - general
PN
)()ln(S
D
WN
asymmetry symmetry
11QCMC’06
outline waves & asymmetry particles & symmetry complementarity
ComplementarityComplementarity
])[()ln()( SDNP )(])[()( SSNW
)()ln()()( SDNN PW
waveparticle
sum
)ln()()( DNN PW Group theoretic complementarity – pure states
PN
)ln(D
WN
asymmetry symmetry
12QCMC’06
outline waves & asymmetry particles & symmetry complementarity
1,0,10,102
12
1 WP NN
group: },{ zG 1
translation: z,1
0
1
wave-like states (asymmetric):
particle-like states (symmetric):
Englert’s single photon interferometry [PRL 77, 2154 (1996)]
a single photon is prepared by
some means
= photon in upper path
= photon in lower path
,1,0
1)()( WP NN
0,1 WP NN
)2( D
)ln()()( DNN PW
13QCMC’06
outline waves & asymmetry particles & symmetry complementarity
Bipartite system a new application of particle-wave duality
2 spin- ½ systems
)(2)()( SNN WP
)4( D
11002
1
group: zyxG 11111 ,,,
translation:
,,,G
wave-like states (asymmetric):
particle-like states (symmetric): 11 2
121 11,00 1)(,0,1 SNN WP
0)(,2,0 SNN WP
G Be
ll
(superdense coding)
)()ln()()( SDNN PW
1
0
14QCMC’06
SummarySummary Momentum prob. density with momentum translations:
pplocalisedde-localised
Information capacity of “wave” or “particle” nature:
Alice Bob. . .. . .
estimate parameter Complementarity
New Application - entangled states are wave like
PN
)()ln(S
D
WN
asymmetry symmetry
particle-like wave-like
)()ln()()( SDNN PW
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