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1 QCMC’06 Joan Vaccaro Centre for Quantum Dynamics, Centre for Quantum Computer Technology Griffith University Brisbane Group theoretic formulation of complementarity
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
