Group theoretic formulation of complementarity Joan Vaccaro Centre

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

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

outline waves & asymmetry particles & symmetry complementarity 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: de-localised p particles are “symmetric” p waves are “asymmetric” Could use either to generalise particle and wave nature – we use (2) for this talk. [Operationally: interference sensitive to ] QCMC’ 06 3

outline waves & asymmetry particles & symmetry complementarity In this talk Tg discrete symmetry groups G = {Tg} measure of particle and wave nature is information capacity of asymmetric and symmetric parts of wavefunction Tg p p Tg balance between (asymmetry) and (symmetry) wave particle Contents: waves and asymmetry particles and symmetry complementarity QCMC’ 06 4

outline waves & asymmetry particles & symmetry complementarity Waves & asymmetry Waves can carry information in their translation: group G = {Tg}, unitary representation: (Tg ) 1 = (Tg ) + Tg symbolically : g g = Tg Tg+ p Information capacity of “wave nature”: Alice Tg 000 001 . . . … Bob 101 . . . g estimate parameter g QCMC’ 06 5

outline waves & asymmetry particles & symmetry complementarity Waves & asymmetry Waves can carry information in their translation: unitary representation: {Tinterferometry g for g G} Example: single photon group G = {g}, = photon in upper path Tg symbolically : ? p g g = Tg Tg+ = photon in lower path Information capacity of “wave nature”: particle-like states: Alice Tg 000 001 . . . g … wave-like states: Bob 101 group: . . . translation: estimate parameter g QCMC’ 06 6

outline waves & asymmetry particles & symmetry complementarity DEFINITION: Wave nature NW ( ) = maximum mutual information between Alice and Bob over all possible measurements by Bob. Alice Tg 000 001 … Bob 101 . . . g = Tg + estimate parameter g Holevo bound increase in entropy due to G = asymmetry of with respect to G QCMC’ 06 7

outline waves & asymmetry particles & symmetry complementarity 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’ QCMC’ 06 Tg’+ = for arbitrary . 8

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. Alice Uj 000 001 … . . . j = Uj Holevo bound Bob 101 Uj + estimate parameter j dimension of state space logarithmic purity of = symmetry of with respect to G QCMC’ 06 9

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

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

outline waves & asymmetry particles & symmetry Englert’s single photon interferometry complementarity [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: QCMC’ 06 12

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

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