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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)] D2 V 2 1 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 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 p Tg p Tg balance between (asymmetry) and (symmetry) wave particle Contents: NW ( ) N P ( ) ln( D) 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 [ ] QCMC’06 … 1 T T g g O(G ) g estimate parameter g 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}, 0 Tg symbolically : ? = photon in upper path g = Tg Tg+ p g 1 = photon in lower path Information capacity of “wave nature”: particle-like states: Alice Tg 000 001 ... g [ ] QCMC’06 … wave-like states: 101 group: ... translation: 1 T T g g O(G ) gG 0,1 0 1 2 G {1, z } 1 , Bob , 0 1 2 z estimate parameter g 6 outline waves & asymmetry particles & symmetry complementarity DEFINITION: Wave nature NW () NW () = maximum mutual information between Alice and Bob over all possible measurements by Bob. Tg Alice 000 001 … Bob 101 ... ... g = Tg Tg+ estimate parameter g Holevo bound S ( ) Tr ( ln ) NW ( ) S ( [ ]) S ( ) [ ] 1 T T g g O(G ) g 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 : Tg Tg probability density unchanged p Tg In general, however, Tg Tg . Q. How can Alice encode using particle nature part only? 1 [ ] T T A. She begins with the symmetric state g g [ ] is invariant to translations Tg : Tg’ QCMC’06 [ ] Tg’+ = [ ] O(G ) g 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 NP ( ) ln( D) S ( [ ]) [ ] 1 T T g g O(G ) g logarithmic purity of [ ] = symmetry of with respect to G QCMC’06 9 outline waves & asymmetry Complementarity particles & symmetry wave particle NW ( ) S ( sum NW ( ) N P ( ) ln( D) S ( ) complementarity [ ]) S ( ) N P ( ) ln( D) S ( [ ]) Group theoretic complementarity - general NW ( ) N P ( ) ln( D) S ( ) asymmetry NW symmetry N P ln( D ) S ( ) QCMC’06 10 outline waves & asymmetry Complementarity particles & symmetry wave particle NW ( ) S ( sum NW ( ) N P ( ) ln( D) S ( ) complementarity [ ]) S ( ) N P ( ) ln( D) S ( [ ]) Group theoretic complementarity – pure states NW ( ) N P ( ) ln( D) asymmetry NW symmetry N P ln( D) QCMC’06 11 outline waves & asymmetry particles & symmetry complementarity NW ( ) N P ( ) ln( D) Englert’s single photon interferometry [PRL 77, 2154 (1996)] 0 = photon in upper path a single photon is prepared by some means N P ( ) NW ( ) 1 ( D 2) 1 = photon in lower path group: G {1, z } particle-like states (symmetric): 0, 1, wave-like states (asymmetric): 1 2 0 translation: 1 , QCMC’06 1 , 1 2 0 N P 1, NW 0 1 , N P 0, NW 1 z 12 outline waves & asymmetry particles & symmetry complementarity NW ( ) N P ( ) ln( D) S ( ) a new application of particle-wave duality Bipartite system 0 2 spin- ½ systems ( D 4) N P ( ) NW ( ) 2 S ( ) 1 group: G 1 1, 1 x , 1 y , 1 z particle-like states (symmetric): 0 0 12 1, 1 1 12 1 N P 1, NW 0, S ( ) 1 wave-like states (asymmetric): N P 0, 12 0 0 1 1 translation: QCMC’06 Bell G G , , , NW 2, S ( ) 0 (superdense coding) 13 Summary Momentum prob. density with momentum translations: de-localised localised p particle-like p wave-like Information capacity of “wave” or “particle” nature: Alice ... Bob ... Complementarity asymmetry NW estimate parameter symmetry N P NW ( ) N P ( ) ln( D) S ( ) ln( D ) S ( ) New Application - entangled states are wave like QCMC’06 14