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Quantum Metrology in Realistic Scenarios

Quantum Metrology in Realistic Scenarios. PART I – Quantum Metrology with Uncorrelated Noise Rafal Demkowicz-Dobrzanski , JK , Madalin Guta – ” The elusive Heisenberg limit in quantum metrology ”, Nat. Commun . 3, 1063 (2012) .

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Quantum Metrology in Realistic Scenarios

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  1. Quantum Metrology in Realistic Scenarios • PART I – Quantum Metrology with Uncorrelated Noise • RafalDemkowicz-Dobrzanski, JK, MadalinGuta –”The elusive Heisenberg limit in quantum metrology”, Nat. Commun. 3, 1063 (2012). • JK, RafalDemkowicz-Dobrzanski –”Efficient tools for quantum metrology with uncorrelated noise”, New J. Phys. 15, 073043 (2013). • PART II – Beating the Shot Noise Limit despite the Uncorrelated Noise • Rafael Chaves, Jonatan Bohr Brask, MarcinMarkiewicz, JK, Antonio Acin –”Noisy metrology beyond the Standard Quantum Limit”, Phys. Rev. Lett. 111, 120401 (2013). JanekKolodynski Faculty of Physics, University of Warsaw, Poland

  2. (Classical) Quantum Metrology Atomic Spectroscopy: “Phase” Estimation N two-levelatoms (qubits) in a separable state unitary rotation output state (separable) uncorrelated measurement – POVM: independent processes unbiased estimator with shot noise always saturablein the limit N → ∞ Quantum Fisher Information (measurement independent) Classical Fisher Information

  3. Quantum Fisher Information • QFI of a pure state : • QFI of a mixed state : • Evaluation requires eigen-decomposition of the density matrix, which size grows exponentially, e.g. d=2N for Nqubits. • Geometric interpretation – QFI is a local quantity • The necessity of the asymptotic limit of repetitions N → ∞ is a consequence of locality. • Purification-based definition of the QFI – Symmetric Logarithmic Derivative Two PDFs : Two q. states : [Escher et al, Nat. Phys. 7(5), 406 (2011)] – [Fujiwara & Imai, J. Phys. A 41(25), 255304 (2008)] –

  4. (Ideal) Quantum Metrology Atomic Spectroscopy: “Phase” Estimation GHZ state unitary rotation output state measurement on all probes: estimator: repeating the procedure k times • Atoms behave as a “single object” with N times greater phase change generated (same for the N00N state and photons). • N → ∞ is not enough to achieve the ultimate precision. “Real” resources are kNand in theory we require k → ∞. Heisenberg limit • A source of uncorrelated decoherenceacting independentlyon each atom will “decorrelate” the atoms, so that we may attain the ultimate precision in the N → ∞ limit with k = 1, but at the price of scaling …

  5. (Realistic) Quantum Metrology Atomic Spectroscopy: “Phase” Estimation with dephasingnoise added: In practise need to optimize for particular model and N optimal pure state distorted unitary rotation mixed output state complexity of computation grows exponentially with N estimator measurement on all probes • Observations • Infinitesimal uncorrelated disturbance forces asymptotic (classical) shot noise scaling. • The bound then “makes sense” for a single shot (k = 1). • Does this behaviour occur for decoherence of a generic type ? constant factor improvement over shot noise achievable with k = 1and spin-squeezed states The properties of the single use of a channel – – dictate the asymptotic ultimate scaling of precision.

  6. Efficient tools for determining d lower-bounding • In order of their power and range of applicability: • Classical Simulation (CS) method • Stems from the possibility to locally simulate quantum channelsvia classical probabilistic mixtures: • Optimal simulation corresponds to a simple, intuitive, geometric representation. • Proves that almost all (including full rank) channels asymptotically scale classically. • Allows to straightforwardly derive bounds (e.g. dephasing channel considered). • Quantum Simulation (QS) method • Generalizes the concept of local classical simulation, so that the parameter-dependent state does not need to be diagonal: • Proves asymptotic shot noise also for a wider class of channels (e.g. optical interferometer with loss). • Channel Extension (CE) method • Applies to evenwider class of channels, and provides the tightestlower bounds on . (e.g. amplitude damping channel) • Efficiently calculable numerically by means of Semi-Definite Programmingeven for finite N!!!.

  7. Classical/Quantum Simulation of a channel as a Markov chain:

  8. Classical/Quantum Simulation of a channel as a Markov chain:

  9. Classical/Quantum Simulation of a channel as a Markov chain:

  10. Classical/Quantum Simulation of a channel as a Markov chain: shot noise scaling !!! But how to verify if this construction is possible and what is the optimal (“worse”) classical/quantum simulation giving the tightest lower bound on the ultimate precision?

