Joan Vaccaro

1
Group theoretic formulation of complementarity
Joan Vaccaro Centre for Quantum Dynamics, Centre
for Quantum Computer Technology Griffith
University Brisbane
2
Outline
Bohrs 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)
3
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
localised
de-localised
p
p
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 ???
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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
Tg
p
p
Tg
Tg
Contents ? waves and asymmetry ? particles
and symmetry ? complementarity
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Waves asymmetry
Waves can carry information in their translation
group G Tg, unitary representation (Tg
)?1 (Tg )?
Tg
symbolically
? g Tg ? Tg
p
?
? g
Information capacity of wave nature
Tg
Alice
Bob
000 001 101
. . .
. . .
? g
?
estimate parameter g
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Waves asymmetry
Waves can carry information in their translation
group G g, unitary representation Tg
for g ? G
Example single photon interferometry
photon in upper path
Tg
symbolically
? g Tg ? Tg
?
p
?
? g
photon in lower path
Information capacity of wave nature
particle-like states
Tg
Bob
wave-like states
Alice
000 001 101
. . .
. . .
group
? g
?
translation
estimate parameter g
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DEFINITION Wave nature NW (?) NW (?)
maximum mutual information between Alice and
Bob over all possible
measurements by Bob.
Tg
Alice
Bob
000 001 101
. . .
. . .
? g Tg ? Tg
estimate parameter g
Holevo bound
increase in entropy due to G asymmetry of ?
with respect to G
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Particles symmetry
Particle properties are invariant to translations
Tg ? G
For pure particle state
probability density unchanged
p
Tg
In general, however,
Q. How can Alice encode using particle nature
part only?
A. She begins with the symmetric state
is invariant to translations Tg
Tg Tg for
arbitrary ? .
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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.
Uj
Alice
Bob
000 001 101
. . .
. . .
? j Uj Uj
estimate parameter j
Holevo bound
dimension of state space
logarithmic purity of symmetry of ? with
respect to G
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Complementarity
wave
particle
sum
Group theoretic complementarity - general
asymmetry
symmetry
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Complementarity
wave
particle
sum
Group theoretic complementarity pure states
asymmetry
symmetry
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Englerts single photon interferometry PRL
77, 2154 (1996)
photon in upper path
?
a single photon is prepared by some means
photon in lower path
group
particle-like states (symmetric)
wave-like states (asymmetric)
translation
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Bipartite system a new application of
particle-wave duality
2 spin- ½ systems
G
Bell
group
particle-like states (symmetric)
wave-like states (asymmetric)
translation
(superdense coding)
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Summary
? Momentum prob. density with momentum
translations
de-localised
localised
p
p
particle-like
wave-like
? Information capacity of wave or particle
nature
? Complementarity
? New Application - entangled states are wave like