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SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston Graham White Howard Wiseman Joan V Griffith Uni. arXive:quant-ph/0501121v2 Quantum Reference Frames superselection rules, reference ancilla & entanglement

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SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 2 Superselection Rules (SSRs) –restricted operations –general symmetry groups Reference & Asymmetry –asymmetry: ability to act as a reference Work - a measure of purity Entanglement - limited by SSR Trade off between resources Etcetera… S Overview

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RQ I W Nov 07 SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 3 Wick, Wightman & Wigner, Phys. Rev. 80, 101 (1952). “We shall say that a superselection rule operates between subspaces … if a selection rule operates between them… and if, … there are no measurable quantities with finite matrix elements between their state vectors.” Superselection Rules (SSRs) Selection rules forbid transitions of a given kind – m = 2 not allowed for optical dipole transitions etc but not transitions of any kind – e.g. electron collisions etc.

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RQ I W Nov 07 SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 4 impose the rule: physical operations conserve local particle number then coherence between subspaces of different particle number are non detectable an imposed superselection rule. E.g. optics: the phase in is unobservable …. Example: local conservation of particle number Y.Aharonov and L.Susskind, Phys. Rev. 155, 1428 (1967). A. Kitaev, D. Mayers, and J. Preskill, Phys. Rev. A 69, 052326 (2004). S.D. Bartlett, T. Rudolph, R.W. Spekkens, Rev. Mod. Phys. 79, 555 (2007)

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RQ I W Nov 07 SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 5 impose the rule: physical operations conserve local particle number then coherence between subspaces of different particle number are non detectable an imposed superselection rule. E.g. optics: the phase in is unobservable …. except relative to a local oscillator (a reference phase) Y.Aharonov and L.Susskind, Phys. Rev. 155, 1428 (1967). A. Kitaev, D. Mayers, and J. Preskill, Phys. Rev. A 69, 052326 (2004). S.D. Bartlett, T. Rudolph, R.W. Spekkens, Rev. Mod. Phys. 79, 555 (2007) Example: local conservation of particle number reference Pegg… PRL 81 1604 (1998) quantum scissors, phase shift

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RQ I W Nov 07 SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 6 Consider set of unitary operators whose effect is not physically detectable: G = {T 1, T 2, T 3, …} Non-detectable operations form a group the effect of a product of two such operators is also non-detectable, thus clearly the identity operator is in G Thus G = {T 1, T 2, T 3, … } is a group which expresses the symmetry of the system if effect of T i is not detectable then effect of time-reversed operator T i 1 is also not detectable, i.e.

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RQ I W Nov 07 SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 7 Accessible state the effective state given the undetectable coherences S no reference G=SO(2) S “crisp“ Bartlett and Wiseman, PRL 91, 097903 (2003). Ex 2: optical phase shifts are non-detectable (without a reference) reduced purity “The Twirl” Ex 1: rotations are non detectable without a spatial reference

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RQ I W Nov 07 SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 8 S “crisp“ Accessible state the effective state given the undetectable coherences reduced purity “The Twirl” Ex 1: rotations are non detectable without a spatial reference Ex 2: optical phase shifts are non-detectable (without a reference) Bartlett and Wiseman, PRL 91, 097903 (2003). no reference equally likely to be any value of effective state has random phase

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RQ I W Nov 07 Work (purity) SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 9 Extracting work (purity measure)

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RQ I W Nov 07 Work (purity) SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 10 subtract initial entropy Extracting work (purity measure) von Neumann entropy dim.

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RQ I W Nov 07 Work (purity) SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 11 1 under G-SSR the extractable work is Extracting work (purity measure) subtract initial entropy von Neumann entropy dim.

