QUANTUM METROLOGY IN REALISTIC SCENARIOSJanek 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 METROLOGYATOMIC SPECTROSCOPY: “PHASE” ESTIMATION

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 saturablein the limit N → ∞

withunbiased

• 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 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

QUANTUM FISHER INFORMATION

– 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)] –

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 → ∞.

(IDEAL) QUANTUM METROLOGYATOMIC SPECTROSCOPY: “PHASE” ESTIMATION

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

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 ?

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

(REALISTIC) QUANTUM METROLOGY

complexity of computation grows exponentially with N

ATOMIC SPECTROSCOPY: “PHASE” ESTIMATION

In order of their power and range of applicability:

o Classical Simulation (CS) methodo 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) methodo 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) methodo 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 !!!.

EFFICIENT TOOLS FOR DETERMINING DLOWER-BOUNDING

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 ”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

Depolarization Dephasing Lossy interferometer Spontaneous emission

CS CS CS N/ACS

N/AQS

QS QS QSCE

CE CE CE

GALLERY OF DECOHERENCE MODELS

inside the set of quantum channels

full rank

on the boundary,extremal

on the boundary, non-extremal,

not -extremal

on the boundary, non-extremal, but -extremal

CONSEQUENCES ON REALISTIC SCENARIOS“PHASE ESTIMATION” IN ATOMIC SPECTROSCOPY WITH DEPHASING (η = 0.9)

GHZ strategyS-S strategy

finite-N boundasymptotic bound

better than Heisenberg Limit

region

worse than classical region

input correlations dominated regionworth investing in GHZ states

e.g. N=3 [D. Leibfried et al, Science, 304 (2004)]

uncorrelated decoherence dominated regionworth investing in S-S states

e.g. N=105 !!!!!!![R. J. Sewell et al, Phys. Rev. Lett. 109, 253605 (2012)]

CONSEQUENCES ON REALISTIC SCENARIOSPERFORMANCE OF GEO600 GRAVITIONAL-WAVE INTERFEROMETER

[Demkowicz-Dobrzanski et al, Phys. Rev. A 88, 041802(R) (2013)]

coherent beam

squeezed vacuum

-

/ns

FREQUENCY ESTIMATION IN RAMSEY SPECTROSCOPY

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(t2) – two Kraus operators SQL-bounding Methods fail !!!

RAMSEY SPECTROSCOPY - RESOURCES: Total time of the experiment , number of particles involved :

⁞ ⁞ ⁞ ⁞ ⁞ ⁞⁞

⁞ ⁞ ⁞⁞

PRECISION:

optimize over t

RAMSEY SPECTROSCOPY WITH TRANSVERSAL DEPHASING

[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

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)].

• 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)].

CONCLUSIONS

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