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, (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, (2013).

(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

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), (2008)] –

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

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

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

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

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

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

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?

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

G ALLERY OF DECOHERENCE MODELS

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, (2012)]

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

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

R AMSEY S PECTROSCOPY WITH T RANSVERSAL D EPHASING [Chaves et al, Phys. Rev. Lett. 111, (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

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, (2001)] squeezed light states in GEO600 - [Demkowicz-Dobrzanski et al, Phys. Rev. A 88, (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, (2010)] MPS states – [Jarzyna et al, Phys. Rev. Lett. 110, (2013)].

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, (2010)]. C ONCLUSIONS T HANK YOU FOR Y OUR ATTENTION