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Sense and sensitivity: ,, robust ’’ quantum phase estimation

Sense and sensitivity: ,, robust ’’ quantum phase estimation. R. Demkowicz-Dobrzański 1 , K. Banaszek 1 , U. Dorner 2 , I. A. Walmsley 2 , W. Wasilewski 1 , B. Smith 2 , J. Lundeen 2 , M. Kacprowicz 3 , J. Kołodyński 1 1 Faculty of Physics , Warsaw University , Poland

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Sense and sensitivity: ,, robust ’’ quantum phase estimation

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  1. Sense and sensitivity: ,,robust’’quantum phase estimation R. Demkowicz-Dobrzański1, K. Banaszek1, U. Dorner2, I. A. Walmsley2, W. Wasilewski1, B. Smith2, J. Lundeen2, M. Kacprowicz3, J. Kołodyński1 1Faculty of Physics, Warsaw University, Poland 2Clarendon Laboratory, University of Oxford, United Kingdom 3Institute of Physics, Nicolaus CopernicusUniversity, Toruń, Poland

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

  3. Classicalphaseestimation detectingn1 and n2 + knowingtheoreticaldependence of n1, n2 on  we canestimate

  4. Classicalphaseestimation n1 and n2aresubject to shotnoise eachmeasurementyields a bit different Shotnoisescaling

  5. Quantum phaseestimation state preparation measurement estimation sensing a priori knowledge In general a 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: 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. In reality thereisloss…

  8. Phaseestimationinthepresence of loss state preparation measurement estimation sensing + loss

  9. Phaseestimationinthepresence of loss state preparation measurement estimation sensing + loss • no analytical solutions for the optimal states and precission • calculating Fisher information not trivial (symmetriclogarithmicderrivative) • phase sensing and loss commute (no ambiguity in ordering) • in the global approach the optimal measurements is not altered – thesolutionisobtained by solving an eigenvalueproblem (fast) • effective numerical optimization procedures yielding global minima • R. Demkowicz-Dobrzanski, et al. Phys. Rev. A 80, 013825 (2009) • U. Dorner, et al.., Phys. Rev. Lett. 102, 040403 (2009)

  10. Estimationuncertaintywiththenumber of photonsused (localapproach) Whatisthescaling? Heisenberg scaling

  11. Estimationuncertaintywiththenumber of photonsused (localapproach) NOON state Whatisthescaling? Heisenberg scaling

  12. Estimationuncertaintywiththenumber of photonsused (global approach) Whatisthescaling?

  13. Do quantum statesprovidebeterscalingexponentinthepresence of loss?

  14. Fundamentalbound on uncertaintyinthepresence of loss (global approach) state preparation measurement estimation sensing + loss J. Kolodynski and R.Demkowicz-Dobrzanski, arXiv:1006.0734 (2010)

  15. Fundamentalbound on uncertaintyinthepresence of loss (global approach) localapproach: global approach:

  16. Fundamentalbound on uncertaintyinthepresence of loss (global approach) localapproach: global approach: analyticalbound for

  17. Fundamentalbound on asymptotic quantum gaininphaseestimation Example: even for moderatelossquantum gaindegradesquickly

  18. Summary • Asymptotically, lossrenders quantum phaseestimationuncertaintyscalingclassical and destroysthe Heisenberg scaling. • Quantum state can be practicallyusefulonly for verysmalldegree of loss (loss <1% impliesgain> 10) • Neitheradaptivemeasurements, nor photondistinguishabilitycan help K. Banaszek, R. Demkowicz-Dobrzanski, and I. Walmsley, Nature Photonics 3, 673 (2009) U. Dorner, R. Demkowicz-Dobrzanski, B. Smith, J. Lundeen, W. Wasilewski, K. Banaszek, andI..Walmsley, Phys. Rev. Lett. 102, 040403 (2009) R. Demkowicz-Dobrzanski, U. Dorner, B. Smith, J. Lundeen, W. Wasilewski, K. Banaszek, andI. Walmsley, Phys. Rev. A 80, 013825 (2009) M. Kacprowicz, R. Demkowicz-Dobrzanski, W. Wasilewski, and K. Banaszek, NaturePhotonics 4, 357(2010) J. Kolodynski and R.Demkowicz-Dobrzanski, arXiv:1006.0734 (2010)

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