  11. The ”Worst” Classical Simulation The set of quantum channels (CPTP maps) is convex Locality: Quantum Fisher Information at a given : depends only on: We want to construct the ”local classical simulation” of the form: The ”worst” local classical simulation: Doesnot work for -extremal channels, e.gunitaries .

  12. Gallery of decoherence models insidethe set of quantum channels fullrank • on theboundary, • non-extremal, • not-extremal • on theboundary, • non-extremal, • but -extremal • on theboundary, • extremal

  13. Consequences on Realistic Scenarios • “phase estimation” in Atomic Spectroscopy with Dephasing(η = 0.9) GHZ strategy S-S strategy finite-N bound asymptotic bound worse than classical region better than Heisenberg Limit region input correlations dominated region uncorrelated decoherence dominated region worth investing in GHZ states e.g. N=3 [D. Leibfried et al, Science, 304 (2004)] worth investing in S-S states e.g. N=105 !!!!!!![R. J. Sewell et al, Phys. Rev. Lett. 109, 253605 (2012)]

  14. Consequences on Realistic Scenarios • Performance of GEO600 gravitional-wave interferometer - coherent beam squeezed vacuum /ns [Demkowicz-Dobrzanski et al, Phys. Rev. A 88, 041802(R) (2013)]

  15. Frequency Estimation in Ramsey Spectroscopy Estimation of t - extra free parameter Parallel dephasing: two Kraus operators – non-full rank channel – SQL-bounding Methodsapply for any t “Ellipsoid” dephasing: four Kraus operators – full rank channel – SQL-bounding Methodsapply for any t four Kraus operators – full rank channel – SQL-bounding Methodsapply, but… as t → 0 up to O(t2) – two Kraus operators SQL-bounding Methods fail !!! Transversal dephasing: Ramsey spectroscopy - Resources: Total time of the experiment , number of particles involved : Precision: ⁞ optimize over t ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞

  16. Ramsey Spectroscopy with Transversal Dephasing Beyond the shot noise !!! (a) Saturability with the GHZ states (b) Impact of parallel component (dotted) – parallel dephasing (dashed) – transversal dephasingwithoutt-optimisation (solid) – transversal dephasingwitht-optimisation [Chaves et al, Phys. Rev. Lett. 111, 120401 (2013)]

  17. Conclusions • Classically, for separable input states, the ultimate precision is bound to shotnoisescaling 1/√N, which can be attained in a single experimental shot (k=1). • For lossless unitary evolution highly entangled input states (GHZ, N00N) allow for ultimate precision that follows the Heisenberg scaling 1/N, but attaining this limit may in principle require infinite repetitions of the experiment (k→∞). • The consequences of the dehorence acting independently on each particle: • The Heisenbergscaling is lost and only a constant factor quantum enhancement over classical estimation strategies is allowed. • (?) The optimal input states in the N → ∞ limit achieve the ultimate precision in a single shot (k=1) and are of a simpler form: • spin-squeezed atomic - [Ulam-Orgikh, Kitagawa, Phys. Rev. A 64, 052106 (2001)]squeezed light states in GEO600 - [Demkowicz-Dobrzanski et al, Phys. Rev. A 88, 041802(R) (2013)]. • However, finding the optimal formof input states is still an issue. Classical scaling suggests local correlations: • Gaussian states – [Monras & Illuminati, Phys. Rev. A 81, 062326 (2010)]MPS states – [Jarzyna et al, Phys. Rev. Lett. 110, 240405 (2013)].

  18. Conclusions • We have formulated three methods: Classical Simulation, Quantum Simulation and Channel Extension; that may efficiently lower-bound the constant factor of the quantum asymptotic enhancement for a generic channel by properties of its single use form(Kraus operators). • The geometrical CS methodproves the cQ/√N for all full-rank channelsand more (e.g. dephasing). • The CE method may also be applied numerically for finite Nas a semi-definite program. • After allowing the form of channel to depend on N, what is achieved by the (exprimentally-motivated) single experimental-shot time period (t) optimisation, we establish a channel that, despite being full-rank for any finite t, achieves the ultimate super-classical1/N5/6 asymptotic – the transversal dephasing.Application to atomic magnetometry: [Wasilewski et al, Phys. Rev. Lett. 104, 133601 (2010)]. Thank you for Your attention

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