Joan Vaccaro Centre for Quantum Dynamics, Centre for Quantum Computer Technology

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

2. Outline Bohr’s complementarity of physical properties mutually exclusiveexperimentsneeded to determine their values. [reply to EPRPR 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 withmomentum translations: de-localised localised p p particles are “symmetric” waves are “asymmetric” Could use either to generaliseparticle and wave nature– we use (2)for this talk. [Operationally: interference sensitive to ]

4. In this talk discretesymmetry groupsG = {Tg}  measure of particle and wave nature isinformation capacity of asymmetric and symmetric parts of wavefunction balance between (asymmetry) and (symmetry) waveparticle Tg p p Tg Tg Contents: waves and asymmetry particles and symmetry complementarity

5. Waves & asymmetry Waves can carry information in their translation: groupG = {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

6. Waves & asymmetry Waves can carry information in their translation: groupG = {g},unitary representation: {Tgfor 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: Bob Tg wave-like states: Alice 000 001 … 101 . . . . . . group: g  translation: estimate parameter g

7. DEFINITION:Wave natureNW () 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 = asymmetryof with respect to G

8. 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  .

9. DEFINITION:Particle natureNP() NP ()= maximum mutual information between Alice and Bob over allpossible 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

10. Complementarity wave particle sum Group theoretic complementarity - general asymmetry symmetry

11. Complementarity wave particle sum Group theoretic complementarity – pure states asymmetry symmetry

12. Englert’s 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:

13. Bipartite systema new application of particle-wave duality 2 spin- ½ systems G Bell group: particle-like states (symmetric): wave-like states(asymmetric): translation: (superdense coding)

14. Alice Bob . . . . . . estimate parameter asymmetry symmetry 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