Precision bounds in noisy quantum metrology

Abstract

In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as $1/N^2$ with the number of particles involved---substantially surpassing the $1/N$-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic $N$ limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large $N$ the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.

PhD Thesis (defended 22.09.2014). 138 pages, 6 chapters (+10 appendices), 20 figures, 6 tables. Final version containing modifications suggested by the referees: Dariusz Chruscinski and Andrzej Grudka. Incorporates and extends the material of arXiv:1006.0734, arXiv:1201.3940, arXiv:1303.7271 and arXiv:1405.7703

References in corpus (65)

• Decoherence, einselection, and the quantum origins of the classical
• Quantum interface between light and atomic ensembles
• On the Improvement of Frequency Stardards with Quantum Entanglement
• A gravitational wave observatory operating beyond the quantum shot-noise limit: Squeezed light in application
• General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology
• Reference frames, superselection rules, and quantum information
• Implementation of Cavity Squeezing of a Collective Atomic Spin
• Quantum limits in optical interferometry
• Quantum speed limit for physical processes
• Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit
• Quantum noise limited and entanglement-assisted magnetometry
• Using entanglement against noise in quantum metrology
• The entanglement of indistinguishable particles shared between two parties
• 'Designer atoms' for quantum metrology
• Microwave quantum logic gates for trapped ions
• Magnetic sensitivity beyond the projection noise limit by spin squeezing
• The creation of large photon-number path entanglement conditioned on photodetection
• Optimal states and almost optimal adaptive measurements for quantum interferometry
• Using entanglement improves precision of quantum measurements
• Uncertainty Relations for Noise and Disturbance in Generalized Quantum Measurements
• Entanglement-enhanced measurement of a completely unknown phase
• Universal Algorithm for Optimal Estimation of Quantum States from Finite Ensembles
• Proof of Heisenberg's error-disturbance relation
• General optimality of the Heisenberg limit for quantum metrology
• Optimal quantum estimation of loss in bosonic channels
• Scaling laws for precision in quantum interferometry and bifurcation landscape of optimal state
• Quantum interferometry with and without an external phase reference
• Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states
• Stability of atomic clocks based on entangled atoms
• Extracting entanglement from identical particles
• Extremal properties of the variance and the quantum Fisher information
• Optimal estimation of joint parameters in phase space
• Quantum Metrological Limits via a Variational Approach
• Fundamental quantum interferometry bound for the squeezed-light-enhanced gravitational-wave detector GEO600
• Covariance and Fisher information in quantum mechanics
• Ziv-Zakai Error Bounds for Quantum Parameter Estimation
• Not all pure entangled states are useful for sub shot-noise interferometry
• Qubit metrology and decoherence
• Quantum Entanglement of Identical Particles
• Phase estimation without a priori knowledge in the presence of loss
• All path-symmetric pure states achieve their maximal phase sensitivity in conventional two-path interferometry
• Parity Measurement is Sufficient for Phase Estimation at the Quantum Cramer-Rao Bound for Path-Symmetric States
• Kind of entanglement that speeds up quantum evolution
• Local asymptotic normality for qubit states
• Near Heisenberg limited atomic clocks in the presence of decoherence
• Ultimate limits to quantum metrology and the meaning of the Heisenberg limit
• Universality of the Heisenberg limit for estimates of random phase shifts
• Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation
• Process POVM: A mathematical framework for the description of process tomography experiments
• Discriminating quantum-optical beam-splitter channels with number-diagonal signal states: Applications to quantum reading and target detection
• Adaptive single-shot phase measurements: The full quantum theory
• Atom correlations and spin squeezing near the Heisenberg limit: finite system size effect and decoherence
• Quantum measurement bounds beyond the uncertainty relations
• Quantum Discord and its Role in Quantum Information Theory
• Local asymptotic normality in quantum statistics
• Beyond quantum Fisher information: optimal phase estimation with arbitrary a priori knowledge
• Measuring Energy, Estimating Hamiltonians, and the Time-Energy Uncertainty Relation
• Optimal Heisenberg-style bounds for the average performance of arbitrary phase estimates
• Phase estimation via quantum interferometry for noisy detectors
• Ancilla-Assisted Enhancement of Channel Estimation for Low-Noise Parameters
• Minimax discrimination of two Pauli channels
• Enhanced Resolution of Lossy Interferometry by Coherent Amplification of Single Photons
• Probing the qudit depolarizing channel
• Entanglement of identical particles and reference phase uncertainty
• Unambiguous comparison of unitary channels

Cited by in corpus (14)

• Thermometry in the quantum regime: Recent theoretical progress
• Quantum optical technologies for metrology, sensing and imaging
• The ultimate precision limits for noisy frequency estimation
• Maximal Quantum Fisher Information for Mixed States
• Probe incompatibility in multiparameter noisy quantum metrology
• Quantum sensing networks for the estimation of linear functions
• Demonstrating quantum coherence and metrology that is resilient to transversal noise
• Cryptographic approach to Quantum Metrology
• Thermometry of Gaussian quantum systems using Gaussian measurements
• Modified Grover operator for amplitude estimation
• Quantum computation speedup limits from quantum metrological precision bounds
• Strongly coupled fermionic probe for nonequilibrium thermometry
• Heisenberg-limited Bayesian phase estimation with low-depth digital quantum circuits
• Catalytic transformations with finite-size environments: applications to cooling and thermometry