1. Q UANTUM M ETROLOGY IN R EALISTIC S CENARIOS Janek Kolodynski Faculty of Physics, University of Warsaw, Poland PART I – Q UANTUM M ETROLOGY WITH U NCORRELATED N OISE 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 – B EATING THE S HOT N OISE L IMIT DESPITE THE U NCORRELATED N OISE Rafael Chaves, Jonatan Bohr Brask, Marcin Markiewicz, JK, Antonio Acin – ”Noisy metrology beyond the Standard Quantum Limit”, Phys. Rev. Lett. 111, 120401 (2013).

2. (C LASSICAL ) Q UANTUM M ETROLOGY A TOMIC S PECTROSCOPY : “P HASE ” E STIMATION estimator shot noise N two-level atoms (qubits) in a separable state unitary rotation output state (separable) uncorrelated measurement – POVM: Quantum Fisher Information (measurement independent) independent processes Classical Fisher Information always saturable in the limit N → ∞ with unbiased

3. 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=2 N for N qubits. 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 Q UANTUM F ISHER I NFORMATION – 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. unitary rotation Heisenberg limit GHZ state output state 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 … measurement on all probes: estimator: 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 → ∞. (I DEAL ) Q UANTUM M ETROLOGY A TOMIC S PECTROSCOPY : “P HASE ” E STIMATION repeating the procedure k times

5. distorted unitary rotation constant factor improvement over shot noise optimal pure state mixed output state measurement on all probes estimator In practise need to optimize for particular model and N O BSERVATIONS 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 ? The properties of the single use of a channel – – dictate the asymptotic ultimate scaling of precision. with dephasing noise added: achievable with k = 1 and spin- squeezed states (R EALISTIC ) Q UANTUM M ETROLOGY complexity of computation grows exponentially with N A TOMIC S PECTROSCOPY : “P HASE ” E STIMATION

6. 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 !!!. E FFICIENT TOOLS FOR DETERMINING D LOWER - BOUNDING

7. C LASSICAL /Q UANTUM S IMULATION OF A CHANNEL as a Markov chain:

8. C LASSICAL /Q UANTUM S IMULATION OF A CHANNEL as a Markov chain:

9. C LASSICAL /Q UANTUM S IMULATION OF A CHANNEL as a Markov chain:

10. C LASSICAL /Q UANTUM S IMULATION 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. T HE ”W ORST ” C LASSICAL S IMULATION The ”worst” local classical simulation: Does not work for -extremal channels, e.g unitaries. We want to construct the ”local classical simulation” of the form: Locality: Quantum Fisher Information at a given : depends only on: The set of quantum channels (CPTP maps) is convex

12. G ALLERY OF DECOHERENCE MODELS

13. C ONSEQUENCES ON R EALISTIC S CENARIOS “ PHASE ESTIMATION ” IN A TOMIC S PECTROSCOPY WITH D EPHASING (η = 0.9) GHZ strategy S-S strategy finite-N bound asymptotic bound better than Heisenberg Limit region worse than classical region input correlations dominated region worth investing in GHZ states e.g. N =3 [D. Leibfried et al, Science, 304 (2004)] uncorrelated decoherence dominated region worth investing in S-S states e.g. N =10 5 !!!!!!! [R. J. Sewell et al, Phys. Rev. Lett. 109, 253605 (2012)]

14. C ONSEQUENCES ON R EALISTIC S CENARIOS P ERFORMANCE OF GEO600 GRAVITIONAL - WAVE INTERFEROMETER [Demkowicz-Dobrzanski et al, Phys. Rev. A 88, 041802(R) (2013)] coherent beam squeezed vacuum - /ns

15. F REQUENCY E STIMATION IN R AMSEY S PECTROSCOPY two Kraus operators – non-full rank channel – SQL-bounding Methods apply for any t Estimation of Parallel dephasing: “Ellipsoid” dephasing: four Kraus operators – full rank channel – SQL-bounding Methods apply for any t t - extra free parameter Transversal dephasing: four Kraus operators – full rank channel – SQL-bounding Methods apply, but… as t → 0 up to O(t 2 ) – two Kraus operators SQL-bounding Methods fail !!! R AMSEY SPECTROSCOPY - R ESOURCES : Total time of the experiment, number of particles involved : ⁞ ⁞⁞⁞⁞⁞ ⁞ ⁞⁞⁞ ⁞ P RECISION : optimize over t

16. R AMSEY S PECTROSCOPY WITH T RANSVERSAL D EPHASING [Chaves et al, Phys. Rev. Lett. 111, 120401 (2013)] (a) Saturability with the GHZ states (b) Impact of parallel component Beyond the shot noise !!! (dotted) – parallel dephasing (dashed) – transversal dephasing without t -optimisation (solid) – transversal dephasing with t -optimisation

17. C ONCLUSIONS 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)].

18. 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 c Q /√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/N 5/6 asymptotic – the transversal dephasing. Application to atomic magnetometry: [Wasilewski et al, Phys. Rev. Lett. 104, 133601 (2010)]. C ONCLUSIONS T HANK YOU FOR Y OUR ATTENTION