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R. Demkowicz-Dobrzański 1 , J. Kołodyński 1 , M. Guta 2

Quantum metrology and the geometry of quantum channels the illusion of the Heisenberg scaling. R. Demkowicz-Dobrzański 1 , J. Kołodyński 1 , M. Guta 2 1 Faculty of Physics , Warsaw University , Poland 2 School of Mathematical Sciences , University of Nottingham , United Kingdom.

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R. Demkowicz-Dobrzański 1 , J. Kołodyński 1 , M. Guta 2

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  1. Quantum metrology and the geometry of quantum channels the illusion of the Heisenberg scaling R. Demkowicz-Dobrzański1, J. Kołodyński1, M. Guta2 1Faculty of Physics, Warsaw University, Poland 2 School of Mathematical Sciences, University of Nottingham, United Kingdom

  2. Interferometry atits (classical) limits LIGO - gravitationalwavedetector NIST - Cs fountainatomicclock Michelson interferometer Ramsey interferometry Precision limited by:

  3. N independent photons thebestestimator: Estimatoruncertainty: Standard Quantum Limit (Shotnoise limit)

  4. Entanglementenhanced precision Hong-Ou-Mandelinterference &

  5. Entanglementenhanced precision NOON states Estimator State preparation Measuremnt Heisenberg limit Standard Quantum Limit

  6. Whatarethefundamentalbounds inpresence of decoherence?

  7. General schemein q. metrology Input state of Nparticles phaseshift + decoherence measurement estimation Interferometerwithlosses (gravitationalwavedetectors) Qubitrotation + dephasing (atomicclockfrequencycallibrations)

  8. Localapproachusing Fisher information Cramer-Raobound: F – Fisher information (dependsonly on theinput state) No decoherence Withdecoherence • - Theoutput state ismixed • - Fisher Information, difficult to calculate • Optimalstates do not havesimplestructure Heisenberg scalingislosteven for infinitesimaldecoherence!!! • - OptimalNphoton state (maximalF=N2): • RDD, et al. PRA 80, 013825(2009), • U. Dorner, et al., PRL. 102, 040403 (2009) - Asymptoticanalyticallowerbound: J. Kolodynski, RDD, PRA 82,053804 (2010), S. Knysh, V. Smelyanskiy, G. Durkin PRA 83, (2011) B. M. Escher, et al.Nature Physics, 7, 406 (2011) (minimizationoverdifferent Kraus representations) Heisenberg scaling J. J. . Bollinger, W. M. Itano, D. J. Wineland, andD. J. Heinzen, Phys. Rev. A 54, R4649 (1996).

  9. Maximal quantum enhancement

  10. Heisenberg scalingislosteven for infinitesimaldecoherence!!! Canyouprove simpler, more general and moreintutive? Yes!!!

  11. Classicalsimulation of a quantum channel Convex set of quantum channels

  12. Classicalsimulation of a quantum channel Convex set of quantum channels Parameterdependencemoved to mixingprobabilities Before: After: By Markov property…. • K. Matsumoto, arXiv:1006.0300 (2010)

  13. Classicalsimulation of Nchannelsusedinparallel

  14. Classicalsimulation of Nchannelsusedinparallel =

  15. Classicalsimulation of Nchannelsusedinparallel =

  16. Precision boundsthanks to classicalsimulation • For unitarychannels Heisenberg scalingpossible • Generlicdecoherence model will manifest shotnoisescaling • To getthetighestbound we need to findtheclassicalsimulationwithlowestFcl

  17. The „Worst” classicalsimulation Quantum Fisher Informationat a givendependsonly on Itisenough to analize,,localclassicalsimulation’’: The „worst” classicalsimulation: Works for non-extremalchannels RDD,J. Kolodynski, M. Guta arXiv:1201.3940 (2012)

  18. Dephasing: derivation of theboundin 60 seconds! dephasing Choi-Jamiołkowskiisomorphism (positivieoperatorscorrespond to physicalmaps) RDD,J. Kolodynski, M. Guta, arXiv:1201.3940 (2012)

  19. Dephasing: derivation of theboundin 60 seconds! dephasing Choi-Jamiołkowskiisomorphism (positivieoperatorscorrespond to physicalmaps) RDD, J. Kolodynski, M. Guta, arXiv:1201.3940 (2012)

  20. Summary • Heisenberg scalingislost for a genericdecoherence channel even for infinitesimalnoise • Simple bounds on precision can be derivedusingtheclassicalsimulationidea • In caseclassicalsimulationdoes not work, chanelextensionmethodcan be used – less intuitive but powerful (implementable as a simplesemi-definiteprogram!) RDD,J. Kolodynski, M. Guta, arXiv:1201.3940 (2012)

  21. Gallery of decoherencemodels • on theboundary, extremal insidethe set of quantum channels fullrank • on theboundary, but non-extremal • on theboundary, non-extremal, • but -extremal, classicalsimulation possible • classicalsimulation • possible channel extensionmethod • channel extensionmethod

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