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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. S. This talk. Superselection Rules (SSRs)

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## Fabio Anselmi

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**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**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**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.**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**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**Properties of asymmetry**i) ii) iff is symmetric: iii) does not increase for G-SSR operations Q iv) Synergy of is given by**Example of Abelian case (particle number)**system: asymmetry reference: Pegg & Barnett (1989). asymmetry combined: -invariant to -coherence is preserved reduction in asymmetry (synergy):**Example of Abelian case (particle number)**system: asymmetry reference: Pegg & Barnett (1989). asymmetry combined: -invariant to -coherence is preserved reduction in asymmetry (synergy):**1**under SSR Extracting mechanical work**Synergy **advantage of acting as a composite system acting as single system acting separately Upper bound asymmetry is a resource**Bipartite systems under SSR**Local action of the group: local G-SSR**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**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**Examples**cross terms represent 1 particle at each site – no particle entropy Super-additivity: • releases “latent” entanglement • a kind of distillation**Extracting local work**Oppenheim et al PRL 89, 180402 (2002)**equivalent method**classically-correlated state with min entropy LOCC classical channel local extraction of work**equivalent method for pure states**pure state classically-correlated state with min entropy LOCC dephase in Schmidt basis classical channel local extraction of work**For pure, globally-symmetric states**classically-correlated state with min entropy dephase in Schmidt basis LOCC classical channel local extraction of work**local asymmetry**local symmetry work can act as a local reference**local asymmetry**local symmetry mechanical logical can act as a local reference**Recall examples for U(1)**R S R ability to act as shared reference super-additivity of accessible entanglement =**Recall examples for U(1)**R LOCC LOCC+LocalG S R ability to act as shared reference super-additivity of accessible entanglement =**Details of general group case**Symmetry group: Unitary representation: Globally-symmetric states Locally accessible states**multiplicity “flavour”**Elemental globally symmetric state “colour” “charge” where and are conjugate w.r.t. : i.e.: 1 dim irrep General globally symmetric state**entropy associated with charge fluctuations**entropy associated with mixing of i & j indices entanglement due to multiplicity indices reduced by entropy associated with**Complementarity of resources**ability to act as a reference frame asym sym W RF ability to perform work M L asymmetry symmetry**Hierarchy of restrictions-resources**- G LOCC for globally-symmetric states GG GG LOCC,**reference frames**accessible entanglement and work complementarity of resources: reference ability versus mechanical work versus logical work Summary triality

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