1. Complementarity of resources: work, entanglement, reference quant-ph/0501121 John Vaccaro Howard Wiseman Kurt Jacobs Fabio Anselmi University of Hertfordshire Hatfield, UK Griffith University Brisbane, Australia

2. S This talk • Superselection Rules (SSRs) • conservation of local particle number • general symmetry groups • Reference frames • asymmetry: ability to act as a reference • Extracting mechanical work • Bipartite systems under SSR • Accessible entanglement • Extracting local work • Hierarchy of restrictions/resources • Complementarity

3. Superselection Rules Conservation of particle number all physical operations conserve particle number cannot observe coherence between subspaces of different particle number effectively a superselection rule.

4. S S Expressed as symmetry group U(1) Operationally accessible states “crisp“ projective measurement of General symmetry group Symmetry group: unitary representation: Operationally accessible states system under SO(2)  reduced coherence

5. S Reference systems (frames) Reference frames break the symmetry preserve coherence of system Any system with asymmetry can act as a reference system (frame) for G A symmetric system cannot act as a reference. R Measure of asymmetry asymmetric system von Neumann entropy induced by G reference system

6. Properties of asymmetry i) ii) iff is symmetric: iii) does not increase for G-SSR operations Q iv) Synergy of is given by

7. Example of Abelian case (particle number) system: asymmetry reference: Pegg & Barnett (1989). asymmetry combined: -invariant to -coherence is preserved reduction in asymmetry (synergy):

8. Example of Abelian case (particle number) system: asymmetry reference: Pegg & Barnett (1989). asymmetry combined: -invariant to -coherence is preserved reduction in asymmetry (synergy):

9. 1 under SSR Extracting mechanical work

10. resource?

11. Synergy  advantage of acting as a composite system acting as single system acting separately Upper bound asymmetry is a resource

12. Bipartite systems under SSR Local action of the group: local G-SSR

13. can act as local & shared reference Local asymmetry i) iff is locally symmetric: ii) iii) does not increase for locally G-SSR operations Q iv) Synergy of is given by

14. Accessible entanglement under SSR Wiseman and Vaccaro, PRL 91, 097902 (2003). 17 fixed total number of N particles f n N B A 11 6 + + N-n n

15. Examples cross terms represent 1 particle at each site – no particle entropy Super-additivity: • releases “latent” entanglement • a kind of distillation

16. Extracting local work Oppenheim et al PRL 89, 180402 (2002)

17. equivalent method classically-correlated state with min entropy LOCC classical channel local extraction of work

18. equivalent method for pure states pure state classically-correlated state with min entropy LOCC dephase in Schmidt basis classical channel local extraction of work

19. Extracting local work under local SSR

20. For pure, globally-symmetric states classically-correlated state with min entropy dephase in Schmidt basis LOCC classical channel local extraction of work

21. local asymmetry local symmetry work can act as a local reference

22. local asymmetry local symmetry mechanical logical can act as a local reference

23. Recall examples for U(1) R S R ability to act as shared reference super-additivity of accessible entanglement =

24. Recall examples for U(1) R LOCC LOCC+LocalG S R ability to act as shared reference super-additivity of accessible entanglement =

25. Details of general group case Symmetry group: Unitary representation: Globally-symmetric states Locally accessible states

26. multiplicity “flavour” Elemental globally symmetric state “colour” “charge” where and are conjugate w.r.t.  : i.e.: 1 dim irrep General globally symmetric state

27. entropy associated with charge fluctuations entropy associated with mixing of i & j indices entanglement due to multiplicity indices reduced by entropy associated with

28. Complementarity of resources ability to act as a reference frame asym sym W RF ability to perform work M L asymmetry symmetry

29. Hierarchy of restrictions-resources - G LOCC for globally-symmetric states GG GG LOCC,

30. reference frames accessible entanglement and work complementarity of resources: reference ability versus mechanical work versus logical work Summary  triality