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RQ I W Nov 07 Ref & Asymmetry SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 12 Reference ancilla (frame) the SSR imposes a symmetry which reduces the purity we need to break the symmetry & preserve the coherence this requires an asymmetric ancilla define symmetric state as one for which define asymmetric state as one for which The Twirl use loss of purity to measure asymmetry von Neumann entropy Asymmetry

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RQ I W Nov 07 Ref & Asymmetry SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 13 iff is symmetric: Asymmetry (reference ability) does not increase for G-SSR operations Q Synergy of is given by i) ii) iii) iv) any ancilla with asymmetry can act as a reference to (partially) break the SSR Properties of Asymmetry: R reference ancilla system S

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RQ I W Nov 07 Ref & Asymmetry SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 14 S R gGgG fGfG acting separately acting as single system Upper bound asymmetry is a resource advantage of acting as a composite system Synergy S gGgG R

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RQ I W Nov 07 Ref & Asymmetry SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 15 Example: local conservation of particles [U(1)] decoherent free subspaces (superselection sectors) coherence is preserved system: ref. ancilla: combined (ref. ancilla + system): Pegg & Barnett (1989). invariant to group:

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RQ I W Nov 07 Ref & Asymmetry SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 16 combined: state A G Synergy of A G : the reduction in entropy due to combined action S R

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RQ I W Nov 07 Ref & Asymmetry SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 17 asymmetric symmetric

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RQ I W Nov 07 Entangle SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 18 Bipartite systems & Enganglement Local action of the group: local G-SSR

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RQ I W Nov 07 Entangle SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 19 iff is locally symmetric: Local asymmetry does not increase for locally G-SSR operations Q Synergy of is given by i) ii) iii) iv) can act as local & shared reference

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RQ I W Nov 07 Entangle SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 20 Super-additivity: Accessible entanglement projection onto n particles at A Examples: N particles shared between A and B Wiseman and Vaccaro, PRL 91, 097902 (2003).

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RQ I W Nov 07 Entangle SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 21 Extracting local work Oppenheim et al PRL 89, 180402 (2002)

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RQ I W Nov 07 Entangle SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 22 classically-correlated state with min entropy LOCC local extraction of work equivalent method transfer using a classical channel

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RQ I W Nov 07 Entangle SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 23 transfer using a classical channel pure state dephase in Schmidt basis equivalent method for pure states LOCC classically-correlated state with min entropy local extraction of work

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RQ I W Nov 07 Entangle SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 24 transfer using a classical channel pure state equivalent method for pure states LOCC classically-correlated state with min entropy local extraction of work dephase in Schmidt basis

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RQ I W Nov 07 Entangle SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 25 transfer using a classical channel Pure, globally symmetric states LOCC local extraction of work classically-correlated state with min entropy dephase in Schmidt basis for each charge Extracting local work under local SSR

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RQ I W Nov 07 Entangle SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 26 transfer using a classical channel Pure, globally symmetric states LOCC local extraction of work classically-correlated state with min entropy dephase in Schmidt basis for each charge Extracting local work under local SSR

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RQ I W Nov 07 Trade off SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 27 mechanical work logical work symmetry asymmetry reference Tradeoff

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RQ I W Nov 07 Trade off SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 28 Recall examples for U(1) SR R ability to act as shared reference super-additivity of accessible entanglement =

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RQ I W Nov 07 Trade off SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 29 Recall examples for U(1) SR R ability to act as shared reference super-additivity of accessible entanglement =

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RQ I W Nov 07 Trade off SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 30 Optimum shared reference states? make zero make maximum

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RQ I W Nov 07 Trade off SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 31 Hierarchy of restrictions-resources LOCC G GGGG LOCC, GGGG - for globally -symmetric states

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RQ I W Nov 07 Etcetera SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 32 Etcetera… Complete reference frame when then system is completely “shielded” from G Normalised synergy of asymmetry: Figure of merit - Quality system state reference: quality (M=30) N

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RQ I W Nov 07 Etcetera SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera 33 repeated use of a reference ancilla with independent systems reduces its reference ability… Consumption of reference ability Complementarity – generalisation S1S1 R S2S2 R’ The symmetry-asymmetry dichotomy is fundamental to a system. Arises from its “geometry”. It may help understanding of the fundamental particle-wave duality in terms of a symmetry-asymmetry dichotomy.

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SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 34 reference ancilla accessible entanglement and work tradeoff of resources: reference ability versus mechanical work versus logical work triality Summary

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SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 35

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