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R. Demkowicz-Dobrzański 1 , J. Kołodyński 1 , K. Banaszek 1 , M. Jarzyna 1 , M. Guta 2 1 Faculty of Physics , Warsaw

Theoretical tools for Quantum-enhanced metrology the illusion of the Heisenberg scaling. R. Demkowicz-Dobrzański 1 , J. Kołodyński 1 , K. Banaszek 1 , M. Jarzyna 1 , M. Guta 2 1 Faculty of Physics , Warsaw University , Poland

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R. Demkowicz-Dobrzański 1 , J. Kołodyński 1 , K. Banaszek 1 , M. Jarzyna 1 , M. Guta 2 1 Faculty of Physics , Warsaw

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  1. Theoreticaltools for Quantum-enhanced metrology the illusion of the Heisenberg scaling R. Demkowicz-Dobrzański1, J. Kołodyński1, K. Banaszek1, M. Jarzyna1, 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. Whatarethefundamentalbounds inpresence of decoherence?

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

  5. General schemein q. metrology Input state of Nparticles phaseshift + decoherence measurement estimation a priori knowledge Veryhard problem!

  6. Global approach Localapproach no a priori knowledgeaboutthephase we want to sensesmallfluctuationsaround a knownphase Tool: Symmetryimplies a simplestructure of theoptimalmeasurement Tool: Fisher Information, Cramer-Raobound Optimal state: Theoptimal N photon state for interferometry: D. W. Berry and H. M. Wiseman, Phys. Rev. Lett.85, 5098 (2000). J. J. . Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649 (1996). Heisenberg scaling

  7. Impact of decoherence? Localapproach Global approach Tool: Symmetryimplies a simplestructure of theoptimalmeasurement Tool: Fisher Information, Cramer-Raobound • Fisher Information, moredifficult to calculate • Optimalstates do not havesimplestructure Heisenberg scalinglost! • No analyticalformulas for theoptimalprecision and states • nontrivialeigenvalue problem • RDD, et al. PRA 80, 013825(2009) • U. Dorner, et al., PRL. 102, 040403 (2009) Analyticallowerbound: Analyticallowerbound: J. Kolodynski, RDD, PRA 82,053804 (2010) • S. Knysh, V. Smelyanskiy, G. Durkin PRA 83, (2011)

  8. Fundamentalbound on quantum enhancement of precision

  9. General methodfor otherdecoherencemodels? Fisher information via purifications System + Environment • B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nature Physics, 7, 406 (2011) • A. Fujiwara and H. Imai, J. Phys. A: Math. Theor.,41, 255304 (2008).

  10. General methodfor otherdecoherencemodels? Fisher information via purifications Kraus representation Equivalent Kraus set Minmizationoverdifferent Kraus representationnon-trivial dephasing • B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nature Physics, 7, 406 (2011)

  11. Canyou do itsimpler, more general, moreintutive? Yes!!!

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

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

  14. Classicalsimulation of Nchannelsusedinparallel

  15. Classicalsimulation of Nchannelsusedinparallel =

  16. Classicalsimulation of Nchannelsusedinparallel =

  17. Precision boundsthanks to classicalsimulation • For unitarychannels Heisenberg scalingpossible • Generlicdecoherence model will manifest shotnoisescaling • To getthetighestbound we need to findthe „worst” classicalsimulation

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

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

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

  21. Summary • Heisenberg scalingislost for a genericdecoherence channel even for infinitesimalnoise • Simple bounds on precision can be derivedusingclassicalsimulation idea • Channels for whichclassicalsimulationdoes not work • ( extremalchannels) have less Kraus operators, othermethodseasier to apply RDD,M. Guta, J. Kolodynski, arXiv:1201.3940 (2012)

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