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

2. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 2 Outline 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)]

3. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 3 Elemental properties of Wave - Particle duality xx localisedde-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  ]

4. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 4 In this talk  discrete symmetry groups G = {T g }  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 pp TgTg TgTg TgTg

5. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 5 Waves can carry information in their translation: Waves & asymmetry TgTg Information capacity of “wave nature”: group G = {T g }, unitary representation: ( T g )  1 = ( T g )  +  g  g = T g  T g + 000 001 … 101 symbolically : AliceBob...  g g p estimate parameter g TgTg 

6. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 6 TgTg TgTg  Waves can carry information in their translation: Waves & asymmetry Information capacity of “wave nature”: group G = {g}, unitary representation: { T g for g  G}  g  g = T g  T g + 000 001 … 101 symbolically : Alice Bob... p estimate parameter g wave-like states: group: Example: single photon interferometry particle-like states: ? translation: = photon in upper path = photon in lower path  g g

7. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 7 D EFINITION: Wave nature N W (  ) N W (  ) = maximum mutual information between Alice and Bob over all possible measurements by Bob. increase in entropy due to G = asymmetry of  with respect to G Holevo bound 000 001 … 101 AliceBob... estimate parameter g  g = T g  T g + TgTg

8. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 8 T g’ T g’ + = for arbitrary . Particles & symmetry Particle properties are invariant to translations T g  G probability density unchanged 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 T g : TgTg

9. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 9 D EFINITION: Particle nature N P (  ) N P (  ) = maximum mutual information between Alice and Bob over all possible unitary preparations by Alice using and all possible measure mts by Bob. logarithmic purity of = symmetry of  with respect to G Holevo bound 000 001 … 101 AliceBob... UjUj estimate parameter j  j = U j U j +... dimension of state space

10. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 10 Complementarity wave particle sum Group theoretic complementarity - general asymmetry symmetry

11. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 11 Complementarity wave particle sum Group theoretic complementarity – pure states asymmetry symmetry

12. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 12 group: translation: 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 

13. QCMC’06 outline waves & asymmetry particles & symmetry complementarity 13 Bipartite system a new application of particle-wave duality 2 spin- ½ systems group: translation: wave-like states (asymmetric): particle-like states (symmetric): G Bell (superdense coding)

14. QCMC’06 14 Summary  Momentum prob. density with momentum translations: pp localised de-localised  Information capacity of “wave” or “particle” nature: AliceBob... estimate parameter  Complementarity  New Application - entangled states are wave like asymmetry symmetry particle-likewave-like