Report

QUANTUM METROLOGY IN REALISTIC SCENARIOS Janek Kolodynski Faculty of Physics, University of Warsaw, Poland 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). • JK, Rafal Demkowicz-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, Marcin Markiewicz, JK, Antonio Acin – ”Noisy metrology beyond the Standard Quantum Limit”, Phys. Rev. Lett. 111, 120401 (2013). (CLASSICAL) QUANTUM METROLOGY ATOMIC SPECTROSCOPY: “PHASE” ESTIMATION N two-level atoms (qubits) in a separable state unitary rotation output state (separable) uncorrelated measurement – POVM: independent processes unbiased estimator with shot noise Classical Fisher Information Quantum Fisher Information (measurement independent) always saturable in the limit N → ∞ QUANTUM FISHER INFORMATION • QFI of a pure state : • QFI of a mixed state : – Symmetric Logarithmic Derivative Evaluation requires eigen-decomposition of the density matrix, which size grows exponentially, e.g. d=2N for N qubits. • Geometric interpretation – QFI is a local quantity Two PDFs Two q. states : : The necessity of the asymptotic limit of repetitions N → ∞ is a consequence of locality. • Purification-based definition of the QFI [Escher et al, Nat. Phys. 7(5), 406 (2011)] – [Fujiwara & Imai, J. Phys. A 41(25), 255304 (2008)] – (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 kN and in theory we require k → ∞. Heisenberg limit • A source of uncorrelated decoherence acting independently on 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 … (REALISTIC) QUANTUM METROLOGY ATOMIC SPECTROSCOPY: “PHASE” ESTIMATION with dephasing noise added: optimal pure state In practise need to optimize for particular model and N distorted unitary rotation mixed output state measurement on all probes complexity of computation grows exponentially with N estimator 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 = 1 and spinsqueezed states The properties of the single use of a channel – – dictate the asymptotic ultimate scaling of precision. EFFICIENT TOOLS FOR DETERMINING LOWER-BOUNDING D In order of their power and range of applicability: o Classical Simulation (CS) method o Stems from the possibility to locally simulate quantum channels via classical probabilistic mixtures: o Optimal simulation corresponds to a simple, intuitive, geometric representation. o Proves that almost all (including full rank) channels asymptotically scale classically. o Allows to straightforwardly derive bounds (e.g. dephasing channel considered). o Quantum Simulation (QS) method o Generalizes the concept of local classical simulation, so that the parameter-dependent state does not need to be diagonal: o Proves asymptotic shot noise also for a wider class of channels (e.g. optical interferometer with loss). o Channel Extension (CE) method o Applies to even wider class of channels, and provides the tightest lower bounds on (e.g. amplitude damping channel) . o Efficiently calculable numerically by means of Semi-Definite Programming even for finite N !!!. CLASSICAL/QUANTUM SIMULATION OF A CHANNEL as a Markov chain: CLASSICAL/QUANTUM SIMULATION OF A CHANNEL as a Markov chain: CLASSICAL/QUANTUM SIMULATION OF A CHANNEL as a Markov chain: 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? 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: Does not work for -extremal channels, e.g unitaries . GALLERY OF DECOHERENCE MODELS 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)] CONSEQUENCES ON REALISTIC SCENARIOS PERFORMANCE OF GEO600 GRAVITIONAL-WAVE INTERFEROMETER /ns coherent beam squeezed vacuum [Demkowicz-Dobrzanski et al, Phys. Rev. A 88, 041802(R) (2013)] FREQUENCY ESTIMATION IN RAMSEY SPECTROSCOPY Estimation of t - extra free parameter Parallel dephasing: two Kraus operators – non-full rank channel – SQL-bounding Methods apply for any t “Ellipsoid” dephasing: four Kraus operators – full rank channel – SQL-bounding Methods apply for any t four Kraus operators – full rank channel – SQL-bounding Methods apply, 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 ⁞ RAMSEY SPECTROSCOPY WITH TRANSVERSAL DEPHASING Beyond the shot noise !!! (a) Saturability with the GHZ states (dotted) – parallel dephasing (dashed) – transversal dephasing without t-optimisation (solid) – transversal dephasing with t-optimisation (b) Impact of parallel component [Chaves et al, Phys. Rev. Lett. 111, 120401 (2013)] CONCLUSIONS • Classically, for separable input states, the ultimate precision is bound to shot noise scaling 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 Heisenberg scaling 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 form of 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)]. 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 method proves the cQ/√N for all full-rank channels and more (e.g. dephasing). • The CE method may also be applied numerically for finite N as 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-classical 1/N5/6 asymptotic – the transversal dephasing. Application to atomic magnetometry: [Wasilewski et al, Phys. Rev. Lett. 104, 133601 (2010)]. THANK YOU FOR YOUR ATTENTION